[SLFM 079] Generalized Recursion Theory - Fens Tad, P.G.hinman [Studies in Logic and the Foundations of Mathematics] (NH 1974)(T)

GENERALIZED RECURSION THEORY

PROCEEDINGS OF THE 1972 OSLO SYMPOSIUM

Edited by

J . E . FENSTAD University of Oslo

and

P. G. H I N M A N University of Oslo

and

University of Michigan, Ann Arbor

1974

N O R T H - H O L L A N D PUBLISHING C O M P A N Y - A M S T E R D A M L O N D O N

AMERICAN ELSEVIER PUBLISHING C O M P A N Y , INC. - NEW Y O R K

0 NORTH-HOLLAND PUBLISHING COMPANY - 1974

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for this volume 0 7204 22760

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PREFACE

The Symposium on Generalized Recursion Theory was held at the Univers- ity of Oslo, June 12- 16, 1972. The Symposium received generous financial support from the Norwegian Research Council and from the University of Oslo. About 50 persons attended the meeting.

This volume contains 12 of the papers presented at the meeting. Of the five remaining papers the contribution of Y.N. Moschovakis replaces the one originally presented, which will be published by North-Holland in 1973 under the title Elementary Induction on Abstract Structures. The paper “Post’s problem for admissible sets” by S. Simpson is a later addition. The Editors asked K. Devlin to write a survey paper on the Jensen theory of the fine structure of the constructible hierarchy. The two remaining papers, by S. Aanderaa and L. Harrington, solve important problems left open at the end of the Symposium, and we are happy to include these papers in the Proceedings. We should finally note that the authors have been free to revise their papers after the Symposium, which in some cases has led to extensions of the results as originally reported.

We hope that the inclusion of a bibliography of papers on generalized recursion theory will increase the usefulness of the present volume. The partici- pants of the Symposium agreed that a bibliography of the field would be useful, and the preparation of it was taken over by Gerald Sacks, who received extensive assistance from Leo Harrington. The Editors are grateful to them for their valuable work. The reader will note that the bibliography carries the disclaimer “uncritical”. This is to emphasize that the purpose was not to present a comprehensive and scholarly bibliography of works relevant to generalized recursion theory, but to provide a useful list of some of the basic papers.

The Symposium was intended to present a broad view of methods and results in generalized recursion theory. We believe that the meeting acheved some measure of success toward this goal so that the published Proceedings also can serve as an introduction for the beginning research student who wants to specialize in this rich and fascinating branch of logic.

The Editors

V

PART I

RECURSION IN OBJECTS OF FINITE TYPE

J E. Fenstad, I? G. Hinman (eds), Generalized Recursion Theory @ North-Holland PubL CornD., 1974

RECURSION IN THE SUPERJUMP

Peter ACZEL University of Manchester

and

Peter G. HINMAN University of Oslo

and University of Michigan

The ordinary jump operator of recursion theory is a function OJ : ww + ww defined by

0, if {m) (a )$ ;

1, otherwise . oJ(a)(m) =

By treating OJ as a (type-2) function: ww X w + w and coding the two arguments into one, we obtain from the schemata of [Kl] a notion of recursion relative to oJ. It is well-known that OJ is of the same degree as 2E, that a set A of natural numbers is recursive in oJ just in case it is hyperarithmetic, and that wyJ, the least ordinal not recursive in oJ, is just wl, the least non-recursive ordinal and the second admissible ordinal.

The same procedure can, of course, be applied to any function Gump) J . For example, W defined by

0, if {m}(a ,oJ)$;

1, otherwise,

is recursively equivalent to the hyperiump and of the same degree as El. The sets of natural numbers recursive in hJ are the recursive analogue of the C- sets of descriptive set theory [Hi], and wtJ is the least recursively inaccessible ordinal.

3

4 P. ACZEL and P.G. HINMAN

The superjump S as defined in [Gal is a type-3 jump defined by

In particular, S(oJ) = hJ . By coding arguments as above we may consider S as a function: (Wo)w+w.Thus from [Kl] we have a notion of recursion relative to S. In this paper we study some properties of this notion. In $5 1 and 2 we discuss a hierarchy of jump operators, due to Platek, obtained by iteratingS over a set of ordinal notations. $3 contains some results concerning the size of as, the least ordinal not recursive in S. In $4 we extend Platek's hierarchy to one with the property that a set of natural numbers is recursive in S iff it is recursive in some jump operator occurring in the hierarchy. Finally, in § 5 we discuss several other type-3 functionals which are in some sense equivalent to s.

In ## 1 and 2 we assume familiarity with [Pl] and conform for the most part to his notation. The rest of the paper does not have this prerequisite, but we recommend to the reader the clear general discussion of [PI, 257-2631 as background.

§ 1. Platek's hierarchy

Modern mathematics must be considered more an art-form than a science, but it is perhaps a harder master than most of the arts: mathematics must not only be beautiful, it must also be correct. Sad to say, even the beautiful can be false and such an occurrance is the starting point of this paper. In [PI], Platek constructs a hierarchy of jumpsJf indexed by elementsa of a set 0' of ordinal notations. Jf = oJ , J ; = h J , and, roughly speaking, the hierarchy is obtained by iterating S over the set of notations. The construction is closely parallel to that of Kleene's setsH, for a E 6, a set of notations for recursive ordinals, and even more closely parallel to that of Shoenfield's sets H c for a E O F , a set of notations for the ordinals recursive in the type-2 function F [Sh] . Since a set A of numbers is recursive in oJ iff it is recursive in some Ha (a E S ) , and, for any F in which oJ is recursive, A is recursive in F iff it is recursive in someHr (a E a"), it would be elegant and satisfying if also A were recursive in S iff it is recursive in some J: (a E 0'). The main theorem

RECURSION IN THE SUPERJUMP 5

of [HI assets that this is true - unfortunately it is not. Before describing the counterexample to Platek’s theorem, we point out

where his argument breaks down. In the sentences beginning at the bottom of p. 265 of [Pl] and continuing at the top of p. 266 he applies the boundedness lemma to a function @(@(a) = so that for a E 6 ’, H{ is recursive in J,$,,) to obtain a d E 0’ such that eachHF is recursive i n J i . The function H i s defined by the recursion theorem over 0 ’ and thus @ is partial recursive with domain including O F . The boundedness lemma applies to total functions + E 32, so to use it here we would need to find such a $ which agreed with @ on OF. The natural + to choose would be

However, there is no apparent way to find aJ-number for (the characteristic function of) 6’ and thus show this $ E 92. In fact we shall see that for the functional K below, to which this step of the argument would have to apply, 6 does nor have a J-number and there is no d E 6’ such that all Hf (a E o K , are recursive in J:.

Our counterexample consists in defining a jump K with the following three properties:

(1) K is recursive in S; ( 2 ) a E as-+ ~f is recursive in K ; (3) for any A 5 o, if A is recursive in K , then for some a E 6’, A is re-

From (1) follows that S(K) is also recursive in S and hence that there

S cursive in J , .

existA C o recursive in S but not recursive in K - for example, {a : {a}(K).l}. Hence from (2) we have immediately that there are A recursive in S but in no J i (a E 0’). Property (3) completes the picture to show that the jumpsJ: (a E Cis) provide a natural hierarchy for the sets recursive in K . Of course, it also follows easily from (2) that O K is not recursive in any Jf (a E 8 ’) and that there is no d E 8 ’ such that all A recursive in K are recursive in J:.

[Pl, p. 2631 ensures that for every u < 16‘1 = sup { la I : a E O’}, u is recursive in some Jf and thus by (2) recursive in K . Hence 10’1 5 of. Conversely, if u is an ordinal

In terms of ordinals, the uniqueness theorem for 0

6

recursive in K , then by (3) there is some a E 0' such that CJ < ola < 10'1. The last inequality may be shown by constructing an order-preserving partial recursive map of oJl into 0' and obtaining an upperbound by use ofJ&. Thus w f = 10'1. Of course OK is recursive in S(K) so of < < of.

We turn now to the definition of K and the proof of properties (1) -(3). K is based on the same idea as the jump T of [Pl, Th. VI] .

P. ACZEL and P.G. HINMAN

J S

Definition 1.1. Fld(y) = (r, : p I , p } and le tp < ,q i f p _< ,q andp f q ;

(a) For any y : w X o + 2 we write p _< ,4 if y ( p , q ) = 0, let

(b) W i s the set of y : o X w -+ 2 such that < , well-orders Fld (7); (c) for y €94, ll yll is the order type of < ,; (d) for any y andr Eo,

Note that for y E W a n d r E o, y r r E W a n d IIy I' rll = 0 if r B Fld(y) while llr r rli < llyll i f r E Fld(y).

Definition 1.2. (a) for y € W a n d llyll= 0, K , =oJ; (b) for y E W a n d llyll> 0, K , = Xa .S(K,r,(o))(a+), where

o;'(m) = a(m + 1);

K,(a), if 7";

Am - 0. otherwise. (c) K ( 7 , =

Theorem 1.3. K is recursive in S.

RECURSION IN THE SUPERJUMP I

then G is partial recursive in S. By the recursion theorem there exists an F such that G(Z, m , y,a) = {Z}’(m, y, a). If F = {F}S, we claim that for all a, y, and m :

Note that the only properties of S used in the preceding theorem are that oJ and hJ are recursive in S. The next lemma records the other (very general) properties of S that are needed in this section. Thus our methods apply in many other situations. For example, they provide an alternative proof to the result of [Mo, 8 Ilthat Kleene’s proposed hierarchy for 3E fails to exhaust even the sets of numbers recursive in 3E.

Lemma 1.4. (a) There exists a primitive rekursive $J such that for all J and a,

44 = #J(G(J)); (b) there exists a primitive recursive f such that for any e, J, and J ’ , i f J is

recursive in J’ with index e, then S(J) is recursive in S (J ‘ ) with index f (e).

Proof. (a) is proved in two lines just as the corresponding fact one type down with OJ in place of S.

For (b), suppose that for all 0 and q

It suffices to show that there is a primitive recursive g such that for any a, m = (mo , ..., mkPl) , and a € w w ,

as then

g is defined by the recursion theorem and by cases depending on the index a.


The only difficult case is when

where b and q are coded into a. Here we define g(a, e) to be the “natural” index such that

It is straightforward to show for this case that

{a } (m, a , J ) = n + {g(a, e)} (m, a, J ’ ) N n .

Conversely, if {g (a , e ) } (m ,a , J ’ ) = n , then by virtue of the first term in the definition, Xp - {g (b , e ) } (p ,m ,a , J ’ ) is total and a l l computations of its values are subcomputations of the computation of {g(a,e)} (m,a, J ’ ) . Hence, the induction hypothesis guarantees that hp * {b} ( p , m, a, J) is the same total function and thus that (a } (m, a, J) is defined with value n .

‘‘natural’’ index c , computed from an index f o rg , such that In the case {a} (b, m, a, J ) = {b} (m, a, J) we take g(a, e ) to be the

Corollary 1.5. There exist primitive recursive y €9’ and any p , q : (a) If llyll> 0 then S(KYlp) is recursive in K , with index f(p); (b) p < ,q + KrrP is recursive in KYr4 with index g (p ,q ) ; (c) ifl lyll= Ily rpll + 1, then K7 is recursive in S(KYrp) and y with index h(p) .

g, and h such that for any

Proof. (a) is immediate from the definition. When p < ,q, y r p = (y rq ) r p so(b)followsfrom(a)and 1.4(a).SupposeIIyll=Ilyrpll + l . I f r < , p , t h e n KYr, may be computed from KrrP by (b). If r Q: , p , then either r = p , so that KYr, is trivially recursive in Krrp, or IIy rrll = 0 and K,,, = OJ so again is recursive in KYrp (with index computable from 7). Hence using y we may compute eachKYrr uniformly from KrrP. By 1.4(b) the same is true ofS(Kyrr)


and S ( K T r p ) and from the definition of K , it is clear that this is sufficient to establish (c). 0

To obtain result (2) it is much more convenient to work with a hierarchy slightly different from that of [Pl] obtained by introducing an ordering relation < and at limit stages 3' 5 e requiring that Am - {e} (m, J,") ascend in the ordering < s. The construction is entirely parallel to that of [Sh] . Let bs and7: denote the set of notations and jumps thus obtained. The new system is a subsystem of Platek's and it is an easy exercise to prove.

Lemma 1.6. There exist partial recursive f and g such that for all a E 6', f ( a ) E 6' and Jf is recursive in Tf(a) with index g(a).

For each a E us, let yo be defined by

Then y, € W a n d llyall = lal. Note that if b < Sa , then yo r b = yb.

Lemma 1 -7. There exist partial recursive f and g such that fur all a E 0'. (a) y is recursive in 7: with index f ( a ) ; (b) 7: is recursive in K , with index g(a).

Proof. (a) is proved by a straightforward induction over 6'. At limit stages one uses the fact that oJ is recursive in 7,". For (b) we define g by the recursion theorem over 6 s as follows. For a = 1,7," = KTa = oJ. If a = 2b, the induction hypothesis yields that J i is recursive in K Y b = KrQrb with index g(b). Then by using 1.5(a) and 1.4(b) we may compute an indexg(a) o fTf=S(y i ) from K,,. If a = 3 in the previous case an index of 7: and hence of @ from KYQrb and then by l.S(a) from K,,. Similarly, for each m we can compute an index ofTi(,) from KY,. Putting these together, we obtain an index g(a) for 7; from K,,. 0

Theorem 1.8. There is a partial recursive h such that for all a E 6 , J , is recursive in K with index h(a).

5 e , let @(m) = {e ) (m,Tf ) . Since b < S a we can find as

-s -s


Proof. We again define h by the recursion theorem over a'. For a = 1,Yf = oJ = Aa K(hm - l , a ) , so it is clear how to pick h( 1). Fo ra = 2b, using h(b) and (a) of the previous lemma we can compute an index for yb from K . By 1.7(b) we have an index of 7; from K and yo. But y, is obviously recursive in yb so we have one from K alone. For a = 3b * 5 e , let @(m) = {e} (rn ,Tf) . Using h(b) we can find an index of @ from K and using h(@(m)) we can find an index of Ti(m) from K . Hence we can find an index of 3; from K . 0

Result (2) now follows immediately from 1.6 and 1.8.

Lemma 1.9. There exist primitive recursive f and g such that for any d E 0' and any y Ecklrecursive in J i with index e, f ( d , e ) E 6' and K , is recursive in Jf(d,,) with index g (d , e ) .

Proof. Let d , e , and y be as described. We first define functions @ and $ recursive in .Ti, with indices from J i depending only on e , such that for all p , @ ( p ) E 6' and KrrP is recursive in J&, wjth index $ ( p ) . Once this is done we can compute from the index of @ from J i by the boundedness lemma a c E O S s u c h t h a t ld l< lc l a n d V p - I@(p)I<IcI.Thenusing1.4(b)andthe uniqueness lemma, for all p , S ( K y P p ) is recursive inJf with index computable from $ ( p ) . Since $ is also recursive inJf it follows that K , is recursive in J f . Hence we may take f ( d , e ) = c andg(d,e) an index formalizing the preceding labyrinthine computation.

I t remains to define @ and $. Given p , we first check from y and oJ, hence from J:, whether p E Fld(y) and IIy f pll > 0. If not, set @ ( p ) = d (a trick useful below) and $ ( p ) any index of K r r p = oJ from@. If so, we assume as induction hypothesis that forq andr such that IIy fq l l < IIy r rll< IIy rpll, @(q) and @(r) are defined and 1@(4)1< I@(r)l. If p has an immediate predecessor q in the ordering<,, this can be determined andq computed fromJ$. Then by 1.5(c) we may compute an index of K r l p from S(K,,,, a ) and y and thus, using the induction hypothesis and 1.4(b), from S(J&,) and y. Since y is recursive inJ: and Id1 5 l@(q)I, it suffices to take @ ( p ) = 2@(4) and $ ( p ) an appropriate index.

Finally, suppose 1Iy rpll is limit ordinal. Let 0 be defined by: e(0) = the unique q E Fld (y) such that II y l' 411 = 0;

O(m +1) = least q * e(m) < ,q < , p .

RECURSION IN THE SUPERJUMP 1 1

It is clear that 0 is recursive in y and oJ, hence i n J i , and that f3 defines a strictly increasing cofinal sequence in < rrp. Let q(m) = @(O(m)) and let a be an index of r ) from J i . Then 3d 5a E Os and we claim that Krr? is recursive

It suffices to show that for all r , S(K(,,.p),.r) is recursive in J f d S 5 . (uniformly in r , of course!). If r 4: , p , then K ( , r p ) r r = oJ. If r < , p , we compute the least m such that r < ,O(m). Then I@(r)l< I@(€J(m)) I = Iv(m) I

which in turn is recursive inJfd.5Q. Thus we take $ ( p ) = 3d - 5“. o SO by 1.4(b) and the uniqueness lemma, S ( K ( , l p ) p r ) is recursive in Jq(ml S

Theorem 1.10. There exist primitive recursive f and g such that for any a, m, and n :

{a} (m, K ) = n + f(a, (m)) E 0 and

Ig(a)I(m,Jfia,( , , , ) -n .

Proof. We define f and g by recursion over computations in K . Most cases are straightforward - we consider only the case where K is applied. Suppose

By the induction hypothesis, V p f (b, ( p , m)) E 0‘ so by the boundedness lemma we can computed E 6’ such that V p - I f ( b , ( p , m ) ) l < Idl. Hence y = Xp * {b} ( p , m, K ) is recursive in Ji with index (say) e , computable from g(b). Then appropriate values for f(a,(m)) andg(a) can be computed using the functions of Lemma 1.9. 0

Corollary 1.1 1. For any A C w, i f A is recursive in K , then for some a E 6 ’, A is recursive in J:.

Proof. Immediate from 1.10 and the boundedness lemma. 0

9 2. The ordinals 5 I6 I

In this section we shall characterize the ordinals ,:$(for a E 6’) and 16’1.


Definition 2.1. For any ordinals u and r: (a) u is 0-recursively inaccessible iff u > w and u is admissible; (b) u is r t 1-recursively inaccessible iff u is .r-recursively inaccessible and

(c) if r is a limit ordinal, u is r-recursively inaccessible iff u is p-recursively a limit of r-recursively inaccessibles.

inaccessible for all p < r.

Our aim is to prove:

JS Theorem 2.2. inaccessible ;

(a) For all a E 6’, wla is the least u which is lal-recursively

(b) I 0’1 is the least u such that u is u-recursively inaccessible.

This follows easily from the following two results: (1) For any jump J and any a E 6’, if Jf is recursive in J , then w i is

(2) for any ordinal u and any a E 0’ if u is la I-recursively inaccessible, la I -recursively inaccessible;

then w p 5 u.

Proof of Theorem 2.2. By ( I ) is la]-recursively inaccessible and by (2) it must be the least such. By (2) of 5 I , J f is recursive in K for alla E 6’ SO by (l), of = 10’1 is lal-recursively inaccessible for all a E 0’. As 10’1 is a limit ordinal, 10’1 is thus 10’1-recursively inaccessible. I f u < 16’1, u = la1 for some a E 8’. Since la1 < wla, it follows from ( 2 ) that u is not u-recursively inaccessible. n

S

S

J S

We now turn to the proof of (1). For any jump J , let

R(J) = {my : F is type 2 and J is recursive in F } .

Lemma 2.3. For any jump J and ordinal u, i f u E R(S(J)), then u E n(f) and u is a limit of ordinals in n(J).

Proof. Let u = w p with S ( J ) recursive in F. By 1.4(a), J is recursive in F so u E R ( 4 . For any r < u there exists a set A C w such thatA is recursive in F and .r < wf (for example, A codes a well-ordering of w in type 7). Then


But clearly ufA € Q(J). Hence u is a limit of ordinals in n(J). 0

Theorem 2.4. For any a E 6‘ and any ordinal u, if u E R(J:), then u is la I - recursively inaccessible.

Proof. We proceed by induction on lal. If la1 = 0, thenJf = OJ and it is well- known that any u € Ci(oJ) is admissible and greater than w, hence O-recursively inaccessible. If a = 2b, n((Jf) = n(S(J,”)). By the induction hypothesis, elements of are I b I -recursively inaccessible so any u E C l ( J f ) is la I - recursively inaccessible by Lemma 2.3. If a = 3b * S e and @(n) = {e} (n , J,”), then any u E n(Jf) is clearly also in Ci(J&H$ for each n and thus by the induction hypothesis is I@(n) I-recursively inaccessible. Since la I =

sup { I@(n) I : n E w} and is a limit ordinal, u is la ]-recursively inaccessible. 0

This establishes result (1) and we now turn to (2). Our proof will require some detailed information about recursion on ordinals. As there is at present no good reference for this material, we digress here to state the facts that we need. First we note that the class of primitive ordinal recursive functions has closure properties similar t o the class of ordinary primitive recursive functions; in particular, if g is primitive ordinal recursive and

then so is f . With each admissible ordinal K and each a < K is associated a k-ary partial function {a}, on K for a natural number k “decodable” from a. A function f on K is K-partial recursive just in case f = {a} , for some a < K . By allowing indices from K (not just a), we include all constant functions with values < K among the K -recursive functions.

(A) (Uniform Normal Form Theorem) For each k > 0 there exists a primitive ordinal recursive relation Tk such that for any K

(i) if K is admissible, then fur any a, p = ( p o , ..., pk- 1), and v < K ,

(ii) K is admissible iff for any a and p < K ,


(B) (Uniform Iteration Theorem). For'each i E w there is a primitive ordinal recursive function Sbi such that for any admissible K and any a, p < K

( C ) (Recursion Theorem). For any admissible K and any K-partial recursive function A there exists an esuch that

As corollaries of (A) we have:

(D) { K : K is admissible} is primitive ordinal recursive.

(E) For any admissible K and X and any a , p, v < A,

h < K A {a},@) = v -+ {a},(p) = v .

(F) For any 1-recursively inaccessible K and any a, p < K ,

(i) {a},(p)$ c-, 3X,,, [A is admissible A { a } , ( ~ ) & ] ; (ii) Vpp<w {a},(p)$ - 3h,,, [A is admissible AVpP<,{a},(p)$].

Lemma 2.5. For any admissible K > w, any K -recursive well-ordering of o has order-type < K .

Proof. Let R be a K-recursive well-ordering of w of type A. By the Recursion Theorem (C) there exists a K-partial recursive function f such that for all m E w ,

The next definition is the key to our proof of (2):

Definition 2.6. For any jump J , any K > w, and any e < K , J is K-effective

RECURSION IN THE SUPERJUMP I5

with index e iff K is admissible and for any a < K such that {a}, r w E w,

J is K -effective iff J is K -effective with some index. E f , , is the set of all K

such that J is K -effective with index e.

Now (2) follows immediately from:

Theorem 2.7. For any jump J and any K ,

(a) if J is K -effective, then w{ i K ; (b) If a E 0’ and K is la J-recursively inaccessible, then J; is effective.

This, in turn, will follow from some lemmas.

Lemma 2.8. There exists a primitive recursive g such that for any K and e such that J is K-effective with index e, any a, any c such that {c} , 1 o E

any m = mo, ..., mk-l < w 0 and

Proof. The proof is very similar to that of Lemma 1.4(b). g is defined as there except for two cases:

(i) if {a} (m,a, J) = a(mj) , then g(a, e) is chosen so that

(ii) if {a)(m,a,J)= J(hp.{b}(p,m,a,J))(q),wechooseg(a,e) to be the “natural” index such that

The proof that g is as required is essentially identical to that of Lemma 1.4(b). 0


Part (a) of Theorem 2.7 follows immediately from Lemmas 2.5 and 2.8. Part (b) will follow from the next three lemmas, which are the induction steps in the definition of a recursive function f such that if a E 6' and K if la I - recursively inaccessible then J: is K -effective with index f (a) .

Lemma 2.9. There exists a co < w such that OJ is K -effective with index co for every admissible K > w.

Proof. Let R be a primitive recursive relation such that

For any admissible K , let

Since the definition off, is uniform with respect to K , there exists an index co such that f , = { c ~ } ~ for every admissible K . Then OJ is K -effective with index co. 0

Lemma 2.10. There exists an primitive ordinal recursive f such that for any K > w, any jump J , and any e, if K E E f J,e and K is a limit of ordinals h E E f , , , then K E Efs(J),fl(e). If e < 0, then also f l ( e ) < w.

We claim first that for K as in the hypothesis of the lemma, if {a}, I' w E w w , then for all m E w,


Since K is 1-recursively inaccessible, it follows from (F)(ii) that h,(e, a ) 4 so that f , (e,a,m) is also defined with value 0 or 1. Then

The second equivalence uses (E) and the third Lemma 2.8. To complete the proof, it suffices to show that f , is uniformly K-partial

recursive, that is, that for some fixed index cl , f, = { c ] } ~ for all K satisfying the hypothesis. Given this, f i ( e ) = Sbl(cl ,e) is the required function. Let

I t follows from (A), (D), and closure under bounded quantification that Ef, is primitive ordinal recursive. We claim that for K E E f , , ,

The implication (+) is immediate from the definition of E f , , . Suppose h E Ef, and {a},, r w E ww. Then by (E), {a},, r w = {a} , w so that

J({a),, w ) = J ( { a ) , r w ) = {Sbl(e , a) ) , r w = {Sbl(e,a)), r w .

Now in the definition of h, we may replace E f , , by Ef, without altering the value when K E Ef,,,. Thus h , and f , are uniformly K-partial recursive for K satisfying the hypothesis of the lemma. 0

Lemma 2.1 I . There exists a primitive ordinal recursive f 2 such that for any K > w, any b, and any indexed set of jumps { J , : n E w } such that for all n , J , is K -effective with index {b},(n), if J = ham * J(,)o(a) ( (m) l ) , then J is K-effective with index f2(b). If b < w, then also f2(b) < w.


Proof. If {a}, r w E w w , then

for an appropriate c2 . Thus it suffices to set f 2 ( b ) = Sb,(c,,b). 0

Proof of Theorem 2.7. (a) follows immediately from Lemmas 2.5 and 2.7. For (b), let 3. = fj r w. Then fi and f 2 are (ordinary) primitive recursive. By the (ordinary) recursion theorem there exists a primitive recursive function f with index f such that:

where h is a primitive recursive function such that for any admissible K

withg as in Lemma 2.8. It is now straightforward to prove by induction on 6" that for a E O S , if K is I a I -recursively inaccessible, then Jf is K-effective with index f ( a ) . 0

$3. S and the first recursively Mahlo ordinal

In this section we extend the ideas of $ 2 to obtain a bound for w f , the least ordinal not recursive in S.

Definition 3.1. sible and for any K -recursive function f from K to K , there is an admissible h < K which is closed under f;

(i) For any ordinal K , K isrecursively Mahlo iff K is admis-

(ii) po is the least recursively Mahlo ordinal.


It is easy to see that po is po-recursively inaccessible and is not the first such ordinal. Hence 16’1 < p,. We shall show here that wf < p o . It has recently been shown by Leo Harrington that wf = p o ; his proof appears in his contribution to this volume. That of = po appeared as a theorem in [Ac], but its purported proof there depended on the fallacious results of [PI].

Definition 3.2. For any K > w and any e < K , S is K -effective with index e iff K is admissible and for any J and d such that J is K-effective with index d , S ( J ) is K-effective with index Sb l (e ,d ) . S is K-effective iff S is K-effective with some index.

The next lemma is analogous to a weak form of Lemma 2.8.

Lemma 3.3. There exists a primitive ordinal recursive function g such that for any K and e such that S is K -effective with index e, any a < w and m = mo, ..., mk-l < w,any c = c o , ..., cI-l < K a n d a = a,, ...,alp, E

such that ai = { c ; } ~ 1 w for i l,(m,c> = n.

Proof. This is similar to that of Lemma 2.8 and we treat only the case

where J = hyq { b o } ( q , m , a , y , S) and0 = hr - { b l } ( r , m , a , S ) , with b,, b , , and p coded into a . Given that { a } ( m , a , S) = n , it follows that J and 0 are both total objects with all of their computations “preceding” the given one. By the induction hypothesis we can compute from a , m and c an index d , such that J is K-effective with index d , and b, such that 0 = {b2IK r w . Then

Hence it suffices to choose g(a,e) such that

Corollary 3.4. For any K , ifS is K -effective, then wf 5 K .


Proof. If S is K-effective, then by the preceding lemma any well-ordering of o recursive in S is also K-recursive, hence of order-type < K by Lemma 2.5. 0

Lemma 3.5. For any J, e, and K , if^ is recursively Mahlo and K E E f , , , then K i s also a limit of ordinals h E E f J ,e .

Proof. Suppose K is recursively Mahlo, K E Ef,,, and let Ef, be as in the proof of Lemma 2.10. Then for any admissible h with o, e < h < K ,

h E Ef J,e ++ h E Ef,. With the help of the uniform normal form theorem (A) we can define a primitive ordinal recursive relation R such that for such A,

Now let h, be any ordinal < K and set

Clearly for any admissible A,

h is closed under f ++ a, ho < h and h E E f , , .

Since K is recursively Mahlo and f is K -recursive, there is an admissible h < K

such that h is closed under f and hence a h E EfJ,, with A, < h < K . 0

Theorem 3.6. For any K , if K i s recursively Mahlo, then S is K -effective.

Proof. Suppose K is recursively Mahlo and let e be an index such that

where fl is the function of Lemma 2.10. We claim that S is K -effective with index e. Let J be any jump which is K-effective with index d : K E Ef,,,. By the preceding lemma, K is also a limit of ordinals A E EfJ,,, so by Lemma 2.10 S(J) is K-effective with index f l ( d ) . But then S ( J ) is also K-effective with index Sbl(e ,d) as required. 0

Corollary 3.7. of < p 0 .


For ordinals K which are projectible to w , the converse of Theorem 3.6 holds also:

Theorem 3.8. For any K which is projectible to o, K is recursively Mahlo i f f S is K -effective.

Proof. Suppose S is K-effective, say with index e, and let h be a 1- 1 K-recursive function which projects K into w. We define a hierarchy of jumps J , for u < K as follows:

and for limit u

We define, by use of the recursion theorem, a K-partial recursive functiong such that for all u < K , J , is K-effective with indexg(u):

and for limit u, g(u) is an index b such that

Now let f be any K-recursive function. We aim to find an admissible h < K

such that h is closed under f . Let T be the jump defined by


Using the function g and the fact that W i s recursive in S , it is easy to see that T is K-effective. Hence S ( T ) is also K-effective so

Then X is admissible and it remains to show that X is closed under f . I f u < X, then u = 1 1 y I1 for some y €9 and recursive in T. Hence Jf(.)+, = ha T(y , a ) is recursive in T and <A. Note that for any u < T < h, S(J,) is recursive in J , so up < a?. Hence f(u) < w{f(O) < w:f(0)+l < h. o

Corollary 3.9. p g is the least K such that S is K -effective.

34. A countable hierarchy for 1-SC ( S )

We next define an extension of Platek’s hierarchy and show that a set of natural numbers is recursive in S iff it appears in our hierarchy. To guide the reader through the inevitable forest of indices and recursions we sketch first the main points of argument. We first define a set of notations 6, and for each u E 6 a jump J,, by adjoining the method of [Ri] to Platek‘s hierarchy - that is, whenever Platek’s inductive definition grinds to a halt but the set of notations is closed under some new primitive recursive function {d } , we add 7 d to tlic set ofnotations, collect a11 previous J,’S to form J,,, and go on. With relatively minor altcrations to the proofs of [Sh] and [PI] we establish uniqueness and boundedness lemmas.

various J1, ( u E 0) as in Theorem 1.10. The difficult point, which failed for Platek’s hierarchy, is to show that from such a method for computing F we can find one for S ( F ) . Here we accomplish this roughly as follows (the details differ somewhat from this sketch). From the instructions for F we can compute a primitive recursive .$ : 0 + 6 such that for any u E 6 and any a recursive in J,, we can compute the value F(a) from Jt( ,) . I f d is a primitive recursive index for g, then 7 d € 0. A computation similar to that of the proof of 1.1 0 shows that computations from F can be done relative to J,d and hence that the “diagonal set” { a : { a } ( F ) $ } is recursive in S ( J 7 d ) . Then to compute S(F)(a) for any a computable from the J,’s we apply this procedure to a recursive join of F and 01.

We aim to show that computations relative to S can be done relative to the


Definition 4.1. We define by ordinal recursion a sequence of sets 6" and, for u E 0, an ordinal Iu I and a jump J, as follows: (a) Let 6(") = u{0' : 7 < u} and for u E 0(") let [a,u] denote the function

(b) Let 0; be the smallest set with the following properties: - {aHm,J,).

(i) O(0) S O ; ; (ii) 1 E 67;

(iii) ~ v [ v E ~ ( o ) -+ 2" E 071 ; (iv) for any v E S(") and any 6, if [b,v] is a total function y such that

y (O)=vandfo ra l lm ,y (m)E 0'") andIy(m)I<ly(m+l) l , then 3".5b E 0:.

(c) Let 0; be the smallest set with the following properties: (i) 0'") c 6" . 23 -

(ii) for any primitive recursive index d, if V v [ v E c!") + {d } ( v ) S'"'] andVvVv'[lvlIIv'I<u-+I{d}(v)I Sl{d}(v ' ) l < u ] then 7 d E 0;.

07, if S ( U ) # 60. 1 ,

05, otherwise. Then 6" =

For u E 0" - d"), Iu I = u and we define J, as follows:

u = l : J , = o J ;

u = 2" : J, = S(J,);

u = 3 " * S 6 : Ju(LY)((m,n))=J~b,"l(,)(LY)(n),

u = 7d : J,(cu)((m,n)) = (~ , (a ) (n ) , if m E O(");

1 0, otherwise.

Lemma 4.2. For any a, b, d , u , and v : (a) (u I < 13".Sbl -+ 12'1 < 13" .Sbl ; (b) l ~ l < 1 7 d l - + 1 2 U I < 1 7 d / ; (c) I u ( < 17dland 3" .Sa E 6andVm(l [a,u](m)l< 17dl)

-+ 13U.5al < 17dl;


(d) d is a primitive recursive index and Vv [ v E 6 + { d } (v) E 61 and

(e) 7 d E 6 and I v I < I 7 d ~ -+ I i d ) (v) I < I 7 d I ; ( f) d is a primitive recursive index and I 7 e 1 < 17 d l + 3v( I v I < I 7 1 5

VvVv”lvl I IV’I < I01 -+ I {d)(v)l I ICdI (v‘) I < I6 I I -+ 7 d E 6 ;

I {d l ( v ) I).

Proof. Most are immediate from the definition and are included to facilitate understanding the nature of 6. (c) depends on the fact that if 17dl = u, then a(”) = 6;. For (0, if there were no such v , then for u = 17el, 7 d E 65 = 6” so I 7dl 5 I 7el . 0

Lemma 4.3 (Uniqueness). There exist partial recursive f and g such that for all v E 6 :

(a) [ f ( v ) , v ] is the characteristic functions of {u : IuI < Ivl}; (b) for any u, i f Iu I 5 I v 1, then J , is recursive in J , with index g(u, v).

Proof. The functionsfandg are defined by the recursion theorem over 6 much as in [Sh, p. 104-106]. Our equivalent of [Sh, (1)-(3) p. 1041 is Lemma 1.4 and the fact that for all v E 6,oJ is recursive in J , with index computable from v. Thus all cases not involving a notation of the form 7 d may be treated as they are there and we consider here only the cases that are new. (1) u = 7 d and v = 2,:

(a) I u I < I v I ++ lu I < I w I & 3e(w = 7 and 1 u I 5 I w I). The first clause can be checked from J , by the induction hypothesis. By 4.2(f) the second holds just in case

which again can be checked from J,.

hence inJ, with index computable from 1.4(a). (b) l u l ~ l ~ l + l u l ~ l ~ l S O J , isrecursiveinJ, withindexg(u,w),

( 2 ) u = 7 d and v = 3 W * 5 a : exactly as in [Sh]. ( 3 ) v = 7 d :

(a) lul<Ivl-3n.J , (hp.O)((u,n))#O. (b) l u I < I ~ I + l u J < I v I o r 3 e ( u = 7 ~ a n d l u I = J v J ) . G i v e n lul<lvl we

can, by (a), decide which is the case. If IuI < IvI, thenJ,(a)(n)= J,(CY)((u,n)). If IuI = IVI,J, =J,. 0


J Corollary 4.4. For any u E 0, w I u > Iu I.

We shal l now prove that for all u E 6 , J , is recursive in S. In earlier instances of this type of hierarchy (e.g. [Sh]) the analogous result was nearly trivial. This is not quite so here and in fact the proof eluded the authors until Leo Harrington provided the key idea of Theorem 4.6 below.

Lemma 4.5. There exists a functional F recursive in S such that for all u, m, a, and T,

J,(a)(m)t 1, i f y ~ ~ ~ u ~ ~ l l y l l ;

0, otherwise. F(u ,m,a ,y) =

Proof. The definition of F is similar to that of the corresponding functional F in the proof of Theorem 1.3. We shall describe intuitively how F is computed and leave it to the reader to define the appropriate auxiliary functional G and apply the recursion theorem. The proof that F is as required will be by induction on llyll for y €9, so to compute F(u,m,cr,y) we may assume y EW and that for all p E Fld (y), F(u, m, a, y r p ) is the correct value. Hence we may decide recursively in S, whether or not a given v belongs to O(llrll) by computing whether or not 3p[F(v ,m,a ,y r p ) > 01, and if the answer is yes we may compute the unique p,, such that I v I = II y t' p,,ll.

Now we proceed by cases on u .

u = 1: F(u ,m,a ,y) = oJ(cr)(m) ;


otherwise,

F (u , (m,n ) , a ,y ) = 0 .

u = 7 d : ifp, exists,F(u,(m,n),cw,y)= F(u,(m,n),a,yEp,);otherwise, if the following three conditions are satisfied: (i) d is a primitive recursive index;

(ii) V v b , exists - + p ~ ( ~ ) ( ~ ) exists]; (iii) VvVv‘[p, and p,,’ exists A pv 5 pv’ + p(d)(”) 5 P { ~ } ( ~ ~ ) I ; then

F(m,n,a,yrp,), if pm exists;

0, otherwise; F(u, (m, n) , a, y) =

otherwise, F(u,(m,n),a,n) = 0. u is not of one of these forms: F(u, m, a, 7) = 0. 0

Theorem 4.6. There exists a functional G partial recursive in S such that for any u, m, and a,

u E 6 -+ G(u,m,a) = J,(a)(rn) .

Proof. We define G by recursion over 6. The clauses for u = 1, 2”, or 3 ” . S b are routine (in the sense that they differ little from the corresponding clauses in [Sh] or [PI]) and we omit them. If u = 7 d , l e t H be defined by:

By the preceding lemma H is partial recursive in S. Furthermore, the conditions of the first clause imply IvI < IuI = 17dl and hence I{d)(v)l< IuI, so that by the induction hypothesis for this stage, G({d}(v) ,m,a) is defined for all m and a. Thus H is a total functional. Let I be a jump recursive in S such that both F and H are recursive in I (F being as in Lemma 4.5).

We claim that w i 2 IuI. If not, then as w i is a limit ordinal one of the following must hold:


(i) u { = 1 3 ” . 5 b ( < l u l : thenIvI<w{sothatthereexistsyEW,yrecursive in I , such that I v I I I 1 yll. Then

So J,, is recursive in I. Similarly, for all m , JFb ,v j (m) is recursive in I with index computable from m. HenceJg,.5b is recursive in I so by Corollary 4.4, 13’- 5bl < w{, contrary to assumption.

I v I < 0: 5 I{d} ( v ) I . Choose y E 94, y recursive in I , such that I v 1 < II yll. Then

(ii) a{ = 17el < Iu I : then by Lemma 4.2(f) there exists a v such that

J { d } ( ” ) ( c w 4 = H ( v , m , f f , 7 ) 7

so J{d) ( , , l is recursive inI , which again leads to a contradiction by use of Corollary 4.4.

Hence, w i 2 lu I.

codes a well-ordering of type 0:. Hence it suffices to set There exists a fixed index a such that for any jump J , hp 0 {i} ( p , S ( J ) )

Corollary 4.7. (a) For all u E 6, J , is recursive in S ; (b) lO l<wf .

We now turn to proving that every set of natural numbers recursive in S is recursive in some J, (u E 6). To this end we need a boundedness lemma for 6. As in [Sh] we shall derive this through the use of a partial recursive “addition” function @ on 6 . Apparently it is not possible to define such a function with the property Iu @ v I = Iu I + IvI. Fortunately, some weaker properties suffice.

Definition 4.8. An ordinal u is superadmissible iff u = I 61, or for some. 7 d E 0 , u = 1 7 d l .

Lemma 4.9. There exists a partial recursive function @ such that fo r any u , v E O :


(a) u@visdefinedandu@vEO; (b) for any superadmissible u, if Iu I , I P I < u, then I u @ v I < u; (c) Iu @ v I 2 max {lu I, Iv 11; (d) v f 1 -+ Iu I < Iu @ v I ; (e) IU’IIIUI’IU’@VI~IU@VI.

Proof. We define @ by recursion on v E 6 (much as in [Sh]) as follows: ( 1 ) u @ 1 = u ; (2) u @ 2v = 2 u e v ;

(3) @ 3”.5b = [email protected] ,where [ c , u @ v ] ( m ) = u @ [b,v](m). S u c h a c exists and can be easily computed by the uniqueness lemma and the induction hypothesis that I v I 5 I u @ v I ; -

(4) u k 7 d = 37d*5C, where,ifO= 1 and% = 2m,

7 d @ r n , if lu1<17dl ;

7 d , if and m = O ;

u @ E , if and m>O.

By (a) of the uniqueness lemma, the cases are recursive in J7. so such a c

We prove (a)-(e) by induction on v E 6. Cases (1)-(3) pose no unusual exists.

problems - for (b) we use 4.2(b),(c). For case (4), let y = [c , 7 d ] . Then clearly y(0) = 7d and for all m,y(m+l) = 27(m) so Iy(m)l< l-y(m+l)l. Hence u @7d E 0. Properties (b), (c) and (d) are obvious. Suppose l u ’ l < l ~ l . I f l u 1 < 1 7 ~ l , t h e n a l ~ o l u ’ l < 1 7 ~ l sou’@v=u@vand(e) issat is- fied. Otherwise, by the induction hypothesis, for all m, Iu’@iiil I Iu @E I so a g a i n l u ’ @ v ) I l u @ v l . 0

Lemma 4.10 (Boundedness). There exists a primitive recursive 5 suck that for any superadmissible u and any a, u suck that I u I < u, [a, u ] is total, and vm( I [a,ul(m) I < u):

( a ) SupII [a,ul(m)l: m E o ) < It(a,u)l< 0;

(b) If IvI 5 Iu I and [b,v] is a total function suck that for every m, [b,v](m) is equal either to 1 or to [a,u](m), then {(b,v) 5 {(a,u).

Proof.Let{(a,u)=3U.5C, where [c,u](O)-u and [c,u](mtl)-- [c,u](m) @ 2[n7u1(m). Then (a) is obvious from 4.9(a)-(d) and 4.2(a).


Suppose b and v are as in the hypothesis of (b), and <(b ,v ) = 3 Y . 5 d . It suffices to show

For m = 0 this is just the hypothesis IvI 5 lu I . Assume it is true form. If [b, v ] ( m ) = 1, then by 4.9(d)

If [ b , v ] ( m ) = [a ,u] (m) , the result follows by 4.9(e). 0

To provide a convenient setting for discussing computations relative to various initial segments of the Ju's, we next define a sequence of computation systems similar to d of [Ri, 283-2841. The intuition is that for a given u, a function is u - computable iff it can be computed effectively except for references to "indexed oracles" for all JU( Iu I < 0).

Definition 4.1 1. For every u 5 I0 I : (a) the relation [ x ] " ( m ) = n is defined inductively as in [IU] for type-2

with a clause S8 for functional application of the form:

(b) 31 7 is the class of partial functions @ such that for some x , @(m) 2 [x]'(m). Such a @ is called u-computable with a-index x .

(c) 32 5 is the class of partial functionals F which are u computable on Wp - that is, F is defined on all (total) a E % and there exists a u-computable @ such that for all x , if [ x ] " is a total unary function, then @(x) % F ( [ x ] " ) . A u-index for @ is also called a u-index for F.

When u = 10 I, we shall usually omit the superscript u.

Lemma 4.12. There exist primitive recursive f and p such that for any x , m, and n, and any v €6, i f O = l v l :

(a) uxnu(m) N- [ f (x7 v), Vi(m); (b) i fu is superadmissible, [[x]"(m)J. - p(x, (m)) E a("), and i fso

T = l p (x , (m) ) l , and [ x ] " ( m ) N- [ x ] T ( m ) .


Proof. f and p are defined by straightforward recursions and the properties established by corresponding inductions. For (b) we use the boundedness properties of superadmissibles. 0

Corollary 4.13. There exists a primitive recursive '17 such that for any superadmissible u and any x, if [x]la is total, then ~ ( x ) E 6") and [ x i =

[ x ] ' q ( x ) ' . h particular, [ x jU is recursive in Jq(x) . Hence the total u- computable finctions are just those recursive in some J,(lu I < u).

Proof. If [[xi" is total, take q(x) to be an element of O(O) such that for all rn, Ip(x,(rn))l< I ~ ( x ) l . Such an 1) is easily defined by use of Lemma 4.10. 0

The next lemma is the key to the succes of our method. For application to the main Theorem it is most convenient to have it formulated in terms of functionals, but the intuition behind it is in terms of jumps: for every J E 7? 2

there exists a superadmissible u < 18 I such that for all 01 E 92 p, J(a) E 37 p. This form can be obtained from the version below by an application of the boundedness lemma,

Lemma 4.14. There exists a primitive recursive O such that for any F E 7? 2

with I 6 I -index x, O(x) E 6, u = I O(x) I is superadmissible, and F E 31 1 with a-index x.

Proof. Since F has I6 I -index x , we have from 4.12(b) that for total [[y 1, F ( [ I y ] ) = o[x](y) = [Ix]'(y) for T = Ip(x,y)l. For each v E 6, let

A, = { y : ny] l v ' is total} .

It is obvious that lvl 5 Iv'I -+A, 2 A"! and it follows from 4.12(a) thatA, is semi-recursive in J , and hence recursive in J,v. Hence there exists an index c computable from x such that for each v E 0 and ally:

Let t(v) = { (c , 2") ({ from 4.10) and set O(x) = 7 d where d is a primitive recursive index for t .


It is clear that t maps 6 into 6, so to show 7 d E 0 we need only check the monotonicity condition of 4.2(d). But if Iv I I I w I, then clearly the functions [c, 2'1 and [c, 2"'] satisfy the hypothesis of 4.10(b) so I ,$(v)l= INC,2") I I 1s(c,2w)l = It(w)l.

The ordinal u = lO(x) I = I 7dl is superadmissible by definition and it remains to show that x is a u-index for F. For anyy such that [ y ] " is total,

[ y p ( j + so thaty € A , @ ) and T = Ip(x,y)I <{(c , 2q(-'") = g(q(y)). Since Iq(y)l< u, by 4.2(e) so is [ (q (y ) ) < u, hence T < u. Thus F ( [ y ] " ) = [ x ] " ( y ) as required. 0

F(uyn0) =F(uyn) = u x n w = u x n w for7 = iP(X,y)i. BY 3.9, uyn" =

Lemma 4.15. There exists a primitive recursive f such that for any superadmissible u and any F E 925 with 0-index x, for all a and m

hoof. The definition off is by a standard application of the recursion theorem similar to those in the proofs of Lemmas 1.4(b) and 2.8 and we consider only the case

By the induction hypothesis

for some primitive recursive (substitution) function g. Then we pick f(a,x) to be an index such that

The first term is, of course, to ensure that Xp {b} ( p , m, F ) is a total function.


Corollary 4.16. For any superadmissible u 5 I 6 I and any F E 92 5, 1-sc ( F ) - C W ? .

In the introduction we defined S as a function on jumps. I t is obviously equivalent to consider it as a function on functionals defined by

0,

1, otherwise.

if (a(0)) (a+, F ) .1 ;

This form is more convenient for treating computations in S .

Lemma 4.17. There exists a primitive recursive 71 such that for any total F and any x , i f F E '32 with I 6 1 -index x , then S(F) E 9? with 1 6 1 -index

.(X>.

Proof. Suppose F E W 2 with 161-index x . Let f be a primitive recursive function such that for any total [ y ] E q2, f ( x , y ) is an 161-index for a recursive join Fy o f F and [ y ] . By Lemma 4.14,0( f (x ,y)) €%, u ( y ) =

l e ( f (x ,y ) ) l is superadmissible, and Fy E %?;Q). By 4.15 and 4.12(a) we can define a primitive recursive g such that

and thus a primitive recursive h such that

Thus the relation {a} (1 y 1, F ) .1 is uniformly semicompu table from J,cf(x,v)) and hence computable from S(J,cfcx,y))) which isJ2eu(x,y)). From this it is easy to compute a I 61-index for S(F). 0

Theorem 4.18. There exists a .primitive recursive f such that for any a < o, m = mo, ..., m k p l <a, x = x o , ..., x I P l <wand a = ao, ..., such that ai = [ [ x i ] E 92 for i < I , i f {a}(m,a, S ) = n then

E wo

nmn (m, = n.


Proof. This is similar to the proof of Lemma 3.3, and we again proceed immediately to the main case where

with F = hp {b}(rn,a, 0, S ) and a = A4 * { c } ( 4 , m,a , S) . The assumption that ( a } ( m , a , S)S guarantees that F and a are total. Furthermore, from the induction hypothesis it is clear that there exists primitive recursive g and h such that F E 92 with I 0 I -index g(b, m , x ) and a E 02 with I 6 I -index h(c,m,x). Hence by 4.17S(F)(a) = [ [n(g(b,m,x))n(h(c,m,x)) . Then it suffices to let f ( a ) be an 16 1-index for this as a function of m and X. 0

Corollary 4.19. For any @ and F : (a) @ partial recursive in S -+ $ E 92 ; (b) F partial recursive in S and defined on all a E ‘2 + F E ‘% 2.

Corollary 4.10. recursive in J,, ;

(a) For any LY, a is recursive in S iff for some u E 6, a is

(b) 101 =.us.

Proof. (a) is immediate from 4.19(a) and 4.13; (b) follows from (a) and 4.7. 0

$5. Functionals equivalent to S

Along with the superjump S , Gandy introduced in [Gal two other functionals and sketched proofs that all three generate the same class of type-1 functions. In [Pl] , Platek made use of some slightly altered versions of Gandy’s functionals. As most details of the proofs of equivalence were not included in either article, we thought it worthwhile to present here (reasonably) complete proofs. To avoid repetition we shall treat in detail only Platek’s versions of the functionals and mention at the end how the proofs may be altered to handle also Gandy’s original versions. For completeness’ sake we also include the equivalence of a fourth functional. The main ideas of this section are due to Gandy.

of Kleene’s schemes SO - S9 in [Kl] . He only considers computations for partial functionals hn cp( a’) where a is a sequence of total objects. Also he

In order to formulate our results we shall need to extend the interpretation


only defines relative recursion relative to such functionals. We wish to consider these notions for partial functionals where some of the arguments may range over partial type 2 objects. We shall continue to use F , G , H , ... to denote total type 2 objects while we shall use F, G, ... to denote partial type 2 objects. Kleene's scheme S8.2 must be supplemented by a new scheme SS.2 introducing the new type of variable:

with appropriate modifications in the indexing to take care of the new scheme and the new type of variable. By using the scheme SO as in [Kl] we may relativise to a sequence may be partial. But it will be necessary to insist that ql, ..., 'kl are consistent in the following sense: if \ k j ( n ) $ and n ' results from n by extending some of the partial type 2 objects occuring in n then 'Pi(.') = \ki(a). Without this restriction the inductive definition for the graph of the enumerating functional (see 3.8 of [KI]) would no longer be monotone, so that a crucial ingredient of [Kl] would be missing. I t is not hard to see that if cp is partial recursive in ql, ..., ql where ql, ...,

texts where this might not appear necessary. For example let 3 O = XF.0 and 30 = hF.0. Clearly these are distinct and both are partial recursive, using essentially the same schemes. But what about 30= hfi.30(ha*(a))? ' 5 is extensionally identical with 30, but it is not partial recursive, and hence must be distinguished from 3 O on the grounds that 30 and 30 have arguments of different type.

In the next two definitions we introduce the three functionals we shall compare with S.

..., ql of functionals, some of whose arguments

are consistent then cp is also consistent. It is important to distinguish between variables F and F even in some con-

Definition 5.1. For any total F ,

0, if ~ ( Y [ ( Y recursive in ' E , F and F(a) = 01;

1, otherwise. & ( F ) =

We recall from [PI] that a partial functional F is called acceptable iff the domain of F includes all (Y recursive in 2E, F. If F is acceptable and F C G,


then G has the same 1-section as F

Definition 5.2. For any acceptable F,

0, if { a ) ( k ) ~ . ;

1, otherwise. S+(LZ, F ) =

0, if 3a [a recursive in 2E, F and k(a) = 01 ;

1, otherwise. &+(F) =

Both St and E+ are undefined for non-acceptable F.

We note that S+ does not seem to be naturally interpreted as a jump since the jump of an acceptable functional would not in general be itself acceptable. Computations relative to &+ are the same as what are called “weak computations from 8’’ in [Pl]. Of course the proof in [Pl, Corollary 111 that the same a are weakly computable from 8 as are strongly computable from 8 depends on the incorrect hlerarchy result.

Note also that both S + and 8’ are consistent. This follows from the fact that if fi is acceptable and F E G, then G is also acceptable, for any a and m, {a} (m, 2E, k) = ( a } (m, 2E, G) (because only values k(a) for a recursive in 2E, F are used in the computation), and 1-sc (2E, k) = 1-sc ( 2 E , G).

Lemma 5.3. (a) 8 is recursive in &+; (b) S is recursive in S + ; (c) & is recursive in S ;

Proof. (a) is trivial as &(F) = E+(XaF(a)). For (b) note that S(F)(a)(a) = S+(p(a) , (F , a)) where p is a primitive recursive function such that {p (a ) } ( (F ,a ) ) = {a}(a,F). For (c) note that

E(F) = 0 - 3a{b}(a, 2E, F ) = 0,

where b E o such that { b } (a, 2E,F) = F(Am{a} (m , 2E,F)) . Hence


w h e r e e E w such that {e) (a , (2E,F))= Oif 2E(ha{b}(a,2E,F))= 0, and is undefined otherwise. Then 8 ( F ) = S((2E, F))(ht.O)(e). 0

Our aim now is to show that Z+ and S+ are both partial recursive in G . From this together with Lemma 5.3 and the appropriate transitivity results it follows that 8 , S , &+, S+ are all partial recursive in each other and hence have the same 1-sections.

We shall use the notion of a computation tree relative to ’E, F. Our notion differs somewhat from that introduced in [Kl]. Let Q[a be the inductively defined set of all tuples (a ,m ,n ) such that (a}(m, 2E, F ) s n . L e t s F be the well-founded partial ordering of Q [PI corresponding to the relation ‘precedes or equals as subcomputation’ among computations. For example, if a is the index for the composition of <b} and {c} , {c ) (m, 2E, 8) = n, then for somep, ( c , m , p ) and (b ,p ,m ,n ) are the unique immediateSF-predecessors of (a ,m,n) . Of course, the precise definition depends on a particular indexing, but we shall not delve into this level of detail.

T&,n is most easily defined by recursion on computations. Note that for any ( a , m, n ) E Q [PI, TCm,n is a well-founded partial ordering with largest element (a ,m ,n ) and no limit points. Any such ordering Thas a natural ordinal 11 T 1 1 associated with it. If T r u = {(s, t ) : T(s, t) and T(t , u)), then for any u other than the T-largest element, II T r u II < I I TII.

Let T:m,n be the restriction o f S F to {s : s<’(a,m,n)}. Formally, of course

Lemma 5.4.Forany ( a , m , n ) € a [ F ] , TCm,n isrecursivein 2E,F .

Proof. Construct in the usual way a primitive recursive f such that for (a ,m ,n )EQ[P] , f ( (a ,m ,n ) ) isanindexofTCm,, f rom2E,k. 0

Definition5.5. (a) dF(T)-for some(a ,m ,n )€a[F] , T=T&,,;

T = To,m,n.

P (b) i f d (T) ,aT, mT, and nT are the unique a, m, and n such that P

Lemma 5.6. There exists a function @ partial recursive in El such that (a) Fis acceptable c=$ AT@(T, F ) is total;


(b) If F is acceptable then XT@(T, F ) is the characteristic function of JF.

Proof. Let cl ‘(T) be the conjunction of the following clauses: (a) T is a well-founded partial order with largest element but no limit points; (b) all elements of the field Tare of the form (a, m,n); (c) for all u other than the T-largest element, J ‘(T ru ) ; (d) for all t = (a,m,n) in the field of T, one of the following holds:

(1) a is an index for an initial function and T = {(t, t)}; ( 2 ) for some b and c, {a}(m, 2E, p ) = {b}( {c}(m, 2E, P ) , m, 2E, &’) and

forsomep,sl=(c,m,p)ands2=(b,p,m,n)are theuniqueimme- diate T-predecessors o f t ;

a = A ~ * { b } ( p , m , ~ E , F ) andVp3!4 such tha t (b ,p ,m,q ) i san immediate T-predecessor o f t and all immediate T-predecessors o f t are of this form; further, if 2E was applied then q = 0 or 1 depending on whether or not one of the immediate T-predecessors o f t is of the

(3) for some b , {a}(m, 2E, P ) = 2E(OI> or k(a) where

form (b ,p ,m,O) ; (4) other cases similarly. J‘ is well-defined by recursion on 1 1 TII, and a proof that it is recursive in

E , follows the lines of the proof of 1.3. Of course, El is needed for (a) and for the number quantifiers.

describes application of F as in (3) , let & = hp * unique 4 [(b, p , m,4) is an immediate T-predecessor o f t ] . Then set

For any T such that J ’ (T) and any t = (a, m, n) in the field of T which

0, if J ’ (T) and for all t = (a, m, n ) E Fld (T) which describe application of F, &3,) = n;

1 , ifeither- d!(T)or d’(T)but f o r s o m e t = ( a , m , n ) E F l d ( T ) which describes application o f F , 3n’ (F(&)=n’ and n’#n).

@(T, F) - Thus @ is partial recursive in

(i) ( a , m , n ) ~ a [ F ] - + @ ( T [ ~ , ~ , P ) . - 0,. (ii) V T [ @ ( T , F ) $ and(@(T,F)---O+dJ(T))] ,andforanyF

(iii) if AT@(T, F) is total then F is acceptable. The proof of (i) is a straightforward induction over Q[&’]. For (ii) if

and it suffices to prove for any acceptable F

- J ‘(T), @(T, F) N_ 1 so assume J ’ (T) . We now proceed by induction on II TI1


k*(T) N

and by cases according to the T-largest element t = (a , m, n). We consider only the case where t describes an application of P. By the induction hypothesis Vp.@(Tr(b,p,m,p,(p)),F)$. If any of these hasvalue 1, thenclearly also @(T,F)= 1.If they all havevalue 0, then Vp. J F ( T r(b,p,m,P,(p))) which implies V p a p t ( p ) ‘v {b} ( p , m, 2E, g). Hence Pt is recursive in 2E, F, so, as F is acceptable, P(&)J and thus @(T, P ) $ . If @(T, #) = 0, then P(&) = n and thus dF(T) .

that ~ ( ( w ) $ . For each 4 E w Te,4,,(4) E J I . Let b describe the application of F t0a ; i . e . (b}(2E,&’)~~(hq(e}(q,2E,.~)).Nowlet Tbe theunique tree in J’ with T-largest element ( b , O).and 7’&,(4), for each 4 E w, as immediate subtrees, i.e. T(s, t ) iff either 34T,&,(,)(s, t ) or (s= t=(b , 0)) or (34T[4,,(4)(s,s) and t = ( b , 0)). If t = (a,m,n>E Fld(T) describes application of F then either t = ( b , O), when when F( &) N- n . Hence

For (iii) let AT@(T, F ) be total and let a = Aq{e}(4, 2E, P). We must show

= a, or else t E Fld(T&,(4)) for some 4 ,

0, if @(T, F ) = 0 and a , is an index describing application of

1 , if @(T, F) = 0 but ( a , is not an index describing application

Fand nT = 0;

o f F o r n, + 0);

So if AT@(T,F) is total it follows that F(a)$ . 0

Definition 5.7. For any F and T:

Lemma 5.8. For any F (a) F i s acceptable - F* is total; and (b) ifk is acceptable then 1-sc (2E, F ) C 1-sc @*).

Proof. (a) follows immediately from the definition of F* and Lemma 5.6. For (b) we define a primitive recursive f such that


{a} (m, 2E, P ) = n -+ hr { f ( a , h))) (r , P * ) is the

characteristic function of T&,n .

The definition off for most cases is routine. We treat only the case when a

describes an application of F, say

{ ~ } ( m , ~ E , @ ) = F ( f l ) = n where P = h p . { b } ( p , m , 2 E , F ) .

For any 4, let Tq be the tree with largest element ( a , m , q ) and all T&,,,D(p) as immediate subtrees. Then dF(Tq) iff 4 = n. From the induction hypothesis we can define a primitive recursive g such that for all 4, hr {g(a, m,4) } ( r , P*) is the characteristic function of Tq. Then n = least q [$'*(Tq) I 1 J and f ( a , m) -g(a, m, n) is a proper value.

Finally for any a, m such that {a} (m, 2E, F)4,

{a} (m, 2E, F) = least n - { f ( a , m)) ((a, m , n ) , F*) = 0 . 0

Theorem 5.9 (Candy). &+(F) = &(F*); hence &+ is partial recursive in 8.

Proof. Suppose first &'(j) = 0 so for some b , F(hp - { b } ( p , 'E, .F)) = 0. Then if a is an index such that {a}(& N_ $(hp { b } ( p , 2E, F)), $'*(T$,o) = 0. By 5.4, T:4,0 is recursive in 2E, F and thus by 5.8 recursive in F*. Hence &(c;'*) = 0.

Conversely, if &(r'*) = 0, say PI(?') = 0, then @(T, F) = 0 and for some b, {aT} (mT, 2E, P ) = P(hp * {b ) (p , mT, 2E, F)). But then d F ( T ) and this is a correct computation from 2E, F. Hence hp - { b } ( p , m T , 2E, P) is a function f l recursive in 2E, F and $(p) = n - 0. Thus &+@) = 0.

Lemma 5.8(a) shows that & (F)$ iff &(F*)$, so that the first part is proved. The second part follows from the fact that E l is clearly recursive in & and the passage from F to F* is uniform. 0

+T :

Definition 5.10. Let 3' be defined like d ' except that no provision is made for application of 2E. Then for any F, T, and a,

0, if @(T, F) = 0 and 3 ' ( T ) and aT = a and mT = $;

1, if Q(T, F) = o but (- S'(T) or aT # a or mT + $1; 2, if @ ( T , P ) = 1.


Lemma 5.1 1. For any a and F (a) F is acceptable - F,** is total ; and (b) ifFisacceptable then l - sc (2E,F)G l-x(F,**).

Proof. As for 5.8. 0

Theorem 5.12. S+(a, F ) = &(Pi*); hence, S+ is partial recursive in I%.

Proof. Suppose first S+(a, c) = 0, so for some n , {a } ( F ) = n. Hence Fl*(T[o,n) = 0. By 5.4 T:Q,n is recursive in 2E, F and thus by 5.1 1 recursive in F:*. Hence &(Pi*) = 0. Conversely, if &(&’:*) = 0, say F i * ( T ) = 0, then @(T, F) = 0 and 3’(T), so T is a correct computation tree involving only F. Hence {aT} ( F ) = nT so S+(a, F) = 0. Lemma 5.1 1 (a) shows that S’(a, F ) $ iff &(pi*)$, so that the first part is proved. The second part follows just as in 5.9. 0

Thus from 5.3 and 5.12 we obtain the equivalence of all four functionals and, in particular, the identity of their 1-sections. Candy’s original functional was

0, if 3a [a recursive in F and F(a) = 01 ;

1, otherwise,

and its natural extension gi defined for all recursively satisfactory F, that is, F such that the domain of F includes all a recursive in F. A minor modification of 5.3(c) shows g is recursive in S and the proof - - of 5.12 actually shows S+ -

partial recursive in E. Z+ is an extension o f & so is partial recursive in I%+. - For the converse of this, let % be defined from 3’ as @ was - from 6’ and define F* from 5 in such a way as to ensure - 2E is recursive in F * . Then for any recursively satisfactory F, g’(F) = z ( P * ) . Hence E and E+ also have the same 1-sections as S. We leave it to the reader to check that $+, the extension of S+ to all recursively satisfactory F, also has the same 1-section.


Bibliography

P. Aczel, The ordinals of the superjump and other related functionals, abstract, Conference in Mathematical Logic-London 1970, Lecture Notes In Mathe- matics 255, Ed. Wilfrid Hodges (Springer, Berlin, 1971) 336-337.

R.O. Gandy, General recursive functionals of finite type and hierarchies of functions, paper given at the Symposium on Mathematical Logic, University of Clermont-Ferrand, June 1962 (mimeographed).

P.G. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math.

S.C. Kleene, Recursive functionals and quantifiers of finite types I, Trans. Amer.

Y.N. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. SOC. 129 (1967)

R.A. Platek, A countable hierarchy for the superjump, Logic Colloquium ’69,

SOC. 142 (August 1969) 111-140.

Math. SOC. 91 (1959) 1-52.

249-282.

edited by R.O. Gandy and C.E.M. Yates (North-Holland, Amsterdam, 1971).

W. Richter, Recursively Mahlo ordinals and inductive definitions, Logic Colloquium 25 7 -27 1.

’69, edited by R.O. Gandy and C.E.M. Yates (North-Holland, Amsterdam, 1971) 273-288.

J.R. Shoenfield, A hierarchy based on a type-two object, Trans. Amer. Math. SOC. 134 (October, 1968) 103-108.

J.E. Fenstad. P. G.Hinman (eds.), Generalized Recursion Theory @North-Holland Publ. Cornp., 1974

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL

Leo HARRINGTON Massnchusetts Institute of Technology

The purpose of this paper is to continue the investigation of the superjump, which was first defined by Candy in [Gal, which was studied despite adversity in [Pl] and [Ac] , and which has recently been examined with notable success in [A-HI . The main result of this paper is that of, the first ordinal not recursive in the superjump, is the first recursively Mahlo ordinal. This result was announced in [Ac], but its proof hinged on some corollaries of a fallacious theorem from [Pl] . These corollaries turn out to be correct.

As prerequisites, a working knowledge of Kleene’s (or some equivalent) formulation of recursion on higher type functionals, [Kl] , would be desirable. The reader would also be advised to acquaint himself with the discussions in [Pl] and [A-HI, and most particularly with $ 3 and $4 of [A-HI.

I should acknowledge here that my debt to [PI], [Ri] and [A-HI is even greater than the derivative nature of this paper would suggest.

A few definitions seem to be in order.

Definition. The superjump, S, is a type 3 functional defined by o if { e l F $ i 1 otherwise

S(F, e ) =

(where F is any type 2 functional). 2E is a type 2 functional defined by

0 if X = @ 1 otherwise

(where X is any real).

43

44 L. HARRINGTON

Let F be a fixed type 2 functional. F will stay more or less fixed till further notice. The rest of this paper will be developed relative to F. This will seeming- ly increase the generality of the results while in no way increasing the difficulty of their proofs. In fact F will in general be tacitly ignored.

Definition. 1 - Sc (F, 2E), [ 1 - Sc (F,S)], is the collection of reds recursive in F, 2~ [recursive in F, SI .

Let u be an ordinal. Define L,(F) by: LJF) = U6<(r {x CL6(F) ;x is first order definable with parameters over (L6(F),e,F 1 (2w nL,(F)))). Let MJF) denote the structure (L,(F), E , F P (2w n Lu(F))).

u is an F-admissible ordinal if M,(F) is an admissible structure. f~ is F-recursively Mahlo if u is F-admissible and if for any function

wf(p:) is the first F-admissible (F-recursively Mahlo) ordinal. f : u + a, A, over M J F ) , there is an F-admissible /3 < u such that f” /3 Go.

In [Sh] , Shoenfield constructed a hierarchy for 1 - Sc(F, 2 E ) which could be used to show that 1 - Sc (F, 2E) = 2w f’ L F ( F ) . In 5 1 Shoenfield’s hierarchy will be extended to a hierarchy for 2w n LpB(F). This is done in a way totally analogous to the extension in [A-HI of the hierarchy from [Pl]. The technique of [Ri] is the guiding influence in both cases. This new hierarchy will then be employed in $ 2 to demonstrate that 1 ~ Sc(F, S) = 2- n L notion of partial recursiveness in S is discussed.

w1

(F) . In $ 3 the possibility of finding a reasonable PF

In this section the hierarchy for 2w n L lemma concerning this hierarchy is proven.

(F) is defined, and the crucial 8

Definition. A set of notations, ‘ K F , a function \ I F : 7ZF +Ordinals, and for each 12 E qF a real, H:, are defined by induction as follows

(i) 1 E qF, I I l F = 0, H: = w .

( i i ) I f x E q F then: 2 ’ E n F , 12’IF=IxlF+ l , andH2F,={(e , a ) ; ( e}~x is total and F({e)H.F) = a}

F

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL 45

(iii) If n E

13” - g e l F = the 1st limit ordinal greater than InlF and I{e}HF(i)lF for all i E w , a n d H S n .

(iv) If there is an ordinal u such that (a) u is a limit ordinal; (b) u # I 3 a * 5b I F foral la ,b;and(c)foral ln , i f In lF<u then I{e}(n)IF<u;then: 7 e € % F , 17elF = the 1st ordinal satisfying (a), (b) and (c), and H&= {(m,O);ImI”Q: 17elF}U {(m,a+l ) ; lml”< 17elFanda€HE}.

and if for all i E w {e}Hg(i) E % then: 3, * 5e E %“,

= {(m,O); ImIFQ: 13n-5elF}U {(m,a+l ) ; IrnIF< 13n-5el Y a n d a E H i } .

(It should be noted here that this hierarchical definition has some atypical features when compared with corresponding definitions in [Sh] , [Pl] or [A-HI. The most glaring dissimilarity is that H: = H L if In I F = I m IF.)

For an ordinal u, let %: = {n E nF; In I F < a}. For n $92 let

Till further notice the superscript F will be systematically deleted. Let I nFl = sup { ~ n I”; n E % ”1.

Definition. For an ordinal u, u is %-admissible if u = 1 7el for some 7e €% ; u is %-inaccessible if u is %-admissible and a limit of %-admissibles.

Notice that for any n €% there is an %-inaccessible ordinal greater than In I.

Remarks. in H,,, uniformly from n , m; if In I < J m I then the Turing jump of H, is recursive in H, uniformly from n , m.

(2) For n €92, if n is %-admissible (or even if In 1 = w -In I) then there is a real I,, recursive in H,, which codes M , , , (F). The 1,’s are in fact recursive in the Hn’s uniformly.

(1) For n, m €% , if ( n 1 5 Im I then %, and H, are both recursive

(3) For any ordinal u, { (n ,a ) ; n E%

By remarks(2) and(3) wehave: { X ; X I H , fo r somen€%}= 2WnLI,I(19.

It should be clear that p c cannot receive a notation, and so t 92 I I p c . To

and a EH,} is C, overM,(F).

demonstrate now that our hierarchy is in fact a hierarchy for 2w n L p c ( F ) , we need only show that 1 % I is F-recursively Mahlo. The key lemma is the following.

46 L. HARRINGTON

Lemma. There is a recursive function p such that i f at least one of mo, m is in %, and $0 is the first %-inaccessible ordinal greater than min ( I mo I, I ml I), then A m o , m 1 ) E 92 9, and Ip(mo, ml) I 2 min (I m0 I , Im I).

Proof. The proof is of course by effective transfinite induction on p = min( lmO1, Iml I), so assume that p(mb,m;) is defined and works whenever, min ( I mb 1, I) < p . The only novel cases are when one of m,, , m is of the form 7e. (As may have been noticed, throughout this paper generalized recursive functions have been preferred to primitive recursive ones. When dealing with potential notations of the form 7 e or 3" * 5e, this could be a cause for worry as {e} or {e}Hn need not be total. This difficulty is easily handled since an oracle for H , or H2,, can always be presumed, and so {e} or {e}Hn may be converted into a total function. For the remainder of this proof, this annoyance will be ignored.)

As a notational convenience define, given a set of ordinalsA, the strict limit sup ofA, slsA, to be the 1st limit ordinal greater than every member ofA.

Case 1: mo = 7eo,ml = 2x Find an e such that for all n E 72

(Such an e can be found: By remark (1) there is a recursive function k such that for n E % {k(n)} 2" = % for all integers n and all reds X

H Let h be a recursive function such that

if either mo or m l is in {k(n)}X

p({eo}(n) ,x) otherwise.

Get e so that {e}(n) = 32n - 5h(n).) Notice that this respects the induction hypothesisas IXl>p*p= 17eol andso Inl<p* I{eo}(n)I< 17e01=~ Let p(mo,ml)= 7e.

Ip(mo,ml)l < p : By the induction hypothesisn €%, * {e}(n)€% ,. Let u = sls({ 1 {e}(n) 1 ; n €92 p } U {p} ) . So u 5 0 . But u has a notation of the form 3m - 5- (where Im 1 = p), and so u < 0 since 0 has no such notation. But fo rnEC) lp - Y p , I{e}(n)I=sls{lnl}= I n l + o . T h u s 1 7 e l i t h e l s t % - admissible ordinal greater than u, which is less than Therefore I 7e I < 0.

since 0 is 92 -inaccessible.

Ip(mo,ml)12p: Supposenot.Then I n [ < 17eI*In1<pandso

THE SUPERJUMP AND,THE FIRST RECURSIVELY MAHLO ORDINAL 47

I{e}(n)l> Ip({eo}(n),x)l, and {eo)(n),x satisfies the I.H. (induction hypothesis).But ln1<17el*l{e)(n)I< 17eI<_IxIandtherefore I {eol(n) I'I Ip({eo)(n),x) I 1 = jslsiln I'

Notice that no,nl €92, *min(l{eo)(no)l, I~el)(nl)I)<min(17eol,l~e11) = p and so the I.H. is maintained. Let p(mo,ml) = 7e.

By an argument entirely similar to that in case 1, we get lp(rno,ml)l < P . p(mo,ml)l 2 - p : Suppose not. Then 17el < 17e01 and so there is

In a way similar to case 1, using the uniformities mentioned in remark (l),

if In l2p

M{InIlu {Ip({eOl(no), {el}(nl))l;no,nlE92,,,)) if In1

no E 92, such that I{eo)(no)l 2 17el. Similarly there isnl E 9217eLsuch that Iler(nl)I 2 17el. But pickn € 7 2 17e, so that In1 > lnOL Inll. T en 17el > I {e)(n)l> Ip({eo)(no), {el)(nl))l ~ m ~ ( l ~ e ~ ~ ( ~ ~ ) I , I ~ e ~ ~ ( ~ ~ ) 1 ) ~ 1 7 e l .

Cuse 3 : mo = 7eo,m 1 - - 3n' - se' Claim. There is a recursive function f such that given 4, if 14 I < p or if Imo I = p, then f(4) € no and I f (4) I 2 min ( Imo I , I4 I )- Proof. Find a recursive function e such that for all n E 9'2

Notice that either 14 I < p, or l ~ l = p = min( lmO1, I4 I ) in which case In1 < lmOl * I{eo)(n)I< ImOI = p . So if 1n1 <min(IrnoI,Iq~) then one ofq, { eo ) (n ) is in %,, and thus the I.H. is preserved.

Let f ( 4 ) = 7e(4). As usual I f ( 4 ) I < P. ~f(4)i~min(Im01,141): Supposenot.Then Inl< If(s)I*

In1 <min(Imol, Iq / ) * the I.H. applies to {eo)(n),q and I{e(q)}(n)l>lp({eo)(n),q)l.But Inl< lf(4)I+l{e(q))(n)I< 17e(4)l<lql, and so I{eo)(n)l I Ip({eo)(n),q)l < I{e(q))(n)l. Thus 17e(4)l 2 17e01.

I f(nl)l 2 min( lrnol, Inl\). Letxi denote the possibility imaginary object Now to finish case 3. Notice that by the claim f ( n l ) €92 and

48 L. HARRINGTON

{el lHn1(i) . If n E % then the xi’s have substance, and the claim implies that f(xi)E%p andf(xi) 2min( imoI , IxiI).

Let m = 2f(n1). Find e such that

Notice that If(nl)l < lmOl =$ lnll I If(nl)l and thus thexi’s have substance, and therefore the I.H. is not violated. Let p(rno,ml) = 3M 5e . As always Ip(mo,m1)I If(nl)l 2 Inll, and thus I{e}Hm(i)I > If(xj)I 2 [xi[. So 13m * S e l 2 13n1 *5e1(.

It is now possible to show that 172 1 is F-recursively Mahlo. The proof will only be sketched as it is precisely the same as Richter’s argument in [Ri]. In fact the 92 hierarchy (for F trivial) is essentially just Richter’s system of notations for po with the onerous 3(x7y) clause eliminated.

Our first job is to show that ;% 1 and the %-admissible ordinals are in fact F-admissible. This should be clear for 72 -admissible ordinals which are not %-inaccessible - the first 92-admissible greater than In I is just wFHn. So let P be either an %-inaccessible ordinal or 1 % 1.

We will place a recursion theoretic structure on the hierarchy restricted to 92, as follows.

31 Definition. [el P(x) ~y means e = (eo,e,), {eo}(x) E 9ZP, and {e l}H{eo}(x) (x ) z y .

I [el %@(x) I = I {eo}(x) I , otherwise I [ eynO(x) t= m.

cp(x> s [el %~(x) .

n [el %P(x).l means [el %O(x) ‘ y for somey. If [el p(x)J. then

A partial function cp is partial 7Zp-recursive if for some e and all x,

We now give a series of theorems which together imply that /3 is F- admissible.

Lemma I . I f cp is a partial 7Zq-recursive finction, and if the range of cp is a

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL 49

subset of %,, then there is an e such that for all x in the domain of cp,

Icp(x)l I I {eI(X>I < P -

Theorem 1 (Bounding). If cp is a total %,-recursive function, and i f the range of cp is a subset of %@, then sup { I cp(x) I ; x E a} < P.

Lemma 2 should be an immediate corollary of the Lemma in 5 1.

Theorem 2 (Selection). There is a partial 92 -recursive function I) such that for all e, if [el %@(x) J. for some x , then [ e] P( + (e)) 4. e,

Theorem 2 is a soft consequence of Lemma 2. For an actual proof of a similar theorem in a different setting, see [Mo] or [Gr]. [Gal is the prototype.

Theorem 3. Let f be C, overMP(F), and assume that f maps a subset of P into 0. Then there is a partial %,-recursive function cp such that for all n , cp(n)$iff InIEdomf,inwhichcasef(InI)= Icp(n)l.

Theorem 3 can be proven by combining remark ( 2 ) of 5 1 with a few appli-

Theorems 1 and 3 should now make it clear that /3 isF-admissible. But this, cations of Theorem 2.

together with Theorem 3 and Lemma 1 for the case /3 = 1% 1, should make it equally clear that 1 % I is F-recursively Mahlo. So to summarize.

Theorem 4 I % 1 = p c . The %-admissible ordinals are just the F-admissible ordinals which are less than &. The partial %-recursive functions are just the partial functions which are C , over M (F).

P{

I t is much easier to show that our hierarchy is also a hierarchy for 1 - Sc (F,S). This can actually be seen from its definition (the Lemma of 5 1 is not needed). Arguments similar to those in 54 of [A-HI give:

50 L. HARRINGTON

Theorem 5. There are recursive functions f and g such that

(b) I f {e}'jF(x) r y then [g(e)ln(x) ' y . (a) [ e I W Z Y ' i f f Cf(e)}S%> 2 . Y

(a) can be proven fairly directly, as in 94 of [A-HI. (b) is best proven in the following form: there is a recursive function h such that, if [eoIn is total and if {el}S>F([eo]q , x ) z y , then [h(eo ,e l ) ]y(x) z y . h should be constructed by induction on the length, in the sense of [Gal or [Gr], of the computation associated with {el}SJF( [ eOln ,x).

Theorems 5 and 4 immediately yield that 1 - Sc ( F , S ) = 2w n L p ~ ( F ) . By letting F become unfixed, it is possible to catch a glimpse of t i e type 3

( F ) k= ~ ( x ) ) ] where functionals recursive in S. They are just the total functionals of the form hF [ [ eIqnF(O)], or equivalently of the form hF [/.tx(M ~ ( x ) is a uniformized C, formula in the language appropnate to the struc- turesM (F) .

& .

P g

§3

In 9 2 we were able to characterize the reds recursive in S. The next natural step would be to investigate the reds r.e. in S. Unfortunately the fact that S is of type 3 swamps all more delicate considerations. The reds r.e. in S are just the l3: reds, which is definitely more than 1 - Sc (S) warrants - there are plenty of reds which are r.e. and co-r.e. in S and yet not recursive in S. This seems to arise as a consequence of the ability of a computation from S to diverge for totally gratuitous, rather than inherent or ineluctable, reasons. (See [Pl] pp. 262-263.) If these gratuitously diverging computations are systematically excluded, then S is seen to have a much more natural r.e. structure, as for example Platek's weak computability [Pl]. Before tlying to justify such systematic exclusions of divergent computations, it would be best to make this notion precise.

Definition. Let G be a higher type functional of unspecified type. A recursive function f is a type n reindexing for G if for any type n a and any e, if {e}'(a) 2 x then {fe}'(a) z x. (Notice that {fe}'(a)$ is possible even if {e}'(a)t.) f is a minimal type n reindexing for G if, for any other type n

THE SUPEFUUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL 5 1

reindexingg, there is a recursive function h such that for all type n a and all e, { feIG(a) k (he ) IG(a) .

Iff is a type n reindexing for G then f gives rise to a natural enumeration of type n+l partial functionals which are recursive in G: the e* partial functional is just ha[{ f e jG(a ) l . AII total type n+l functionals recursive in G appear in this enumeration, but some partial functionals recursive in G may be omitted. Any partial functional omitted in this way may be viewed as being gratuitously recursive in G. (For example: for many functionals G, it is possible to find H G such that there are more type 1 partial functionals recursive in H v G than there are recursive in G. These partial functionals have little right to be recursive in H v G, and this intuition is upheld by the fact that there is a reindexing for H v G which omits them.)

Call a partial functional strongly recursive in G if no reindexing for G ex- cludes it. This may seem a stringent requirement, but for the least pathological higher type functionals, the functionals of type n in which “E is recursive, this is no restriction since the identity type n reindexing is minimal. I t is possible to have minimal reindexings different than the identity. The situation seems to be the following:

Definition. f is an acceptable type n reindexing for G if there is a type n + l partial functional p recursive in G such that for any type n ao,al and any eo 3 el

(ii) if {f(ei)IG(aj)4 for some i, i = o or 1, thenp(ao,eo,al,el)J.

( i ) i fp(ao,eo,al ,e l )=i theni=Oor 1 and { f e j } G (aj).l,and

An acceptable reindexing supports a very reasonable recursion theory, which in some cases is more reasonable than the usual one. Type 0 selection, (the analogue of Theorem 2 of 52, see [Gal or [Gr]), though not necessarily true of the identity reindexing, is true for acceptable reindexings. Acceptable reindexings also possess some degree of naturalness:

Theorem. An acceptable type n reindexing for G is minimal. I f f and g are tun acceptable type n reindexings for G then there is a recursive permutation h such that { f e IG(a) E {g(he)}G(a) for all type n a.

Returning to S, Theorem 5 of 52 may now be summarized by saying that

52 L. HARRINGTON

S has an acceptable type 2 reindexing, which may now be viewed as giving S a natural partial recursive structure. The type 3 partial functionals which are strongly recursive in S are the partial functionals mentioned at the end of $ 2 .

The results of this paper can be generalized to other type levels. Define the type n +3 superjump, n+3S, by

o if {elG+ 1 otherwise

"'3S(G,e) =

(where G is any type n + 2 functional). There is an acceptable type n +2 reindexing for n+3S, and this reindexing satisfies Grilliot's formulation of type n-1 selection (see [Gr] and [Ma]). There seems to be no model theoretic characterization of (n +1) - Sc ("'3S), but it is possible to characterize ( n + ~ ) -

the characterization of (n+l) - S C " ( ~ + ~ E ) in Chapter 3 of [Ma]. There is also a hierarchy for (n +1) - Sc ("'3S) which may be gotten by appropriately extending a hierarchy similar to the one for n+2E described in [Sa]. In all this n+3S behaves remarkably like a type n +2 object.

= u, of type ( n + ~ ) - sc (a, n + 3 ~ ) dong lines similar to

References

[Ac]

[A-HIP. Aczel and P.G. Hinman, Recursion in the Superjump, this volume. [Gal

P. Aczel, The ordinal of the superjump and related functionak, in: Conference in Mathematical Logic - London '70 (Springer, Berlin, 1972) pp. 336-337.

R.O. Gandy, General recursive functionals of finite type and hierarchies of functions (A paper given at the Symposium on Mathematical Logic held a t the Univer- sity of Clermont Ferrand, June 1962). T.J. Grilliot, Selection functions for recursive functionals, Notre Dame Journal of Formal Logic 10 (1969) 225-234. S.C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Am. Math. Soc. 91 (1959) 1-52; 108 (1963) 106-142. R.A. Platek, A countable hierarchy for the Superjump, in: R.O. Gandy, C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 257- 271.

[Ma] D.B. MacQeen, Post's problem for recursion in higher types, Ph.D. dissertation, MIT (1 972).

[ Mo] Y.N. Moschovakis, Hyperanalytic predicates, Trans. Am. Math. Soc. 129 (1967) 249-282.

[Ri] W. Richter, Recursively Mahlo ordinals, in: R.O. Gandy, C.E.M. Yates (eds) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 273-288.

[Sa] G.E. Sacks, The 1-Section of a type n object, this volume. [Sh] J.R. Shoenfield, A hierarchy based on a type 2 object, Trans. Am. Math. Soc. 134

[Gr]

[Kl]

[PI]

(1968) 103-108.

J.E.Fenstad, P . G.Hinman (eds.J. Generalized Recursion Theory @ North-Holland Publ. Comp., I974

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

Yiannis N. MOSCHOVAKIS University of California, Los Angeles

For each object U of finite type over the integers and for each k 2 1 , put

ken(U) = the k-envelope of U

= the set of all relations with arguments of type < k which are semirecursive in U .

We will give a fairly simple, indexing-free characterization of ,en(U) when U is of type m 2 2, m - 1 5 k 5 m + 1 and U is normal, i.e. the existential quantifier mE over the objects of type m - 2 is recursive in U .

Our method will also give simple structural characterizations of some other interesting classes of relations, e.g. the E and the (positive) second order inductively definable relations with arguments of type 0 and 1 . But our main aim is to make a start in the study of structural properties of envelopes. There is evidence that this study will increase our understanding of recursion in higher types. For example, we will also show that the 1-envelope of a normal object of type greater than 2 is not the 1-envelope of any normal type-2 object. Also, the class of II,!, (n 2 2) or E& ( m 2 1 ) relations on the integers is not the 1-envelope of any normal type-2 object (this result is independently due to A.S. Kechris). Thus the I-envelope of a normal object u codes substantially more information about U than the 1-section of U which according to Sacks [ 19731 fails to determine whether type(U) = 2 or type(U) > 2.

During the preparation of this paper the author was partially supported by a Sloan Foundation Fellowship and by a grant from the National Science Foundation.

53

54 Y.N. MOSCHOVAKIS

1. Preliminaries

The set T o of objects of type j over the integers is defined by the induction

A ppint

x = ( X I , ..., x, )

is a tuple of objects of finite type and the type of x is the largestj such that some xi is of type j . A (product) space is a Cartesian product

X = X , X ... xx,

where each Xi is some T ( j ) and the type of 31 is the largest j such that some Xi is T ( j ) . If x = ( x l , ..., x,) andy = (y,, ..., y,), we write ( x , y ) = (x,, ..., x,, y l , ..., y,). Similarly, if 31 = X , X ... X X , and y = Yd X ... X Y,, then % X y = X I X ... X X , X Y , X ._. X Y,.

A relation of v p e k > 0 is any subset R C 3c of a space of type k - 1. We also call these pointsets of type k and we write interchangeably

R ( x ) o x E R .

Following Kleene, we let d,pi, y j vary over T ( j ) . It is also customary to reserve the variables n , k , m, ... for naming integers and the unsuperscripted Greek letters a, 8, y, ... for naming objects of type 1, or Peals.

facts of Kleene.[ 19591. In particular, it is defined there what it means for a (total) function

We assume that the reader is familiar with at least the basic definitions and

to be primitive recursive. This is by means of eight natural schemes and involves no indexing. A function

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS 551

is primitive recursive if there is some primitive recursive

g : T(i)X x - + w

f ( x ) = haig (a i ,x ) .

f : x + y = Yr,X ... XY,

such that

Similarly,

is primitive recursive, if there exist primitive recursive functions

If 5X and Y are of type <_ 1, then these definitions agree with the classical definitions of primitive recursive functions and functionals.

A partial function

f : x + w

is a function with domain some D E %which takes values in w. We write

to indicate that f is defined at x and has value n. We can think of a (total) function f : % -+ w as a partial function which happens to be totally defined on .

The definition of recursive partial functions in Kleene [ 19591 adds only one more scheme to those needed for primitive recursion but is substantially more complicated in its interpretation. An inductive definition is given of a set of triples

56 Y.N. MOSCHOVAKIS

where e, n vary over w and a varies over tuples of arbitrary length of objects of arbitrary finite types. A partial function f : X + w with domain (a subset of) some fixed space X is then recursive, if there is some integer e (an index of f ) such that for all x E X and all n E o,

f (x )=n- (e ,x ,n )E%.

A relation R C X is recursive if its characteristic function

is recursive and semirecursive (or recursively enumerable) if there is a recursive partial f : X + w such that

R =Domain (f) = { x : f(x) is defined}

These notions relativize to a futed object U of type m > 0 by substitution, i.e. a partial function f : X -+ w is rentrsive in U if

with some recursive partial g : T(m) X X -+ w . Similarly, a relation R is recursive in U if xR is recursive in U , and it is semirecursive in U if there is a partial function f : X -+ w , recursive in U , such that R =Domain (f).

It is simple to check that a total functionf : 5X + w is recursive (in U ) if and only if the graph off,

Graph (f) = {(x,n) : f(x) = n }

is recursive (in U ) as a relation. This is not always true for partial functions, which is why it is not convenient in this context to identify partial functions with their graphs.

Since an object F of type k > 0 is a function of type k , it makes sense to ask whether F is recursive in U. This relation is transitive. We agree that the objects of type 0, the integers, are recursive in every object. If F is recursive in G and G is recursive in F , we say that F and G are (recursively) equivalent


or that they have the same Kleene degree. It follows from one of Kleene's basic substitution results that if U and V are equivalent, then for every point- se tR,

R is recursive in U R is recursive in V .

Also, if U and V are equivalent and of the same type, then

R is semirecursive in U R is semirecursive in V .

Following Kleene [ 19631, we define for each object U and each k 2 1, the k-section of U ,

k x ( U ) = {R : R is a pointset of type 5 k

and R is recursive in U ) .

We also define the k-envelope of U as in the introduction,

ken ( U ) = {R : R is a pointset of type 5 k

and R is semirecursive in U ) .

The object mE ( m 2 2) representing quantification over T(m-2) is defined by

0 if (3p"-2)[orz- ' (p"-2) = 01,

1 otherwise. m E ( ( p - 1 ) =

We call an object U of type m normal i f m E is recursive in U . (All objects of type 0 or 1 are normal.) Almost all reasonable structure results that are known about recursion relative to an object U have been proved on the hypothesis that U is normal, and many are known to fail if U is not normal.

such that {e}(a) is d e f i e d an ordinal le, 0 1 , the stage of the induction at which we first recognized that (e , a ,n ) EX, for some n. The following theorem is the correct general version of Theorem 6 of Moschovakis I19671 and can be established by a variation of the proof given there:

The Stage Comparison Theorem for Kleene Recursion. Let x, y vary over

Kleene's inductive definition of recursion naturally assigns to each pair e, a

15 8 Y.N. MOSCHOVAKIS

the spaces X, y respectively, both of type 5 m ( m 2 2). There is a recursive partial function f ( a m , e , x , z , y ) such that

{e ) ( x ) isdefinedand le,xl< lz,y I * f ( m E , e , x , z , y ) = 0 ,

{z} ( y ) is defined and I z , y I < I e , x I * f (mE, e , x, z , y ) = 1.

(Here le,x I I Iz,y I is true if { z } ( y ) is not defined.)

This result is due to Candy [ 19621 for m = 2 and to Platek [ 19661 for m 2 3, independently of Moschovakis [ 19671. (The proof in Moschovakis [1967] is given only form = 3 and Grilliot [1967] extended it to all m.) It is the basic theorem about recursion relative to normal objects.

One of the easy consequences of the Stage Comparison Theorem is the existence of selection operators: for each recursive partial function

f : w X X + w

with type (X) 5 m (m2 2) there is a recursive partial function g ( a m , x ) such that

( 3 n ) [ f ( n , x ) i s defined] --g(mE,x) is defined,

( 3 n ) [ f ( n , x ) is defined] + f (g(mE,x) ,x) is defined.

From this follows immediately that a partial function f : 35 + w with type ( X) 5 m is recursive in a normal object U of type m if and only if the graph o f f is semirecursive in U . In these circumstances we often identify a partial function with its graph.

or a k-pointclass, if every pointset in I' is of type 5 k, e.g. if r is a k-section or a k-envelope.

defined by

A pointclass is any collection of pointsets. We call a pointclass I' of type k ,

A pointclass I' is closed under 3J if whenever R 5 T ( j ) X X is in l?, so is P

P(x ) - (3ai)R(ai,x) .

Closure under V1, & and v is defined similarly. A k-pointclass I' is closed under primitive recursive substitution if it con-


tains all primitive recursive pointsets of type <_ k and if whenever X, y are of type < k R C ?J is in r and f : 5X + y is primitive recursive, then f- ' [R] = {x : f(x) E R } is also in r.

proves that i f U is normal of type m 2 2 and 1 <_ k 5 m + 1, then ken ( U ) is closed under primitive recursive substitution, &, V, 3' and V j for every j l m - 2 .

is a relation G 5 o X 5X in r such that for every R E X,

Using the basic definitions and the Stage Comparison Theorem, one easily

A k-pointclass r is o-parametrized if for every space X of type < k there

R isin forsome e E o , R = G, = {x E X : G(e ,x ) } .

I t is immediate from the definitions thatfor each object Uand each k 2 1, ken ( U ) is o-parametrized.

Finally we look at a more complicated structural property that pointclasses may posses. A norm on a pointset R is any function

u : R + Ordinals

which assigns ordinals to the members o f R . (Sometimes it is convenient to insist that u is onto an initial segment of the ordinals, but we do not do t h s here.) With each norm u on R there are naturally associated two relations,

I fR is in a pointclass r, we call u a r-norm i f both 5; and <: are in r. Notice that both 5: and <: are relations of the same type as R .

A pointclass r is normed (or satisfies the Prewellordering Property) if every pointset in r admits a r-norm.

The Stage Comparison Theorem implies immediately that if U is normal of type m and k 5 m f 1, then ken(U) is normed. This is the Prewellordering Theorem for semirecursion,relative to a normal object and implies directly most of the known structural properties of envelopes and sections, e.g. the existence or hierarchies on sections.

60 Y.N. MOSCHOVAKIS

2. The Main Lemma

The key to our results is the closure of reasonably rich o-parametrized,

Let normed classes under monotone inductive definability .

: Power(%) +Power(%)

be an operator on subsets of % which is monotone, i.e.

Using the notation for induction of Moschovakis [ 19731, we define for each ordinal t the stage I$, by the recursion

I$, = Q , ( U q < E I z ) .

It follows that

v 5 t * I : G z &

I , = u& and that the set

is the least (under inclusion) fixed point of Q,.

A k-pointclass r is uniformly dosed under a monotone operator

Q, : Power(%)+Power(X),

with type (X) < k , if for every space y of type < k and every P C in r, the relation Q C % X y defined by

X y

is also in I?. This says that if for eachy E y we define

Py = (x’ : P(x’ , y ) }


by fixing a parametery in a I?-set P, then each set a@,,) is also obtained by f i i n g y in some r-set Q,

It may happen that

@ : P o w e r ( X X w ) + P o w e r ( X X o )

preserves single-valuedness, i.e. whenever R 2 X X w is (the graph of) a partial function, then so is @(R). Since the empty set 0 is a partial function, it follows in t h ~ s case that I , is a partial function too. For such operations @, we say that a k-pointclass r is uniformly closed for partial functions under a, if for each space y of type < k, whenever P E X X w X y is in r and for every y E 9, the set P,, = {(x’,n‘) : P(x’, n’, y)} is a partial function and Q C X X w X y is defined by

then Q is also in I‘.

Main Lemma. Let r be a k-pointclass which is w-parametrized, normed and closed under primitive recursive substitution, let X be a space of type < k and assume that r is uniformly closed under the monotone operator

: Power (X) +Power (X)

Then I , is in r.

primitive recursive substitution, i f type (X) < k and Similarly, i f r is o-parametrized, nonned and closed under &, V, V’ and

@ : Power (X X w ) -+ Power ( X X w)

preserves single-valuedness and if r is uniformly closed for partial functions under @, then the partial function I , is in r.

Proof. Choose G C w X w X X in r which parametrizes the subsets of

‘6 2 Y.N. MOSCHOVAKIS

o X X in r, let

u : G + Ordinals

be a r-norm on G and put

is in r by closure under prinitive recursive substitution, so by the uniform closure under a, the relation

and define a norm r on R by

~ ( x ) = u(e*,e* , x )

Immediately from the definitions, we have the equivalence

(*I R ( x ) - x E a( {x’ : x ’ <; x } ) .

We now show that (*) characterizesZ@, i.e. any relationR wluch admits a norm r satisfying(*) must bel,. This will complete the proof, since we have found a relation in r which satisfies (*).

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS 6 3

Step 1 . Zf R satisfies (*), then R ( x ) 3 x E I , .

Proof of Step 1 is by induction on ~ ( x ) . By (*), from R ( x ) we get x E @( {x’ : x’ <: x } ) and by the induction hypothesis, {x’ : x ’ <: x } s Z, , so by the monotonicity of @, x E @(Z,). But @(Z@) = I , , since Z, is a fured point of @.

Step 2. Zf R satisfies (*), then x E I , * R(x) .

Proof of Step 2. We show by induction on l, that

x E Z & * R ( x ) .

For this, it will be enough to obtain a contradiction from the hypotheses

From the definition of <:, if l R ( x ) , then

From the definition ofZ$,, we have

and from l R ( x ) and (*), we have

x 4 @ ( { x ‘ : x’ <; X I ) = @ ( R ) .

Finally, from the induction hypothesis we have

which together with the two preceeding displayed formulas directly contradicts the monotonicity of @.

To prove the second part of the theorem, we first show that under the hypotheses on r, for each y of type < k there is a partial function on UXY,

64

G C w X Y X w

Y.N. MOSCHOVAKIS

which is in r and which parametrizes the partial functions in r, on y . (We could have substituted this in the hypotheses instead of the extra closure properties of r.)

Choose

H C o X Y X w

in r‘ which parametrizes the r-subsets of y X w and take

u H + Ordinals

to be a r-norm on H . Put

G(m ,Y > n ) - ff(m ,Y, n )

& o(m,y,n) = infimurn{u(rn,y,n’) : H(rn,y,n’)}

& n = infimum{n’ : H(m,y,n ‘ )& a(m,y,n‘) = u(rn,y,n))

It is immediate that G parametrizes the partial functions on y which are in r. That G itself is in r, follows from the equivalence

Using this, choose G C w X w X ‘x X w in r, parametrizing the r-partial functions on w X ‘x, choose a r-norm u on G , put

and argue exactly as in the proof of the first part.

This proof is a small variation on the proof of Theorem (9A.2) in Moschovakis [ 19731, which in turn was patterned on the proof of the First Recursion Theorem of Moschovakis [ 19711.


As a first and quite typical application of this Lemma, consider the following two simple but useful results.

Proposition 1 . If a k-pointclass r is w-parametrized, normed and closed under vand primitive recursive substitution, then r is also closed under 3'.

Proof. Suppose R 5 w X % is in r, define @ by

Clearly r is uniformly closed under a, hence by the Main Lemma I , is in r From the definition of @ it is immediate that

Conversely, a very simple induction on E shows that for all n ,

so that, in particular,

and r is closed under 3'

Proposition 2. If a k-pointclass r is w-parametrized, normed and closed under primitive recursive substitution, v and &, then r is also clased under bounded universal number quantification, i.e. i f R 5 w X % is in r, so is P(m , x ) defined by

P ( m , x ) a ( V n < m ) R ( n , x )

66 Y.N. MOSCHOVAKIS

Proof. Given R 5 o X 'x in F, define CP by

( m , x ) E @ ( S ) - [m = 0 &R(O,x)]

v [m>O&R(m,x )&(m- 1,x)ESI

It is easy to check that

( m , x) E I , - (Vn 5 m) R(n, x)

3. Characterizations by closure under logical operations

We start with a very simple characterization of the L (recursively enumerable) relations.

(I) For k = 1, 2 , the class of C y k-pointsets is the smallest k-pointclass which is o-parametrized, normed and closed under primitive recursive substitution arid V.

Proof. It is well known that the class of Ey k-pointsets(k = 1,2) has all these properties, e.g. to prove that it is normed, write each E form

relation R in the

R ( x ) - (Ym)P(m,x)

with P primitive recursive and define u on R by

u(x) = least m such that P(m,x)

Conversely, if IT satisfies all these conditions, then I' is closed under 3' by Proposition 1, and since r contains all primitive recursive relations, it must contain all L? relations too.

The next characterization is equally simple but more interesting.

(11) Fork = 1, 2 , the class of Il k-pointsets is the smallest k-pointclass which is w-parametrized, normed and closed under primitive recursive substitution. vand v'.


Proof. It is again well known that the class of IT: k-pointsets ( k = 1,2) has all these properties. Here, the Prewellordering Property is a nontrivial classical result and it is proved using the fact that everyn; re1ation.k of the form

wi thP primitive recursive. (We are using the standard notation,

In view of the Main Lemma, the converse is just the classical theorem of Kleene that every KI relation is inductively definable, see Spector [ 19611. Briefly, to prove that every k-pointclass r satisfying the conditions of the theorem must contain all II; relations, assume that R satisfies (*) with a primitive recursive P and put

( u , x ) € @ ( S ) . - P ( u , x ) V ( V t ) ( U h ( t ) , X ) € S ,

where u v is the obvious primitive recursive concatenation function on sequence codes. Clearly I' is uniformly closed under.@, so by the Main Lemma I , is in r. Now verify

by showing directly from the definition of CP that

and then proving

by a reductio ad absurdum.

68. Y.N. MOSCHOVAKIS

From this we get immediately the next result.

(111) The class of X.$ pointsets of type 2 is the smallest 2-pointclass which is w-parametrized, nonned and closed under primitive recursive substitution, v ,Voand 3 ' .

Proof. That the class of E i pointsets has these properties is again well known - for the Prewellordering Property see Rogers [ 19671 or Moschovakis [ 19731, section 9B. Conversely, if I' has all these properties, it must contain all ni 2- pointsets by (II), and hence it must contain all E $ 2-pointsets by closure under 3 ' .

We do not know a structural characterization of this sort for the class of

(I)-(111) and the trivial closure properties of Z7,II and E determine completely a 2-pointclass r on the hypothesis that it is w-parametrized, normed, closed under primitive recursive substitution and v and also closed under the operations in a set

relations on w.

in fact r must be Ef,n or E i . To study closure under the full set of second order operations on w , including V ' , we now consider induction in analysis.

Let d: be the customary two-sorted language of analysis (or second order number theory), with variables n, k , m , ... over w,a,P, 7, ... over T ( ' ) = with prime formulas

m = n , a=P, m+r?=k , m . n = k , a ( m ) = n ,

the logical connectives &, V, 1, + and the quantifiers 3', V o , El1, V 1 . We take it here with the standard interpretation. For each space % of type 1, we obtain the extension L(S) by adding a relation variable S to d: and the prime formulas S(x), where x is the proper sequence of variables so it denotes a point of % . An operator

:Power(%) -+Power(%)


is positive, second order definable if there is a formula (x, S) of L(S) with only positive occurrences of S such that for all S C X, in the standard interpre t ation,

@(S) = { x : Q ( x , S ) } .

Such operators are monotone. We call R 2 % absolutely second order inductive if there is some positive, second order definable operator

and integers n l , ..., n k , such that

These are the absolutely semihyperprojective relations of Moschovakis [ 19691. In Moschovakis [ 19731 we restricted ourselves for simplicity to induction on one-sorted structures and we allowed substitution of arbitrary constants from the domain of that structure in the language. One verifies trivially that a 2-pointset R is inductive on the structure of analysis in the sense of Moschovakis [ 19731 if and only if there is an absolutely second order inductive relation P in the present sense and a real ao, such that

(IV) The class of absolutely second order inductive pointsets of type 5 2 is the smallest 2-pointclass which is w-parametrized, normed and closed

1 under primitive recursive substitution, V, 3 and V .

Proof. That the class of absolutely second order inductive pointsets of type 2 has these properties follows easily by the “lightface versions” of theorems (lD.l), (3A.3) and (5D.2) of Moschovakis [ 19731.

To prove the converse, we show first that if ?7 satisfies the hypotheses, then r is also closed under &, 3’, Vo. This is completely trivial for 3’ and Vo. For &, let

P ( x ) - G ( e o , x ) ,

C?(x>-G(e l ,~ ) ,

70 Y.N. MOSCI-IOVAKIS

where eo, el are fixed integers and G S w X X parametrizes the subsets of 5X in r a n d put

( (eo,x> if a(o)= 0 , f ( a , x ) =

(el , x ) if a(0) > 0 .

Then

so that the conjunction of P and Q is in r.

under every positive, second order definable operator a, so by the Main Lemma r must contain every I , and hence every absolutely second order definable pointset.

The results in (I)-(IV) can be summarized in the table below where each line is to be read as follows: the smallest 2-pointclass which is w-parametrized, normed and closed under primitive recursive substitution and the operations in the first column is the pointclass in the second column; it is also closed under the operations in the third column.

I t follows that a pointclass r with all these properties is uniformly closed

Operations Smallest Class Additional Closure (9 v x: v, &, 30,3l

(11) v, vo n: (111) v, v0, 31

v, &, 30, vo, v' v, &, 30 , v0, 3 l v, &, 30, vo, 31, v' (IV) v, 31, v1 abs. second

order inductive

Using this table and very trivial arguments we can easily compute the smallest w-parametrized, normed 2-pointclass which is closed under primitive recursive substitution, v and any set of operationsK C {&, 3 0 0 ,V ,3 l , V'}.

It is obvious from the proofs that the hypothesis of closure under primitive recursive substitution can be substantially relaxed in these characterizations. One can in fact put down characterizations that are almost algebraic in flavor, but which are naturally a bit more complicated.

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS 7 1,

4. Characterization of envelopes

We now turn to the characterization of envelopes promised in the introduction.

(V) Let U b e a normal object of type m 2 2and let m - 1 5 k I m + 1. m e n the k-envelope of U is the smallest k-pointclass r which is w-parametrized, normed, closed under primitive recursive substitution, v and & and which has the following two additional closure properties:

then the partial function (1) I f 2 <_ j < k and g ( d , x) is a partial function with graph in r,

f (a' ,x) = ai(Api-2g(ai, p i - 2 , x))

also has graph in r.

j h c t i o n (2 ) If g(~w"-~, x) is a partial function with graph in r, then the partial

f ( x ) = U(ham-2g(am-2, x))

also has graph in r.

Proof. That ken ( U ) has all these properties under the hypotheses on U, k is immediate from the discussion in section 1.

To prove the converse, assume that r satisfies all the conditions of the theorem. By Propositions 1 and 2 we also know that r is closed under go and bounded universal number quantification.

Let (ah, ..., CIA), (a')i be the primitive recursive tuple-coding functions defined in Kleene [1959], and relativizing Kleene's {e} [ao, a ' , ..., & ' I , put

where the number ni of variables in each type i < k is recoverable primitively recursively from the index e in the coding used by Kleene. The inductive definition of the relation {e} (U , a) = n determines a monotone operator

@ : Power(w X 5X X w) *Power(w X X X u ) ,

72 Y.N. MOSCHOVAKIS

The definition of (e ,x ,n) E @(S) is a disjunction of ten clauses, S1 -S9 and S8(U), the last corresponding to the clause for {e} (U , a) = n which applies (I,

S8(U) f(b) = U(ham-2g(am-2, 6) )

Certainly @ preserves single-valuedness. Now the closure properties of r imply that r is uniformly closed under @: we use 3" to express clause S4 for composition we use both 3' and bounded universal number quantification to express clause S5 for primitive recursion, we use (1) to express clause S8 and (2) to express clause S8(U) and of course we use closure under primitive recursive substitution all over. The explicit computation is a bit messy but direct and we omit it.

{e)'[a", ..., = n is in r, hence by closure under primitive recursive substitution, every pointset of type <k which is semirecursive in U is in r.

It is perhaps worth putting down explicitly the result form = 3, k = 2. (The statement is very similar for the 1-envelope of a type-2 object.)

The 2-envelope of a normal type-3 object U is the smallest class r of pointsets of type 5 2 , which is w-parametrized, normed, closed under primitive recursive substitution, v and &and such that whenever g(a,x) is a partial function in r, so is f ( x ) , defined by

By the second part of the Main Lemma then, I , is in r, hence the relation

The Main Lemma does not yield a characterization of this sort for the

The special case U = 3E makes precise the difference between the 2-envelope I-envelope of a normal type-3 object.

of 3E, the class of semihyperanalytical relations of Kleene and the class of absolutely second order inductive relations which we characterized in (IV). I t is trivial that 2en (3E) is closed under V1 . Hence the only difference between

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS ,73

these two classes is that the collection of absolutely second order inductive pointsets is closed under 3 ' , while 2en(3E) is only closed under a very restricted kind of existential quantification on T('); ifg(a,x) is a partial function (with graph) in 2en (3E), then

R(x) - (Va) [g(a, x) is defined] & ( 3 4 [g(a, x ) = 01

is also in 2en(3E). This is a kind of "restricted" 3 ' . It is known from Corollary 10.2 of Moschovakis [ 19671 that 2en (3E) is not closed under 3 ' .

The hypothesis that U is normal is essential in (V), as otherwise ,en(U) need not be normed or closed under V. However, a similar characterization of the class of partial functions in ,en ( U ) for an arbitrary U of type m 2 2 (m - 1 5 k 5 m + 1) can be read off the results in Moschovakis [ 197 1 1 . (To obtain such a characterization was one of the motivations for introducing axiomatic computation theories.) Briefly, the key property of these partial functions is that for each X of type < k there is an w-parametrization G : w X 5X -+ w of those with domain in X, which is in ken ( U ) and which admits a norm u : G -+ Ordinals having the natural properties of a measure of complexity (or length) of computations, as expressed in the axioms for a computation theory. This length function need not satisfy any definability conditions.

5. The type of a 1-envelope

We show here that for sufficiently rich 2-pointclasses F, the class of 1-

As usual, we let 1 F be the dual class of F, pointsets in r cannot be the envelope of a type-2 normal object.

ll?= ( % - R : R R C , R i s i n F }

and we put

We say that A isclosedunder 3' n V', if wheneverP(a,x), Q ( P , x ) are in A and for all x ,

(3a)P(a,x) (VP) Q(P,x)

74 Y .N. MOSCHOVAKIS

then the relation

is also in A . For each real (Y put

I, = {(n,rn) : a((n,m)) = 0)

and let WO be the set of reds which code wellorderings of w ,

a E WO 5, is a wellordering with field w

If a E WO, we let It 1, be the rank o f t in the wellordering 5 a.

theorem for the 1-section of a type-2 object proved in section 2 of Moschovakis [ 19671 (see also Corollary 7.1 of [ 19671 and section 8).

function f , : w + w by the following simultaneous induction:

f( l ,q,(t) = q . (Introduction of constants.)

f ( 2 , w ) ( t ) = U(f,). (Introduction of U.)

totally de.fined when (Y = f w and i f for all t,

The tool for the proof of the theorem is the following representation

Let U be an object of type 2 and define a set N E o and for each z E N a

(1) For each integer q , (1 ,q ) E Nand for all t ,

(2) If w E N , then ( 2 , w ) E N and for all t ,

( 3 ) If w E N and if the (absolutely) recursive partial function {e}(a, t ) is

then ( 3 , w, e ) € N a n d for all t , ft3,w,e)(t) = fecw,t)( t ) . (Diagonalization.) Then, a unary function is recursive in U if and only if it is f z for some z E N . Moreover, if U is normal, then N is semirecursive in U and the partial func-

tion

u(z) = n z E N & U( f,) = n

is recursive in U.

usual way, so that We understand the definition (1) - ( 3 ) as a transfinite recursion in the


where for each ordinal .$, N ( f ) consists of the integers put in N in 5 f steps. The induction clearly closes at some countable ordinal.

Theorem. Let r be a 2-pointclass which is closed under primitive recursive substitution, V, &, 3O, Vo, and assume further that the set WO of codes of wellorderings is in A = r fl 1 r and that A is closed under 3 n V 1 . Then the class of 1-pointsets in r is not the l-envelope of any normal type-2 object.

Proof. Assume towards a contradiction that U is normal of type 2 and that len ( U ) consists precisely of the relations on w in r, let N , { f Z l z E N be the canonical hierarchy of the 1-section of U as described above. Put

P(a,B, y) - a E WO & ( 0 , ~ ) code the definition of N , { f z } z E N along I,.

and then set

76 Y.N. MOSCHOVAKIS

Q(ac,P,r,x) -(YE W O & ( p , y) locally seem to code the definition o f N ,

Cf,),EN lzt 1x1, .

It is trivial to check by transfinite induction along<, the two implications establishing

P(a,P,r) Q W X ) Q ( a A r , x ) .

We now substitute for Q in this equivalence two approximations in r and 1 r. Define Ql(ac,P,y,x) by replacing

in the definition of Q by

where u ( w ) is the partial function, recursive in U , which is associated with the canonical hierarchy on sc (U). Clearly Q, is in r and again it is trivial to check by transfinite induction along 2, the two implications establishing


so t h a t P is in r. Now define &(a,P,y,x) by replacing

in the definition of Q by

Clearly Q2 is in 1 r and the proof of

is exactly as before, completing the proof that P is in A . To complete the proof of the theorem, put

R(a,P,y,x) e P ( a , P , y ) &the recursion definingN, { f z ) z E N closes by 1x1,

Clearly R is in A and

so by the closure of A under 3' n V ' , N is in A . A similar trivial argument shows that the function

f z ( z ) + 1 if 2 E N ,

0 otherwise

has graph in A , which is a contradiction since g is different from all thefz's, z E N . and hence not recursive in U.

78 Y.N. MOSCHOVAKIS

Corollary 1. I f U is normal of type m 2 3 , then len(U) is not the lenvelope of any normal type-2 object.

Corollary 2. The class of C A (n 2 1) or IIk ( m 2 2) relations on w is not the I-envelope of any normal type-2 object.

Both corollaries are immediate, except for the case of in the second, where the result is well known, e.g. it follows from the fact that C is not normed, while the 1-envelope of every normal type-2 object is normed. The second Corollary was independently formulated and proved by A.S. Kechris.

Several interesting problems suggest themselves. We state two of them as conjectures, for sharpness, although we do not have now any hard evidence that these statements rather than their negations hold.

Conjecture 1. If U is normal of type m 2 3 , then the I-envelope of U is not the 1-envelope of any normal object of type less than m.

Conjecture 2. The class of X object U of finite type such that 2E is recursive in U.

relations on w is not the I-envelope of any

Of course it is quite probable that there are perverse, forcing-type counter- examples to these conjectures. We would consider positive answers (to these or similarly motivated questions) very interesting, as they would tend to show that there is a real notion of “type” in Kleene semirecursion, even if the types are lost when one looks only at the recursive cbjects.

6i bl iogra phy

Gandy, R.O. [ 19621

Grilliot, T.J. [ 19671

General recursive functionals of finite type and hierarchies of functions, Proceedings of Logic Colloquium CLermont Ferrand, 1962,5--24.

Recursive functions of finite higher types, Ph.D. Tbesis, Duke University, 1967.

KIeene, S.C. [I9591 Recursive functionals and quantifiers of finite types I, Trans. Arner. Math.

SOC. 91 (1959) 1-52.


[ 19631 Recursive functionals and quantifiers of finite types 11, Trans. Amer. Math. SOC. 108 (1963) 106-142.

Moschovakis, Y.N. [ 19671 [ 19691

[ 19711

Hyperanalytic predicates, Trans. Amer. Math. SOC. 129 (1967) 249-282. Abstract fiist order computability 1, and 11, Trans. Amer. Math. SOC. 138 (1969) 427-464.and 465-504. Axioms for computation theories - first draft, in: R.O. Candy, C.E.M. Yates (eds) Logic Colloquium '69 (North-Holland, Amsterdam,

Elementary Induction on Abstract Structures (North-Holland, Amsterdam, 1973).

1971) 199-255. [ 19731

Platek, R. [ 19661 Foundations of recursion theory, Ph.D. Thesis, Stanford University, 1966.

Sacks, G.E. [ 19731

Spector, C. [ 1961 1

The I-section of a type-n object, this volume.

Inductively defined sets of natural numbers, Infinitistic Methods (Pergamon, New York, 1961) 97--102.

Added in proof. I thank D. Normann, A.S. Kechris and L. Harrington for noticing that Conjecture 2 was formulated in the first draft of this paper so that it was trivially false.

ing (independently) that the 1-envelope of every normal object U of type 2 3 is also the 1-envelope of some normal type-3 object. (July 9, 1973.)

L. Harrington and J. Moldestad recently disproved Conjecture 1 by show-

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

THE 1-SECTION OF A TYPE n OBJECT

Gerald E. SACKS Harvard University, Massachusetts Institute of Technology

1. Introduction

This paper is a puffing up of the proof of the plus-one theorem for the case k = 1. Let nE be the representing function of the equality predicate for all objects X , Y of type less than n : n E ( X , Y) = 0 if X = Y , and = 1 if X # Y .

Plus-One Theorem. Let U be of type n. Suppose nE is recursive in U and k < n. Then there exists a V of type k+l such that the objects of type k recursive in Vare the same as those recursive in U. Furthermore kilE is recursive in V.

Recursion in objects of finite type was discovered by Kleene [ 2 ] . An equivalent formulation, needed for the forcing argument of section 4, is given in section 2. The proof of the plus-one theorem for the case k > 1 will be given in [ 3 ] . It is largely a consequence of a stability lemma described in section 5 .

kscU is the set of all objects of type k recursive in U , and is called the k- section of U. The plus-one theorem states that all k-sections generated by finite type objects (in which the appropriate equality predicates are recursive) are generated by type k + l objects. The first result on k-sections was Kleene’s [ 2 ] : I sc2E is the set of hyperarithmetic reals. He asked if the A: reals consti- tuted the 1-section of some type 2 object. They do by virtue of the complete characterization of 1-sections of type 2 objects (in which 2E is recursive) developed in section 4.

The preparation of this paper was partially supported by NSF contract GP-29079. Its principal results were announced in [ 11. The author is grateful to T. Grilliot for steer- ing him towards k-section problems.

81

82 G.E. SACKS

Section 2 redoes some of the elements of recursion in objects of finite type. Section 3 introduces the notion of abstract I-section and shows that many familiar collections of reals, among then the I-section of nE ( n 2 2 ) and the set of lightface A; reals (min (i , j) 2 I), are countable abstract l-sections. Section 4 proves every countable abstract l-section is the 1-section of some type 2 object in which 2E is recursive by means of a forcing argument of the sort associated with generic classes rather than sets. Section 5 describes some further results based on the technique of section 4, and speculates on the nature of abstract k-sections when k > 1.

2. Recursion in higher types

The objects of type 0 are the nonnegative integers. An object of type n > 0 is a total function whose arguments range over all objects of type < n and whose values are objects of type < n. Any object of type i can be inflated to an equivalent object of type j > i by adding dummy arguments. Any object of type n > 0 is equivalent to one of type n whose values are either 0 or 1; e.g. the function F(x) is equivalent to the representing function of the predicate F(x) = y . Any finite sequence of objects is equivalent to a single object; e.g. the pair F,(x), F l ( x ) is equivalent toH(n,x), whereH(n,x) = Fn(x) for n < 2 , and = 0 otherwise. The previous three sentences should make the ambiguities in what follows tolerable.

Fix n 2 0, and let F , G, ... be objects of type n+2 called functiorzals. The definition of G 5 F , n+2E (read G is recursive in F, n+2E) is given by means of a hierarchy inspired by those of Shoenfield [4] and Grilliot [ 5 ] . D. Mac- Queen (cf. [6]) has shown that G < F , n+2E if and only if G is recursive in F, n+2E i n the sense of Kleene [ 2 ] . MacQueen's argument is not unlike that of Grilliot [5]. The hierarchical approach via total objects is equivalent to Kleene's schematic approach via partial objects because of the presence of n+2E.

Furictions f, g, h, ... and sets R , S , T , ... are objects of type n+ 1 . Individuals a , b , c , ... are objects of type at most n. Subirzdividuals r. s, t , ... are objects of type less than I Z when n > 0 and are nonnegative integers when n = 0.

Fix 1; and g. The hierarchy {So} of sets, defined by induction on u, is designed to define "G is recursive in F, n+2E, g via e" by induction on n.

Stage 0. So contains ( 1, a ) for eveiy a, and ( 1, a, m ) whenever ga = m. ( 1, a ) is a n index for So for evely a.

THE 1-SECTION OF A TYPE n OBJECT 83

Stage~+l.(2~,a)isanindex forS,+l if(i) (2e,a)$S,,(ii) (m,a>isan index for S , for some integer m, and (iii) there is a functionf recursive in S,, n + l ~ , a via e.

Clause (iii) makes sense by induction on n. If n = 0 the clause states f is Turing reducible toS via e. Thefof clause (iii) is denoted by Xb{e}ScJ3"t'EJ"(b).

S,+l is S, augmented by: (1) all indices for So+l ; (2) all quadruples (3e,a,b,rn)if(2e,a)isanindex forS,+, and

{e)Su~"+"~"(b) = m ;

and (3) all triples (5e,a,rn) if (2e,a) is an index for S , and

F(Xb{e}S~~"+'E~"(b)) = m .

I(e,a)l= u means (e ,a ) is an index for S,.

for some Ss+l < X and Xb {e)Su,"+lE'u(b) is the characteristic function of a set T of indices such that

StaKe X, where X is a limit. (7e,a) is an index for S, if (2e .a) is an index

S, is U{Ss16<X} augmented by all indices for S,. f is said to be recursive in F, n+2E,g,a via e, written

f = { e l F,n+2E,g,a

if (2e, a> is an index for some recursive in F, ni2E, g via e, written

andf i~Xb{e}~~~"+ '~ ' ' ( b ) . G is said to be

if G(h) is {e}F>"'2Erg,h(0) for all h.

the set of all objects of type k recursive in F, n+2E. If n+2E is recursive in F in the sense of Kleene [ 2 ] then k ~ c ( F , n+2E) = kScF. R is recursively enumerable in F, n+2Evia e if R = { 0 1 ( 2 ~ , a ) is an index}.

This notion of recursive enumerability coincides with Kleene's [ 2 ] . Thus R is

R is recursive in F, n + 2 ~ if its characteristic function is.ksc(F, is.

84 G.E. SACKS

recursive in F, n+2E if and only if both R and its complement are recursively enumerable in F,n+2E.

The principal fact needed in the next section is the existence of a selection operator discovered by R. Candy [ 7 ] , who had to wrestle with Kleene's definition. His result follows more readily from the hierarchial definition of this section.

Candy's Selection Operator. There exists a recursive function hele* such that if T is a nonempty subset of w recursively enumerable in F, rz+2E via e, then {e*}F>"'2E(0) is defined and belongs to T.

3. Abstract 1 -sections

Let A be a nonempty transitive set. A is said to be an abstract 1-section if it is closed under the operations of pairing and union, and satisfies axiom ( I ) and schemas (2)-(3), where a E A and a ( y ) and a(x,y) are A 0 formulas of ZF (i.e. formulas whose quantifiers are restricted, cf. Levy [lo]) with parameters in A .

( 1 ) Local countability: (x) (x is countable). (2) A, separation:

( 3 ) A. dependent choice: ( E X ) ( Y ) [ Y E X -.Yea 9 ( Y ) l .

(~)(E.Y) 2 (x,Y) -, (E h)(n) 9 (hn, h(n + 111, where h: o + A .

I f A is an abstract I-section, then A is an admissible set as defined by Platek [8], and every member o fA is hereditarily countable.

Each hereditarily countable set x can be encoded by a real number Y . Let m be the function that takes a code Y to the set mY encoded by Y . The relation

Y is a code & nzY = x

Not surprising, since his argument is based on a shadowy hierarchy of computations arising from the Kleene schemas of recursion. The existence of Candy's selection operator was proved by him when n = 0 , by Moschovakis 1161 when n = 1, and by Platek [8) when I 1 > 1 .


is defined by induction on the rank of x : Y is a code for {m( Y,) I n <a}, where Y, = {a I 2 , - 3a E Y } . The set of all codes is Il:, since a code is little more than a wellfounded relation on w .

Proposition 3.1. Let d be a code and a(.) a A 0 formula of ZF. The predicate P(e), defined by

is recursive in d , =E.

Proof. If sets correspond to codes, then restricted set quantifiers correspond to number quantifiers.

For each K C 2- let W K be the set of all sets with codes in K . It is convenient to ignore the distinction between K and W K (e.g. to say K is an abstract 1-section instead of%K is an abstract 1-section) when every member o f K is coded by some member o fK . The next proposition extends a result of Platek [ 8 ] to the effect that the least ordinal not i n m ( , scnE) is XI admissible.

Proposition 3.2. Suppose n > 1 and U is a type n object in which nE is recursive. Then the 1-section of U is a countable abstract 1-section.

Proof. It is immediate that W ( sc U) is closed under the operations of pairing and union. To check A. separation let 9 ( y ) be a A. formula and d E scU be a code. It suffices to find a code c E sc U such that

( y ) [ y E m c - y Emd & a ( y ) ] holdsin%(lscti).ThepredicateP(e), defined by

is recursive in U by Proposition 3.1. The desired c is such that {c,ln<w> is an enumeration of all the e's that satisfy P(e).

To check A. dependent choice let a ( x , y ) be a A. formula such that

86 G.E. SACKS

holds in W( sc v). Suppose { P } ~ is a code. Let Q, be the set of all n such that

{n}’ is a code & 9 (m{p}O, rn{n}u)

Qp is recursively enumerable in U (uniformly in p ) . Gandy’s selection operator yields a partial function t recursive in is a code. Define g recursively in U by:

U such that t p E Q, whenever { P } ~

go = eo ( { e o } u is a code for 0) ,

g(n+1) = tdgrz) .

For each ordinal a let L , be the set of sets constructible in the sense of Giidel [ 1 I ] via ordinals less than a. a is said to be El admissible if L , satisfies the E l replacement axiom schema of ZF.

Proposition 3.3. If a is a El admissible ordinal, then L , r\ 2w is an abstract 1 -section.

Proof. Godel [ 1 11 shows L n 2w = Lwl n 2 w , where w1 is the least ordinal not countable in L . His argument restricted to L , shows

L , ~ ~ w = L n 2 w , W P

where LJ? is the least ordinal not countable in L,. It follows L local countability. Godel’s wellordering of L restricted to L

W P

satisfies YP IS El over LwP.

Let A; be the set of all lightface A; reals.

Proposition 3.4. I f min ( i , j ) 2 1 , then Af is a countable abstract I-section.

Proof. If i = j = 1, then the proposition follows from Spector’s boundedness theorem [9] for E l subsets of Kleene’s 0 and Kreisel’s selection operator [ 121 for KI, predicates of numbers.

1 I

1.e. the graph o f t is recursively enumerable in U.

THE 1-,SECTION OF A TYPE0 OBJECT a i

Assume max ( i , j ) > 1. Let HC be the set of hereditarily countable sets. The Kondo-Addison uniformization of Il: predicates of reals implies HC satisfies A. dependent choice. Consequently HC is an abstract I-section. I t suffices to show%(Aj) is a El substructure of HC. Let a(.) be a A. formula with parameters in%(Af) such that

There exists an arithmetic predicate A( Y) such that

[Yisacode & A(Y)]+-+HCk9(mY).

The set parameters occurring in a(.) correspond to Af codes occurring in A(Y) . The Kondo-Addison uniformization supplies a code Z such that 2 satisfies A( Y) and is A: in the A; codes occurring in A( Y). Since max ( i , j ) > 1, Z €A:. HenceW(Aj) k (Ex)g(x).

4. Generic type 2 objects

Let K be a countable abstract. 1-section. Suppose F maps ow into w and is 0 off K . If F is generic in the sense of the following forcing relation, then the 1-section of (F, 2E) is K .

Let p be a partial function from ww into w. p generates a hierarchy {T,} of reals as defined below. If p is total, then the T,’s are equivalent to the S,’s of section 2 when n = 0 and F = p . If p is not total, then there may be a CJ

such that T,,+l has an index but is not total. Stage 0. To = { l}. 1 is an index for To. To is total. Stage o t l . 2 e is an index for if T, has an index, T, is total, 2e 4 T,,

and {e}Tu(m) is defined for all m. ({e}To is the unique partial function from o into w recursive in T, via Godel number e.) TO+l is total if it has an index and p(Xm{e}To(m)) is defined whenever 2e

is an index for T,,+l. Tu+l is T, augmented by: all indices for Tu+l ; all triples (3e ,m, n ) such

that {e}To(m) = n and 2 e is an index for Tu+l; and all pairs ( 5 e , n ) such that p ( X m { e } T ~ ( r n ) ) = n and 2e is an index for Tu+l.

Im(= CJ means m is an index for T, .

88 G.E. SACKS

Srage h (limits). 7 e is an index for TA if 2e is an index for Ts+l for some 6 < X and hm{e}Ts(m) is the characteristic function of a Set R of indices such that

h = sup {Iml(m E R ) .

T, is total if it has an index. Th is u {T , 16 < A} augmented by all indices for

p is said to gerierafe T , if To has an index and is total. Fact H is easily TA.

proved by induction on u.

Fact H. If u < y and p generates T, and T y , then To has lowei Turing degree than Ty .

Suppose p is total and S is a real. The arguments of Shoenfield [4] show S is Turing reducible to some T, generated by p if and only if S is recursive in p , 2E in the sense of Kleene [ 2 ] .

o f p . p is afircitig cotidition if it meets two requirements. I f Tu+l has an index but is not total, then Tu+l is said to be the maximum

( I ) p E77ZK and has a maximum. (2) X is in the domain of p if and only if X is Turing reducible to T, for

Requirement (2) is not as limiting as it may appear, because Fact H implies some 6 < u, where To+, is the maximum o f p .

that the generation of To by p utilizes the vaiue of p ( X ) only if X is Turing reducible to T6 for some 6 < u .

Froin this point on p , q , r, ... denote forcing conditions. p is extended by q (in symbols p 3 q ) if the graph of p is contained in the graph of q .

The language L(K) will be used t o define the desired generic F's. The individual constants of L(K) are: m for each m € w ; f for each f E ww n W K ; _. u and To for each ordinal u E 9 " K ; a n d 3 for each S E 2w n W K . The vari- abies o f k ( K ) are: x,y, ... (numbers); p , v, ... (ordinals); and Tp, T,, ._. (sets). The atomic formulas of L(K) are of the form: Ix I = p, Ifx I = p, p < v, and - S < T,. The sentences of L(K) are built up from the atomic formulas by substitution of appropriate individual constants for variables and by application of propositional connectives (& and -) and existential number and ordinal quantifiers.

is the least ordinal greater than every ordinal occurring in 9 . Such an 9 is r ruein {T6 i6<o] i f i t is true w h e n 6 < y i s i n t e r p r e t e d a s 6 < ~ , - l m I = 6 - - as

9 is a ranked sentence of rank u if 9 contains n o ordinal quantifiers and u


m is an index for T,, and - S <_ T, as S is Turing reducible to T, . The forcing relation It is defined inductively.

@,I6 < 0). (i) p It- 9 if 9 is of rank u, p generates T, for all 6 < 0, and 9 is true in

(ii) p It- (Ex) a(x) if p IF 9 (m) for some m. (iii) p IF ( ~ p ) 9 (p> i f p 11- 9 ( 3 for some u. (iv)plt- 9 & 9 i f p I t 9 a n d p I t - 9 . (v) p It-- 9

A sentence 9 is said to be El if it is in prenex normal form and contains - [4 IF 91 and 9 is not ranked.

no universal quantifiers.

Proposition 4.1. The relation p It-9, restricted to El g’s, is E, over%K.

Proof. Suppose w € W K and is a partial function from ww into w . The set of To’s generated by w is El over-K uniformly in w. (This last is a consequence of the Z, admissibility of%K and the autonomous fashion in which indices are assigned to T, when h is a limit ordinal.) Thus if w has a maximum, then that maximum belongs t o W K . It follows that the set of forcing conditions is El over3flK.

Let F map ww into w and be 0 off K. F satisfiesp (in symbds F Ep) if the graph of p is contained in the graph of F. F is generic if for each sentence 9 of the language L(K) there is a p such that F E p and either p IF 9 or p 11 - 9 . Generic F’s exist because there are only countably many sentences to be forced. Standard arguments [ 151 show: if F is generic, then each true statement about F (expressible in the language %(K)) is forced by some p satisfied by F.

Lemma 4.2. If F is generic, then K C 1sc(F, 2E).

Proof. Suppose S: w + w belongs to K. Fix p ; since F is generic it suffices to find a 4 C p such that

for some u. Since p has a maxirmm there is an e and a y such that p generates Tr, 2e is an index for Tvl and

90 G.E. SACKS

is undefined. Let hn le, be a recursive function such that

{en = n t {e}*

for all X C w and n E w. Clearly { e n } X is total if and only if { e } X is. It follows 2en is an index for TV1 because 2e is. In addition

p(Xm c e, IT%))

is undefined, because the domain of p is an initial segment of Turing degrees. Choose 4 C p so that the domain of 4 consists of all functions Turing re-

ducible to T7, and so that

for all n E o. Thus q generates Ty+l but not Te2, since q(TP1) is undefined.. And S is Turing reducible to TPl since

for all n and r.

Lemma 4.3. If F is generic, then sc(F, 2E) C K .

Proof. Suppose S E sc(F, 2E) ~ K for the sake of a reductio ad absurdum. Then S is Turing reducible to some To generated by F but not in 'XK; D @ W K since F is generic. Let a be the least ordinal not i n W K . Then F generates some T, with index 7 e . Thus 2e is an index for some Ts+l generated by F in%K, and {e}T6 is the characteristic function of a set R such that

a = s u p { I n J ( n E R } .

Let f € % K enumerate R . Since F is generic there is a p satisfied by F such that:

THE I-SECTION OF A TYPE n OBJECT 91

(a) P It- (X)(EP) t IfX I = PI ;

(c p Il- 8 L T, ;

(a*> (m) (4 )p>q(Er )q . , (Ea) [ r I~ lfml -- = E l >

(b) p IF (P)(Ex)(Ev)[P < v &k Ifx - I = ~ 1 ;

(4 p IF - (Ell) [ IZ“ I = P I . (a) is equivalent to

and (b) is equivalent to

I t follows from (a*), (b*), Proposition 4.1 and the validity of the Z1 dependent choice schema in%K that there exist functions X m I p , and hmlu, in W K such that

(b*) (0)(4)p>q(EY>q. , (Em)(Ey) ,<, [Y It- I f m I = 11.

and (m) (En)[6 < u, < un] . Let h = sup {uml m E w } and w = = U { p , I m E w } . Clearly h, w E W K . w generates T,+l with index 2“ thanks to (c) and the fact that 6 < A. Consequently w generates some Th with index 7e. None of the pm’s generate Tk by (d). I t follows from requirement ( 2 ) of the definition of forcing condition and Fact H that w(Tk) is undefined. Thus w has a maximum (namely Tk+l) and so is a forcing condition. But then w It 17el = - h, an impossibility according to (d).

Theorem 4.4. K is a countable abstract I-section i f and only i f K is the 1- section of some type 2 object in which =E is recursive.

Proof. If G is a type 2 object in which 2E is recursive, then lscG is a countable abstract 1-section by Lemma 3.2.

Suppose K is a countable abstract 1-section. Let F be generic in the sense of Lemma 4.2, and let G be the recursive join of F and *E. Then K = lscG by Lernmas4.2 and4.3.

The following three corollaries of Theorem 4.4 are consequences of Pro- positions 3.2-3.4.

Corollary 4.5. Suppose n > 2 and U is a type n object in which nE is recursive. Then there exists a type 2 object V such that

92 G.E. SACKS

and 2E is recursive in V.

Corollary 4.6. Suppose a is a countable El admissible ordinal. Then there exists a type 2 object V such that

and 2E is recursive in V.

Corollary 4.7. If min (i, j ) 2 1, then the set of all lightface A; reals i s the 1- section of some type 2 object in which =E is recursive.

5. Further results

The method of section 4 is applicable to the study of Gandy's superjump [ 7 ] . Theorems 5.1 and 5.2 are typical results of [ 141 and were inspired by some questions raised by P. Hinman at the 1969 Manchester Logic Colloquium. Let F and G be objects of type 2, G' the superjump of G, and El the superjump of 2E. scG is said to be closed under hyperjump if

l s c ( E 1 , x ) c 1scG

for every X E scG.

Theorem 5.1. Suppose sc G is closed under hyperjump. Then there exists an F such that

] scC= , sc(F ' ) .

Theorem 5.2. (Assume 2w = w1 .) There exists an H such that ( C ) ( E F ) [ H I G + F ' = G ] .

The method of section 4 does not appear to suffice for the proof of the plus-one theorem when k > 1. A stability result is needed to overcome prob-


lems caused by gaps in the hierarchy of section 2 , gaps that fall between objects recursive in F,n’2E when n > 0. Call R subrecursive in F, n+2E if R is recursive in some So (as defined in section 2) with an index of the form (2e , r ) , where r is a subindividual. The stability result in question says: each nonempty recursively enumerable (in F, n+2E) collection of subrecursive (in F,n+2E) sets must have a recursive (in F,n+2E) set among its members.

At this writing it is not known if there exists a decent notion of abstract k-section when k > 1. Decency requires that Theorem 4.4 remain true when; “I-section’’ is replaced by “k-section” and “2” by “ k t l ” .

References

[ l ] G.E. Sacks, Recursion in objects of finite type, Proceedings of the International Congress of Mathematicians 1 (1970) 251-254.

[2] S.C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Amer. Math. SOC. 91 (1959) 1-52; 108 (1963) 106-142.

[3] G.E. Sacks, The k-section of a type n object, to appear. [4 ] J . Shoenfield, A hierarchy based on a type 2 object, Trans. Amer. Math. SOC. 134

[ 5 ] T. Grilliot, Hierarchies based on objects of finite type, Jour. Symb. Log. 34 (1969)

[6 ] G.E. Sacks, Higher Recursion Theory, Springer Verlag, to appear. [7 ] R. Gandy, General recursive functionals of finite type and hierarchies of functions,

[8] R . Platek, Foundations of Recursion Theory, Ph.D. Thesis, Stanford (1966). [9 ] C. Spector, Recursive well-orderings, Jour. Symb. Log. 20 (1955) 151-163.

(1968) 103-108.

177-182.

University of Clermont-Ferrand (1962).

[ 101 A. Levy, A hierarchy of formulas in set theory, Memoirs of the American Mathe-

[ 111 K. Godel, The Consistency of the Axiom of Choice and of the Generalized Con-

[ 121 G. Kreisel, Set theoretic problems suggested by the notion of potential totality, in:

matical Society, Number 57 (1965).

tinuum Hypothesis (Princeton University Press, Princeton, 1966).

Infinitistic Methods (Pergamon Press, Oxford, and PWN, Warsaw, 1961) pp. 103- 140.

[13] J . Shoenfield, The problem of predicativity, Essays on the Foundations of Mathe- matics (Magnes Press, Jerusalem, 1961 and North-Holland, Amsterdam, 1962).

[ 141 G.E. Sacks, Inverting the supejump, to appear. [ 151 S. Feferman, Some applications of the notion of forcing and generic sets, Fund.

[ 161 Y. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. SOC. 129 (1967) Math. 56 (1965) 325-345.

249-282.

PART I1

SETS AND ORDINALS

J.F.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

ADMISSIBLE SETS OVER MODELS OF SET THEORY

K. Jon BARWISE University of Wisconsin, Madison and Stanford University

9 1. Introduction

The addition of urelements gives a new dimension to the theory of admissible sets, a dimension which has applications in several parts of logic. To see why this addition is an obvious step to take we begin by reviewing the development of Zermelo-Fraenkel set theory, ZF, as it is usually presented (see for example Shoenfield [ 19671, 39.1).

The fundamental tenet of set theory is that given a collection of mathematical objects, subcollections are themselves perfectly reasonable mathematical objects, as are collections of these new objects, and so on. Thus we begin with a collectionM of objects called urelements which we think of as being given outright. We construct sets on the collectionM in stages. At each stage a, we have available all urelements and all sets constructed at previous stages. A collection is a set if it is formed at some stage in this construction; the collection of all sets built onM is denoted by V,.

at each stage a, and if we assume that there are enough stages, then the urelements become redundant in that all ordinary mathematical objects occur, up to isomorphism, in V , i.e. in V, for the empty collection M . It is for this reason that the axioms of ZF explicitly rule out the existence of urelements; the combination of the power set and replacement axioms are so strong as to make urelements unnecessary.

So formulated, ZF provides us with an extremely elegant way to organize existing mathematics. It does this at a cost, though. The principle of parsimony, historically of great importance in mathematics, is violated at almost every

Now it turns out that ifwe allow strong enough principles of construction

' Research for and preparation of this paper were supported by NSF GP 27633 and NSF GP 34091X, respectively.

97

98 K.J. BARWISE

turn. And one of the main advantages of the axiomatic method is lost since ZF has so few recognizable models in which to interpret its theorems. For these reasons, and others familiar to anyone versed in generalized recursion theory, it eventually becomes profitable to look at set theories weaker than ZF, weaker in the principles of set existence which they allow us to use. The theory we have in mind here is the Kripke-Platek theory KP for admissible sets.

We now come to the main point. As we weaken the principles allowed in the construction of sets (in the move from ZF to KP) we destroy the earlier justification for throwing out urelements. In this paper we put them back in by “weakening” KP to a theory KPU which does not rule out the existence of urelements. KP will be equivalent to the theory KPU + “there are no urelements”. The result is worth the trouble.

admissible sets with urelements. A large portion of our talk at Oslo was devoted to a review of this folk material. When it came to writing it soon became obvious that neither time nor space would permit a complete treatment in this paper. We are currently at work on such a treatment, however, and plan to publish it as a textbook on admissible sets.

In this paper, then, we abandon once again any reader ignorant of the basics of admissible sets, and discuss the material from the last third of our Oslo talk: admissible covers of nonstandard models. We have chosen this topic because it offers nice examples of the new degree of freedom afforded by urelements in admissible sets, examples in recursion theory and in the model theory of set theory. Proofs not given here will be found in the book referred to above.

There is a great deal of folk literature about admissible sets as well as about

52. The axioms of KPU

Let L be a first order language and let 137 = ( M , ... > be a structure for the language L. We wish to form admissible sets which haveM as a collection of urelements; these admissible sets are the intended models of the theory KPU, other models of KPU being so called nonstandard admissible sets on M .

The theory KPU is formulated relative to a language L* = L(€, ...) which extends L by adding a membership symbol E and, possibly, other function, relation and constant symbols. Rather than describe L* precisely, we describe

ADMISSIBLE SETS OVER MODELS OF SET THEORY 99

its class of structures, leaving it to the reader to formalize L* in a way that suits his prejudices.

2.1. Definition. A structure for L* consists of (1) a structure

possibility, (2) a nonempty setA disjoint fromM. (3) a relation E C ( M U A ) XA which interprets the symbol E, (4) other function, relation and constants on M U A to interpret any other

= (M, ... ) for the language L, M = 6 being kept open as a

symbols in L(E, ...). We denote such a structure by 'u, = (5JX ; A , E , ...).

We use variables of L* subject to the following conventions: Given a structure % ~ = ( ~ J x ; A , E , . . . ) for L*,

p , q , r , p l , ... range overM (urelements) a,b,c ,d ,al , ... range overA (sets) X,.Y,Z, ... range over M U A .

We use u,u, w to denote any kind of variable. This notation gives us an easy way to assert that something holds of sets, or of urelements. For example, Vp 3a Vx (x € a ++ x = p ) asserts that { p } exists for any urelement p , where as Vp 3aVq(q € a ++a=p) asserts that there is a set a whose intersection with the class of all urelements is { p } .

The axioms of KPU are of three kinds. The axioms of extensionality and foundation concern the basic nature of sets. The axioms of pair, union and AO-separation deal with the principles of set construction available to us. The most powerful axiom, AO-collection, guarantees that there are enough stages in our construction process. In order to state the latter two axioms we need to define the notion of AO-formula of L(E, ...), due to LCvy [ 19651.

2.2. Definition. The collection of Ao-formulas of a language L(E, ...) is the smallest collection Y containing the atomic formulas of L* and closed under

(1) if @ is in Y then so is l@ (2) if @, $ are in Y so are (@ A J / ) and (@ v J / ) (3) if @ is in Y then so are

VuEu@ and 3 u € u @

for all variables, u and u.

100 K.J. BARWISE

The importance of A O-formulas rests in the fact that many useful predicates can be defined by AO-formulas and that any predicate defined by a Ao- formula is very absolute.

2.3. Definition. The theory KPU (relative to a language L(E, ...)) consists of the universal closures of the following formulas:

Extensionality : Foundation :

Pair: Union : Ao- Separation :

Ao-Collection :

2.4. Definition. 1 .

V x (x Ea a x E b ) +a = b 3a @(a) -+ 3 a [ @ ( a ) A Vb € a l @ ( b ) ] for d l formulas @(a) in which b does not occur free. 3a ( xEa A y € a ) 3bVyEaVxEy(xEb) 3bVx(x€b++xEah@(x) ) fora l l Aoformulasin which b does not occur free. VxEa3y@(x ,y )+ 3bVxEa3yEb@(x ,y ) for all A. formulas in which b does not occur free.

KPU' is KPU plus the axiom:

3aVx [xEa ++ 3p(x=p)]

which asserts that there is a set of all urelements. 2. KP is KPU plus the axiom

V x 3a(x=a)

which asserts that there are no urelements.

One word of caution. There are some axioms built into our definition of structure for L(E, ...). For example, the sentence

follows from 2.1.3, and

follows from 2.1.2.


In a systematic treatment one would now develop axiomatically a large part of elementary set theory in KPU. It is done a lmst exactly as it is for KP, the only trouble being that there is no such axiomatic development in print for KP. We must therefore leave it to the reader to work most of this out for himself. In particular, he should verify that the following are provable in KPU.

“There is a unique set a with no elements” “Given a, there is a unique set b = Ua such that x E b iff 3y € a (x Ey).” “Given a, b there is a unique set c = a U b such that x E c iff x E a or x E b.” “Given a, b there is a unique set c = a n b such that x E c iff x E a and

We define, as usual, the ordered pair of x,y by x E b.”

and prove that ( x , y ) = ( z , w ) i f fx = z andy = w, and then prove in KPU that

that “for all a, b there is a set c = a X b, the Cartesian product of a and b, such

c = { ( x , y ) : x E b and y E b}.”

53. Some useful principles provable in KPU

A E l formula is one of the form 3u @(u) where @ is a AO-formula. I t turns out that a wide class of formulas are equivalent to XI formulas in KPU.

3.1. Definition. The class of 2: formulas is the smallest class of formulas Y containing the A. formulas and closed under 2.2.2, 2.2.3 and

if 4 is in Y so is 3u@, for all variables u .

Thus, for example, the predicate, “x is a set of urelements” can be written

3a(x=a) A V u E x q p ( x = p )

which is E but, as written, is not XI. We will show, however, that for every

102 K.J. BARWISE

X formula I#J there is a XI formula 4' with the same free variables such that KPU k 4 +-+ 4'.

each unbounded quantifier in 9 by a bounded quantifier: Given a formula 4 and a variable w we write q5(w) for the result of replacing

3u by 3 u E w

Vu by V u E w

for all variables u . Thus @(w) is a A, formula. If 4 is A, then @(w) = 4, since there are no unbounded quantifiers 4.

3.2. Lemma. For each X formula 4 the following are logically valid (i.e. true in all structures %w):

where u E u abbreviates the formula V x [x Eu + x f u ] .

3.3. 2 reflection principle. For all Z formulas @ the following is a theorem of KPU:

(We assume u is any set variable not occuring free in 4, and stop making such assumptions explicit in the remainder of this paper.)

Proof. We know from the previous lemma that 3a@(') + 4 is just plain valid, so the axioms of KPU come in only in showing @ + 3a4('). The proof is by induction on 4, the case for A, formulas being trivial. We take the three most interesting cases, leaving the other two to the reader. Case (i). 4 is $ A 0 . Assume


and

KPU i- e - 3a@

as induction hypothesis and prove

KPU I- ($ A s) --f 3a [ J / A el(") .

Let us work in K P U , assuming + A 0 and proving 3a [I)'") A @(')I. Now there are a l , a 2 such that $("I), O('2) so let a = al U a2. Then Q(") and +(") hold by the above lemma. Case (ii). Q is Vu Eu +(u). Assume that

Again, working in KPU assume Vu E u +(a) and prove 3aVu € u $(u)("). For each u E u there is an b such that +(u)@), so by AOcollection there is an a. such that Vu € u 3 b E a o $(u)@). Let a = Ua,. Now for every U E U 3b Sa +(u)@') so Vu E u $(LA)(") by the above lemma. Case (iii). Q is 324 +(u). Assume +(u) * 3b +(u)@) proved and suppose 32.4 J / (u) true. We need an a such that 3u € a +(u)("). If +(u) holds, pick b so that +(a)@) and let a = b U {u}. Then u E a and $(u)(") by the above lemma.

In his original development of admissible sets, Platek took the X reflection principles as one of the axioms, since it is more useful than Ao-collection.- AO-collection, however, is usually easier to verify in a particular admissible set. We list below some of the consequences of the X refle'cfimprinciple.

3.4. Z-collection. Fbr every Z formula Q the following is a theorem of K P U : Zf V x € a 3y @(x,y) then there is a set b such that

V x € a 3y E b @(x ,y)

and

V y Eb3xEaQ(x ,y )

3.5. A-separation. For any two I: formulas @(x), $(x), the following is a

104 K.J. BARWISE

theorem of KPU: If for x E a, $(x) * 1 $(x) then there is a set b ,

b = { x E a : $ (x ) } .

3.6. Lreplacement. For each Z formula $(x,y) the following is a theorem of KPU: If Vx E a 3!y$(x ,y ) then there is a function f with domain a such that v x E a 4 6 , f (XI).

The above is sometimes unusable because of the uniqueness condition in the hypothesis. In these situations it is 3.7 that often comes to the rescue.

3.7. Strong Lreplacement. For each Z formula $(x,y) the following is a theorem of KPU: If Vx E a 3y $(x, y ) then there is a function f with domain a such that for evely x E a

A set a is transitive if for all x E a and ally E x , y E a. Thus if a is a set of urelernents it is transitive. An urelement is never transitive since only sets are transitive. We can prove in KPU that for every x there is a unique transitive set a with x E a such that if b is any other transitive set containing x , then a C: b. This set a is called the transitive closure of x , TC(x). Using TC one can go on to justify recursive definitions over E. For example, the support function can be defined by

Thus, Sp (a) = { p : p E TC (a)}. A pure set is a set a with empty support, Sp ( a ) = 0.

following form. We will also need the second recursion theorem which for KPU takes the


3.8. Second recursion theorem for KPU. Let @(x,y, R) be a Z formula of L(E, ..., R), where x = xl... xfl , y = y l... y k and R is an wary relation symbol occuringpositively in @. There is E l formula $(x,y) of L(E, ...) such that

Proof. We are using @(x,Y, $(* ,%)) to denote the result of replacing R(z l...zfl) by $(z l...z,,yl...yk) wherever it [email protected] notation we consider the case where n = k = 1. Let O(x,y,z) be

where Sat(z,ulu2u3) is the XI satisfaction relation for El formulasz of 3 = n t k + 1 variables (cf. L6vy [ 196.51). Let m be the Godel number of this formula B(x ,y , z ) and let $(x ,y ) be O(x,y,m), or rather the Zl formula equivalent to it where m has been replaced by its definition. Then

$4. Admissible sets over M

I t facilitates matters if we fix notation and let V, denote the most generous possible universe of sets built o n M so that our admissible sets onM will be substructures of V,. Thus, we define

V,(O) = 0

VM(a t 1) = Power set ( I/,(@) U M )

V,(h) = U,, V,(a) if h is a limit ordinal

106 K.J. BARWISE

where the latter union is taken over all ordinals. If we need to keep things straight for some reason we subscript notions with a n M to denote their inter- pretations in VM. For examples, EM denotes the membership relation of VM where each p E M is taken as having no elements (even though in some other contextM might be a set of sets) and “a is transitiveM” means “x EM y EM a implies x EM a”.

4.1. Definition. A structure ‘u, = (%TI ; A , E, ...) for L (E, ...) is admissibk if l!lm is a model of KPU, ifA is a transitiveM subset of V, (where !lx = ( M , ... )) and E is the restriction of € M to M U A .

If ordinary admissible sets are pictured as in fig. la, as they often are in informal discussions, then admissible sets with urelements should be pictured somewhat as in fig. 1 b.

( a ) ( b )

a) An admissible set A without urelements b) An admissible set %m over M

Fig. 1.

The small cone in ’urn represents the pure sets of %,, i.e. those a €Am with empty support. It is easy to verify that this collection of sets is an admissible set (without urelements).

4.2. Example. For any infinite cardinal K define H ( K ) ~ = (YJI ; A , E) where A = { a E VM : TCM(a) has cardinality ITCM(a) 1 < K } . For any such K , H ( K ) ~ is admissible.H(K>m is a model of KPU’ iff K > IMI. We usually denote H ( u ) m by HFm since it consists of the hereditarily finite sets relative to In.


A subset R of the admissible 'urn is C, if it is definable on by a El formula with parameters from A U M . R is Al if both R and its complement (A UM) - R are L1 on 2lm.

4.3. Example. If 9?l is an acceptable structure then a subset of 9?l is semi- search computable on 9?l iff it is El on HFw , the notations of acceptable and semi-search computable being those of Moschovakis. The result is due to Gordon.

The ordinal of an admissible set is the least ordinal not in the admissible set.

4.4. Example. Given any structure 9?l there is a smallest admissible set over which is a model of KPU', i.e. where the set M itself is an element of the admissible set. Denote this admissible set by HYPm . The proofs in Banvise- Candy-Moschovalus [ 197 11, if carried out in this setting, show that if m is acceptable then a relation S is inductive on 9?l iff it is C on HYP, and is hyperelementary on 93 iff it is an element of W m . The ordinal of HYPm is the closure ordinal of the class of first order positive inductive definitions over m. (We are using inductive and hyperelementary, as in Moschovakis [ 19731, for what was called semi-hyperprojective and hyperprojec tive in Barwise-Candy-Moschovakis [ 197 11 .)

4.5. Example. The results in infinitary logic of Barwise [ 1969al all go through in this more general setting without change. To see how this may nevertheless be a significant extension, suppose am= ( $ n ; A , E , , ...) is countable and admissible with ordinal w. (We will see many examples of such in the following sections.) Let TI, T, be theories of the admissible fragment LA of LU1,, C1 on am, such that every @ E T, is a pure set (and hence finitary). If every finite subset of T, is consistent with T, then T, U T, is consistent, even though T, may have infinitary sentences in it. The proof is a simple consequence of the compactness theorem for LA.

$5. Properties of the admissible cover

Admissible sets am = (m ; A , € , ...) embody certain principles of set contribution, the ordinals a inA give us the stages, the sets of rank OL the principles of set formation available at stage a. What then is to be made of non-

108 K.J. BARWISE

standard models of KPU, KP or ZF. The results we discuss here shows that there is a hard core of admissibility in the heart of even the most non-standard models.

5.1. Definition. Given a model %I= ( M , E , ... >, where E is binary, the covering function CE for is the function which assigns to each x E M the set

xE = { y E M : yEx} .

The name “covering function” is new but the function itself is basic in the study of models of set theory. For example, if $%?= W , E ) is a substructure of (n = ( N , F ) then an x E M is fixed by (n if xE = xF ; otherwise x is enlarged by 9. If x E = x F for all x E M then % is an end extension of %I, written 9.KCend (n.

The covering function f o r m = ( M , E ) maps elements o f M to subsets o f M hence to elements of V,. If $%? + Extensionality then this function is one-one. There are many admissible ’urn which are admissible with respect to the covering function for (XQ. For example, anyH(K),m for K > ] M I . If %I satisfies enough axioms of set theory, however, there is ohe admissible 21n which really lives over m. This set is called the admissible cover of %I and is the object of study of this paper.

5.2. Theorem. Let T be some set theoiy containing KP and let pi = ( M , E ) be a model of T, standard or nonstandard, with covering function CE. There is an admissible set ’ 2 l ~ over 8, called the admissible cover of m, with Properties I-IX listed below.

Property I. C, maps M into ’21, and yiM is admissible with respect to C, ; i.e. aM = (M;A,,E 1 A,,CE) is admissible.

This is equivalent to saying that (%I;A,,€,,CE) is admissible since E can be recovered from CE.

Property 11. There is a function * : A, U M -+ M satisfying:

p * = p forall p E M

(u*)E = {b*: bEa) for u € A , .


One might call * an €-retraction of "2IM onto m. It is Property I1 which insures that the admissible cover of really lives over W. To be more precise:

Property 111. a, is uniquely determined by Properties I and II. In fact, 'u, is contained in any admissible '$3, satisfying I and contains any admissible BM satishing II.

Since I and I1 characterize the admissible cover of m, all other properties could be derived from them, but such a procedure would cause us to duplicate many steps in the proofs of I and 11. For example, the following is obvious from the proof of 11.

Property IV. The cardinality of 'u, is the same as that of M

The well founded part of a model m = (M,E) of KP, WF(m), is the transitive set (in the sense of V ) which is the range of the following collapsing function clpse. The domain of clpse is the set of all x E M for which E is well founded on TCm(x), so that the following makes sense:

clpse ( x ) = {clpse ( y ) : y EX^}

If is well founded then clpse: m zz (WF(m),E).

Property V. The pure sets of the admissible cover of in WF(SB), the well founded part of m. In particular, the ordinal of A , is just the ordinal of the well founded part of m.

are exactly those sets

We will attempt a picture of m and its admissible cover at this point. The represents the point at which 93 becomes non-standard, the dotted line in

lower portion being isomorphic to WF(S?I).

( a ) ( b )

a) = ( M , E ), a model of set theory b) The admissible cover 2 l ~ of (%I

Fig. 2.

110 K.J. BARWISE

This gives a hint of the new dimension now available to us. Consider, for example, a model !J?l of ZF with nonstandard integers. The admissible cover of SB has many infinite sets in it (aE for any infinite integer a, for example), but only the natural numbers for ordinals. This is in stark contrast to old fashioned admissible sets where ifA # HF then w € A .

The next three properties are of a more recursion theoretic nature. We say that an inductive definition r on rrn is a E inductive definition if

the clause x E r (R) is given by a C formula $ ( x , R ) with R occuring positively in 4:

x E r(R) f--f $ ( x , R ) .

We use Z@ for the futed point of I?, as in Moschovakis [ 19731. A relation S of n variables is E inductively definable on !J?l if there is a E inductive definition I@ of n + k variables, k 2 0, and x1 ... xk in %? such that

For example, the domain of the function clpse is E inductively defined on (B by the formula $(x ,R) :

A theorem of Candy shows that if tively definable set on !J?l is El on !J?l. If SB is nonstandard, however,Io need not be even first order definable over in, as the domain of the function clpse shows.

is an admissible set then any C induc-

Property VI. A relation S on !)J is E inductively definable on m i f and only i f it is X on the admissible cover of m. The closure ordinal of a E inductive definition is at most the ordinal o f A M .

In Barwise [ 1969b] we showed that the strict-IIi relations on an admissible set A coincide with the s.i.i.d. relations of Kunen [ 19683, and hence ifA is countable the strict l l i relations coincide with the I;, relations. The proofs of these results carry over verbatim to admissible sets with urelements.

In Aczel [ 19701 it was announced that if Sn is a countable nonstandard

ADMISSIBLE SETS OVER MODELS OF SET THEORY 1 1 1

model of KP then the s-ni relations on tions, but with the x inductively definable relations on fm . This follows from Properties VI and VII, and the above paragraph.

coincide, not with the El rela-

Property VII. A relation S on sible cover of m.

is s-IIi on $I i f f it is s - IIf on the admis-

The non-trivial half of VII and half of 111 can be derived from VIII.

Property VIII. The admissible cover of Kreisel) of the class of all models of KPU of the form

is the €-hard core (in the sense of

where '37 = ( N , F ) is an end extension of 93 satisfying KP.

In the next section we are going to discuss the use of admissible covers in constructing models of set theory. The first application uses only properties I and 11. In some applications, however, we need the following. Given m 5 % and Mo C M we write

m X l % [wrt M o ]

to indicate that every x sentence with parameters from X has the same truth value in m as in % . IfM, = M we write m< '37 and if X = 8 we write sm El 91 .

Let $I= ( M , E ) and '37 = ( N , F ) be models of KP with admissible covers 21M and 91N. I t follows from VIII that !Dl S e n d % iff%21, C B1,. In this case if x E 'u, then x* has the same value regardless of whether the €-retraction * is taken in the sense of 'u, or aN.

Property IX. Given k t A o = { x E A M : x* E M O } . Then

and '37 and % as above with I%? Cend 3 andM0 C M ,

!Dl<l% [wrtMo]

ifand only if

21(M<1tYN [wrtAO] .

112 K.J. BARWISE

In particular, if %rl G1 '31 then aM ?iN and if in4 '% then ylM '?IN. There are a number of other relations (e.g. =,<) which lift up from (m, 3) to ( a M , aN) but the above is what is needed in the applications we have in mind.

KP and all of the properties would go through with slight modification. We restricted ourselves to models

We could have considered admissible covers of models of KPU, rather than

of KP for pedagogical reasons only.

86. lnfinitary methods in the model theory of set theory, revisited: Applications of the admissible cover

We first came across the admissible cover in trying to understand what was behind some of the proofs in Barwise [ 19701. The general situation was this. We had a countable model = ( M , E ) of set theory and we wanted to construct an end extension of % with certain properties. I fM was a transitive set with E = E r M then we were able to exploit the completeness and compactness theorems for LM, the admissible fragment of LU1, associated withM. If % was nonstandard, however, we were forced into some strange considerations. I t is the admissible cover which unifies these two cases and allows the simpler proofs to go through whether m is standard or not.

We give two examples of this here. In the first we extend a theorem of Friedman's from standard m to arbitrary m. This result uses only Properties I and 11 of admissible covers, I in order to form sentences like

inAM and I1 to know that if a theory T is an element ofAM then the set of a E M with amentioned in T is a set of the form d E , d EM, and hence is bounded in 1137.

Friedman's result is the following: Let m= ( A , € ) be a countable admissible set and let T be a theory of LA which is XI on A , contains KP, and is true in some end extension of 1137. (For example, T might be ZF and it might be false in m.) The conclusion is that 'm has an end extension % with new ordinals which is a model of T but where all the new ordinals are nonstandard. We can extend his result as follows.


6.1. Theorem. Let $fl= ( M , E ) be a countable model of KP with admissible cover aM. Let T be a (finitaiy or infinitary) theory C definable on '21M which is true in is an end extension of %17 with new ordinals but such that no ordinal of '% is the least upper bound of the ordinals of I f l .

or some end extension of 11?7. Then T has a model !Jl which

Proof. Friedman's proof used "supercomplete" theories and the infinitary compactness theorem. We could give an extension of Friedman's proof, but for variety we replace supercomplete theories by the forcing version of the omitting types theorem, Theorem 2.2 in Keisler [ 19731. Let LAM be the admissible fragment of LwIw associated with mM and let LB be the smallest fragment containing L and the sentence 9:

AM

where e(x) is

Ord('%') = {aEM: used to denote a. Let @ be the class of formulas of LB which are either in LAM or of the form8(x) or B(y). Let W be the class of all models of T which are end extensions of

a is an ordinal}, and ;is a constant symbol inAM

with new ordinals; i.e. models of T':

T - Vx[xEa*VbEaEx=b] for all a € M

T' is nonempty by the compactness theorem for LAM. The sentence I) can be written as an V V 3 (a) sentence:

So by the version of the omitting types theorem given in Theorem 2.2 of Keisler [ 19731, we will know that 9 is true in some model % E % if we can show that: if p ( c l ... cn) is a finite subset of @(C) satisfiable in W then for each ck E C one of the following is satisfiable in 91'2 :

114 K.J. BARWISE

p’(a) = p ( c l ... cn) U {ck <a} for some a E Ord (B)

Assume that none of the p’(a) are satisfiable in W so that T’ U p ( c l ... cn) + e(c,). We need to show that T’ U p ” is consistent. I t suffices to show that the following C1 theory of LAM is consistent:

T - Vx[x€a+-+VbtaEx=b] forall a € M

Ck >a all a € O r d ( m )

4 ( C l . . . Cn) all 4 E ~ ( c ~ ... cn> n A ,

C i > i all a E Ord (m) if e(cJ Ep(cl ... cn) .

Ck > d

d > a all a E O r d ( B )

where d is a new constant symbol in C, different from c1 ... cn,ck. But any A T U p ( c l ... cn) , since then we need only have d > a for a in some set X € A M , X C Ord (m) and any such set is bounded in m . 0

finite subset of this theory is clearly satisfiable in any model of 111-

For our second example we modify (by strengthening both the hypothesis and conclusion) one of the results in Barwise [ 19701. This result shows how property IX comes into play. A subset of an admissible set is C r if it is definable by a C, formula without parameters.

6.2. Theorem. Let m be a countable model of T, where T is a theoly containing ZFC which is Er on the admissible cover 9lM of B. Let K be a cardinal of m, regular or not, such that

holds in ‘%I. There is a model (n of T which is an end extension of PI such that :


1) no subsets of any h < K are added in % (hence all cardinals X < K of

2) all cardinals of m greater than K have cardinality K in ‘32 (hence many are still cardinals in %), and

new subsets of K are added by ‘32 ).

Proof. Let m= ( M , E ) and let 91, = ( M , , E 1 M,) where Mo = {x EM: Since m, is transitive in m, mo send m. Let a, and aM0 be the admissible covers of h < K in m we let p(h) denote the power set of h in the sense of m. We need to show that the following theory T’ is consistent:

k “TC(x) has power 5 ~ ” ) . Then m,<, m by Levy [1965]

and m, respectively, so that %,o<l %, by Property IX. For

i) T ii) V X ( X E Z * V ~ ~ ~ ~ X = ~ )

iii) AxEKE~x(xGX +. V a , = p ( X ) E ~ = ; ; ) iv) Icl=K v) n € c all a f M .

The theory T’ is El definable using as parameter the sequence

for all a E M

in the X 1 definition (in order to write the sentence (iii)) and this set is an element ofM, by the hypothesis A < K =, 2h I K . If T’ is inconsistent there is a proof in A , of a contradiction from T’, by the completeness theorem for countable admissible fragments. This El property ofA, is thus true inAMo, so there is proof inAMo of a contradiction. This proof can use only 2 K

axioms from T’ and each constant a i n these axioms has a EM,. Thus there is an inconsistent subset T, T’ which asserts (v) only for 5 K setsa, each of them inMo. But this is absurd since fm itself can be made into a model of any such To by the proper assignment of an element ofM, to the constant symbol c. 0

Actually we see that the theory T can be X 1 using parameters fromM, and the proof still goes through. We could combine the above two theorems to make % have no least upper bound for the ordinals in ‘17. We can also combine Theorem 6.2 with a theorem of Keisler and Morley (Corollary A on page 137 of Keisler [1971]) to get

116 K.J. BARWISE

3 “A is a cardinal > K ”

implies

so that H, subsets of K are added by 3 without adding new subsets to any h<K.

We should point out that some hypothesis like our

is needed for the result. For example, if the sentence

2”o = H2

is in T and K = Hy then any end extension of satisfies

which is a model of T and

I K F I = H,

must add new sets of integers (the old power set of o is enumerated by a sequence of length an ordinal (Y with I C Y ( = N, true in %).

57. The construction of the admissible cover and related admissible sets with urelements

In this section we will sketch a construction whch gives the admissible cover, but we shall not verify all of its properties. On the other hand, we shall carry out the construction in a general setting which gives many other examples of admissible sets with urelements.

For a simple such example, consider an admissible set A without urelements, with a C, subsetM ofA. There is a natural admissible setAM overM withA as the collection of pure sets ofA,. The general situation needs the following definition.

117 ADMISSIBLE SETS OVER MODELS OF SET THEORY

7.1. Definition. A structure % = ( N , R l... R,) is Z1 on a structure Bm = ( m ; B , F , ...) i f N i s E l o n B m andif thereare El relations R i ... R ; , R [ ... RF on Bm such that for all i l l and allx l . . . xn i E N

Ri (xl ... xni ) - R f ( x l ... xni)

e l R f ( x 1 ... x,.)

where Ri is ni-ary.

7.2.Theorem.Let BW = ( n ; B , F , ...) beamodelof KPUandlet 2 = (N, R ... R,) be E on Bm. There is a largest admissible set a = ( % ; A , E N ) s u c h that themap *: %,-+%m iseverywheredefined:

p * = p for P E N

(a*), = { b * : b EN a } .

For b EN, if bF E N then bF EA.

We shall sketch a proof of this for the case where there is just one binary relation R (i.e. 1 = 1, nl = 2).

The structure 2 3 ~ ~ is a model of K P U as formulated in some language L(E, ...) (where is an L structure) whereas %,will be a model of K P U as formulated in LO(€), where Lo has just the binary symbol R . To keep things straight we denote this second KPU by K P U , .

Choose E l formulas (of L(E, ...)) u(x), @ ( x , y ) , $ ( x , y ) so that for all x E M U B

118 K.J. BARWISE

which, of course, is true inBm,. The heart of the proof of the theorem consists of constructing a suitable interpretation I of KPU, in the theory KPU, . Toward this end define ( 0 , ~ ) and; for (1,x).

predicates of L(E, ...) as follows, using x for

Point (a) - 3x [u(x) A a = X] se t (a ) -gb[a=I ; A ~ x ~ b ( ~ e t ( x ) vpoint(x>>l x & y a 3 z ( y = ; A x E z ) R ’ ( a , b ) a 3x3y(a=i A b=y^ A @(x,y)) R”(a,b)- 3 x 3 y ( a = x A b = j A $(x,y))

The predicate Set (a) needs the second recursion theorem for KPU in its definition, the predicate x & y can be written as a A,, formula. The interpretation I of LO(€) into KPU, is given by the following instructions: replace each

positive occurrence of R negative occurrence of R occurrence of x E y

quantifier over urelements Vp( ...)

quantifier over sets Va( ...) 3a( ...)

undetermined quantifiers Vx( ...)

3P( - )

3x (...)

by R’ by R”

by Va[Point (a) +(...)I by 3a[Point(a) A (...)I by Va[Set (a) +(,..)I by 3a[Set(a) A (...)I by Va[Point(a) v Set(@)+(...)] by 3a[(Point(a) V Set(a)) A (...)I .

by X E Y

Equality and the propositional operations are not altered. One must carry this out in a sensible way so that no clashes of variables occur. Denote the translation of a sentence @ of Lo(€) under the above by @I.

7.3. Lemma. For each axiom @ of KPU,, d is a theorem of KPU, .

Proof. We run quickly through the axioms.

Extensionality: The translation reads: for all Setsa and b , if x & a -x & b for all Points and Setsx, then a = b. Now if Set (a), Set(b) then a = z o , b = go, every x E a, is a Point or a Set and x 8 a t--, x E a, ; similarly for b and b,. Hence a,, = b , so a = b.


Foundation: If there is a Se ta such that @'(a) then there is a Se ta so that d(a) but for all Sets b g a, l@'(b). Note that x & y + rk ( x ) < rk 01) so choose a of least rank.

Pair: Let co = {a , b} and c = 2,. If a, b are Points or Sets then Set (c) and x G c iffx = a v x = b.

Union: Given a Set a let b = {z : 3 y C% a, z & y } , by AO-separation, and let c = 6. Then z d c iff z C% y I% a for some y .

do-separation: We wish to show that given a Set a there is a Set b such that for all x ,

x E b O x a A d ( x ) .

If a = 2, let bo = { x E a0 : @'(x)) by 3.5 and let b = J0. (The point here is that d is only a I3 formula but (1 @? is also a 2 formula.)

AO-coUection: Assume Set (a) and for every x d a there is a y , either a Point or a Set such that ~ ( x J ) , where @ is A. in LO(€). Now the whole statement $

is a I3 formula (since x c!% a + x E TC (a)) so, by I3-reflection, there is a w , a C w, such that $ ( w ) holds. Let

bo = u { y E w : (Point(w)(y) v Set(w)(y)) A (@)tw) (x ,y ) } x B a

and b = So: One easily checks that Set (b) and V x E a 3y d b @'(x,y). 0

Now when one has an interpretationl of one theory To in another theory T, then any model '23 of T, has associated with it a model 8-I of To in the usual way. Applying this to our situation the structure !&n gives rise via l to a structure 8 -I which is a model. of KPU,. The structu;e'Xj$ has the form m

%m -I = ( % ' ; A ' , E ' )

where g' is isomorphic to % via the map a + a, A' is the set of b E B such that

'Xjmk Set@)

120 K.J. BARWISE

and E' = & 1 A'. If % morphic to the desired admissible set. In general, however,!ljm need not be admissible but only a model of KPU in whch case E' need not be well founded. Thus, we need to apply the following lemma to !By<'.

is admissible then E' is well founded and %@' is iso- rn

7.4. Lemma. The well founded part of a model o f KPU is again a model of KPU.

The proof is like that given in A6 of Barwise [ 19721. Combining these two lemmas, with a natural isomorphism 77 ( ~ ( i ) = a and extend 77: VNr -+ VN via ?(a) = { g ( b ) : b € a } ) gives an admissible set follows:

The mapping * arises as

Let us work again in KPU,. Define

a* = a. if Point (a) and i, = a

a* = {b*: b I% a} if Set (a)

using the second recursion theorem for KPU. This mapping, interpreted, maps !Em-' + maps 9' t o 9 and satisfies $m'

(a*)F = {b*: bE'a}

Restricting this map to the well founded part of % -' and then mapping it over t o%

To see that 21% is the largest admissible set with such a map *, suppose '% ' where some other admissible over )37 with a map *: !& '

%i via the natural isomorphism gives the conclusion of the theorem. %

9 %+%El

p * = p far p € N

(a*)F = {b*: b EN a} .

Consider the map " defined in %; by recursion over EN:

p" = ( O , p )

a " = ( I , { b " : b E a } ) .

If one chases an x the long way around the following diagram, one sees by


induction on EN that all the maps are defined and that 2, the result of all the composite functions, is x for all x E

’ 9’

I I t Id I

Hence %h C ‘?lW Findly, if b E N and b, C N then let a = (l,{(O,c): cFb}) in%m. This concludes the proof of the theorem. When one applies this to obtain

the admissible cover, one has 23 = 9 = a model of KP. Most of the properties of the admissible cover shouf’d be at least plausible from the above proof. Note that in order to show that the admissible cover is admissible with the covering function, one has to have KPUo formulated in Lo(E,f) wherefis a 1-ary function symbol. In the interpretation Z one then has to replace f(x) = y by f(x) = y where f is defined in KPU, by

m

f(x) = { j : y EX}Y .

References

Aczel, P. [ 19701

Barwise, K.J.

Implicit and inductive definability (abstract), J. Symbolic Logic 35, p. 599.

[ 1969al Infinitary logic and admissible sets, J. Symbolic Logic, 34, pp. 226-252. [ 1969b] Applications of strictn: predicates to infinitary logic, J. Symbolic Logic

[ 197 11

[ 19721

34, pp. 409-423. Infinitary methods in the model theory of set theory, Logic Colloquium ‘69, North-Holland, 1971, pp. 53-66. Absolute logics and L,,, Annals of Mathematical Logic 4, pp. 309-340.

The next admissible set, J. Symbolic Logic 36, pp. 108-120. Barwise-Gandy -Moschovakis

[ 19711

Friedman, H. [ 19721 Countable models of set theories, to appear.

Keisler, H.J. [ 197 1) Model Theory for Infinitary Logic (North-Holland, Amsterdam).

122 K.J. BARWISE

[ 19731

Kunen, K. [ 19681

Forcing and the omitting types theorem, to appear.

Implicit definability and infinitary languages, J . Symbolic Logic 33, 446-451.

Ldvy, A. [ 19651

[ 1973 J

Shoenfield, J. [ 1967 J

A hierarchy of formulas in set theory, Memoirs of Amer. Math. SOC. No. 57

Elementary induction on abstract structures (North-Holland, Amsterdam) Moschovakis, Y.N.

Mathematical Logic (Addison-Wesley, Reading, Mass.)

J.E.Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Pu bl. Comp. ~ 19 74

AN INTRODUCTION TO THE FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

(Results of Ronald Jensen)

Keith J. DEVLIN University of Oslo

$0. Introduction

We shall work in Zermelo-Fraenkel set theory (including the axiom of choice) throughout, and denote this theory by ZFC. We shall adopt the usual, well-known, notations and conventions of contemporary set theory (e.g. an ordinal is defined to be the set of all smaller ordinals, cardinals are initial ordinals, etc.).

The paper is entirely self-contained, but some familiarity with the usual definition of the constructible universe, L, in terms of definability, and the proof that L is a model of ZFC + GCH + V = L, will be helpful.

The exposition is based, with permission, very strongly on a set of notes written by Ronald Jensen and entitled “The Fine Structure of the Construc- tible Hierarchy”. Except where otherwise stated, the results are entirely those of Professor Jensen. I t is convenient at this point for us to express our appre- ciation of several dluminating discussions with Professor Jensen on his work in general.

’ Since we wrote this paper, a slightly revised version of these notes has been p u b lished as a research paper. See R.B. Jensen, “The Fine Structure of the Constructible Hierarchy”, Annals of Mathematical Logic, Vol. 4 [ 19721, p. 229. The present paper represents a lengthy discourse on an expansion of the earlier parts of Jensen’s paper, and it is hoped that the somewhat more leisurely pace we adopt here (as opposed to Jensen’s paper) will be of benefit to those not predominantly interested in the set theoretical consequences of the Fine Structure Theory. For those who are so inclined, the notation we use is almost identical to that of Jensen, so this paper should provide a good introduction to Jensen’s.

123

124 K.J. DEVLIN

Previously, Jensen worked, as did most other people, with the usual “constructible hierarchy”. Thus, one defines, inductively, sets La, a E OR, by setting Lo = 0, LA = U,<ALa if lim(A), and = the set of all x C L, such that for some €-formula cp and some a l , ..., a, E La, x = { z E La/ L, + cp [ z , a l , ..., a,]}. One then defines the constructible universe as the class L = U,,oRL,. Now, the important facts concerning this definition which one uses when studying L, are, firstly, that the construction is (in a strong way, to be made precise later) X -definable, and thus has certain absoluteness properties, and, secondly, that La+1 contains all and only those subsets of L, which are L,-definable (and which, therefore, must be in L if L is to be a model of ZFC). But if, indeed, these are the only conditions which we require (and loosely speaking they are), then it is clear that our above definition is unnecessarily restrictive. For instance, there are many simply definable functions on sets under which L must be closed, but which increase rank - and these functions will lead out of the sets L,. For instance, unless lim(a), L, will not be closed under the formation of ordered pairs. Since this function plays a central role in even the most elementary parts of set theory, we see that this defect becomes quite important (though not unavoidable) when we try to study the fine structure of L rather than L itself. So, following Jensen, we define a new hierarchy of “constructible sets”, which is sufficiently like the L-hierarchy to preserve the two properties mentioned above, but which has the extra property that each level in the hierarchy is closed under ordered pairs, etc. More precisely, we first define a certain class of set functions (called “rudimentary functions”), and then define a hierarchy (J,I a E OR) (the Jensen hierarchy) such that each J, is closed under the rudimentary functions, L = UarORJa, and the two properties above hold for this hierarchy. In most cases, J, will be a “constructibly inessential” extension of La, and in fact, if (Val a E OR) denotes the familiar rank-hierarchy, the precise relationship between the J- and the L-hierarchies is easily seen to be J, = Lo = 6 and L,+, = V,+, n Jl+, for all a. Hence we have J, = L, iff wa = a.

In $ 1 we give some basic definitions. In 32 we define the class of rudimentary functions and develop the elementary theory of this class. The reader may, if he wishes, safely skip all the proofs in this section without affecting the reading of the later parts. 93 is devoted to a very brief discussion of the concept of an admissible set. In $4 the Jensen hierarchy is defined and its elementary properties discussed. In 9 5 we investigate the fine:

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY 125

structure of the Jensen hierarchy. A corresponding theory may also be developed for the L-hierarchy, the only difference being that some awkward complications arise because of the above mentioned defects in this definition.

3 1. Preliminaries

We shall be concerned with first-order structures of the form M = ( M , E,A >, where M is a non-empty set and A C M. In general, we shall write ( M , A ) for ( M , E, A ). The (first-order) language for such structures consists of the following:

(i) variables u j , j E w (generally denoted by u, w , x , y , z , etc.) (Vbl.). (ii) predicates =, E, A .

(iii) bounded quantifiers (Vui E uj) , (3ui E uj), i,j E w, i # j .

(iv) unbounded quantifiers ( h i ) , (3u j ) , i E w. (v) connectives A, v, 1, +, -.

Finite strings of variables (or of elements ofM) are denoted by v, X, etc. We w r i t e u E X f o r u l E X A ... A u , E X , w h e r e w e h a v e ~ = a ~ , ..., a,.Similarly for 3v, V v , etc.

The notions of primitive formula (PFml), formula (Fml), free variable, statement, and satisfaction are assumed known. A formula of this language is Co (orn,) if it contains no unbounded quantifiers. Let n 2 I , and let Q, denote V if n is even and 3 if n is odd. A formula is C,(ll,) if it is of the form 3 x 1 V x 2 3 x 3 ... Q , x ~ ~ ~ ( V X , ~ X ~ V X ~ ... Qn+l~,cp) where cp is C,. A formula in wluch the predicate A does not occur is called an E-fomulu.

kM denotes the satisfaction relation for M . Thus, kM is the set of all (p, (2)) such that p is a formula of the above language and 2; E M and p holds

z in M at the point (2). We generally write k M cp[~] for (cp, (2)) E kM. ‘t=Mn

denotes the set of all (cp, (2)) E bM such that cp is En. Let N C M. A set R C M is E f;l(N) (IIF(N)) iff there is a En (n,) for-

mula cp(u,v) and elements a E N such that for all x EM, R(x) - The set of all such R is also denoted by X r ( N ) (II,”(N>>.

q [ x , u ] .

s e t X z ( N ) = u,,;x,M(N),A,M(N)=x,M(N) nn,M(N). Write C If cp is a formula, cpM denotes the relation { x EM1 bM cp[x]}. Similarly,

Let F be a class of structures of the form M = ( M , A ). A relation R is

for XF(@ and E,(M) for C p ( M ) . Similarly for II, A.

and more generally, define cpM [a] for u E M as {x EM1 b~ ~ [ x , a]}.

126 K . J . DEVLIN

uniformly E n ( M) for M E F iff there is a Z, formula q(u, v ) and elements a E n { M I M E F } s u c h tha twheneverMEF,R nM=cpM[a].

52. Rudimentary functions

A function f : Vn -+ V is rudimentary (rud) iff it is generated by the following schemata:

(i) f(xl, ..., x, ) = xi , I I i I n. (ii) f ( x l , ..., x , ) = xi - xi,

(iii) f(xl, ..., x , ) = { x i , xi}, (iv) f ( x l , .. . , x , 1 = h (81 ( X I , .. ., x , ), .. . , gk(x 1 , . . ., x, 11, where

g l , ..., gk, h are rud. (v) f(xl, ..., x , ) = U y E x l h ( y , x 2 , ..., x, ) , where h is rud.

1 I i, j i n. 1 I i, j In.

For example, the following functions are clearly rud: f ( x ) = u x i f(x) = xi u xi (= u {Xi ,Xi})

f@)= {XI

f@)= (x )= { t q } , t X l , ( X 2 , . . . , x , >)I. And i f f (y ,x) is rud, so i sg(y ,x)= ( f ( z , x ) l z E y ) (= U z E y { ( f ( z , x ) , z ) } ) .

We say that R C Vn is rudimentary (rud) iff there is a rud function r : Vn + V such that R = {(x) I r(x) # 8). For example, 6 is rud, since .Y~x+-+I-xf!J.

We list some basic properties of rudimentary functions and relations.

(1) Iff , R are rud, so is

Proof. By definition, there is a rud r such that R(x) ++ r (x ) # 0. Then = U y E r ( x ) f ( X ) .

(2) Let xR be the characteristic function o fR . R is rud if xR is rud.

Proof. By (1).


(3) R is rud iff 1 R is rud.

Proof. By (2), since xqR(x) = 1 - xR(x).

(4) Let fi : V" -+ V be rud, i = 1, ..., m. Let R j C V n be rud and mutually disjoint, i = 1 , ..., m , and such that U E I R j = V n . Define f : V n + V by f ( x ) =fi(x) iffRi(x). Thenfis rud.

By (l), fi is rud. But f(x) = uEl fi(X).

( 5 ) I fR(y,x) is rud, so isf(y,x) = y n { z l R ( z , x ) ) .

Then h is rud. But f ( y , x ) = UZE, ,h(z ,x) .

( 6 ) SupposeR(y,x) is rud and (Vx)(3!y)R(y,x). Set

the unique z € y such that R ( z , x ) , if such a z exists.

8, otherwise.

Then f is rud.

Proof.f(y,x)= U [ y n { z I R ( z , x ) ) ] .

(7) IfR(y,x) is rud, so is (32 € y ) R ( z , x ) .

Proof. Take r rud so that R ( y , x ) - r ( y , x ) f 8. Then

128 K.J . DEVLIN

(8) IfRi(x) are rud, i = 1, ..., rn, then so are m rn

U R i , n R i , (Trivial). i=l i= 1

(9) Let (-)o,(-)l, denote the inverse functions to (-, -). Then (-)*, (-)1 are rud. More generally, let (-);, ..., (-):-1 denote the inverse functions to ( x l , ..., xn). Then (-)", ..., (-):-1 are rud.

Proof. the uniquez E Ux such that ( 3 u l , u z E U x ) ( x = ( u l , u 2 ) A u1 = z )

6, if no such z exists. (XI0 =

etc.

(10) The function the unique z E u u x such that (z ,y ) E x

6, if no such z exists. f ( X J > = X(Y> =

is rud. (By definition.)

(1 1) dom and ran are rud.

Proof. dom (x) = { z E U Ux I (3w E U Ux)((w, z ) E x)} ran(x)= {ZEUUX~(~~EUUX)( (Z ,W)EX)) .

(12) f(x,y) = x Xy = UuExUvEy ( ( u , u ) } is rud.

( 13) f(x, y ) = x r y = x n (ran(x) X y ) is rud.

( 14) f(x, y ) = x"y = ran (x f y ) is rud.

(15) f(x) = xpl is rud.


Proof. Set h(z) = ((z)l,(z)o). Then h is rud. But clearly, f(x) = x-’ = h”(x n (ran (x) X dom (x))).

Recalling our preliminary discussion ( S O ) , we observe that though rud functions increase rank, they only do so by a finite amount. More precisely, by induction on the rud definition * of a given rud function f, we see that there is a p € w such that for all x 1 , ..., x,, rank (f(x ... , x,)) < max{rank(xl), ~ ..., rank(x,)} + p .

* Note : In future, we shall often refer to “the rud definition o f f ” , or simply “the definition off ”. We mean an arbitrary such definition, the actual choice being irrelevant, and hence assumed made once and for all.

We now prove that the rud functions do in fact encompass all of the “simply definable” functions we spoke about in SO. First, let us call a function f : V n + V simple iff whenever cp(z, y) is a Co €-formula, there is a Co €-formula J / such that k cp(f(x),y) ++ $(x,Y). A useful characterisation of this concept is given by the following:

Proposition. A function f : Vn -+ V is simple iff (i) the predicate x € f(y) is Z:; and

(ii) wheneverA(x) is z;, so is (VX ~ f ( y ) ) ~ ( x ) .

Proof. (+) By definition. (+) Let f satisfy (i) and (ii), and le‘t cp(x,y) be a To €-formula. An easy induction on the length of cp shows that (p(f(x),y) is equivalent to a Lo E- formula; so f is simple.

Using this proposition, and easy induction on the definition off yields:

Lemma 1 . Iff is rud then f is simple.

Now, since there are C x functions which increase rank by an infinite amount, it is clear that the converse to the above lemma is false. However, we do have:

Lemma 2. R C Vn is XriffR is rud

130 K.J . DEVLIN

Proof. (+) Let R be Zx. By (3 ) , (7), and (8) above, an easy induction on the Cx definition of R shows that R is rud. (+) Let R be rud. Then xR is rud. SO by Lemma 1, xR is simple. Using our above proposition, an easy induction on the rud definition of xR shows that xR ~ and hence R , is Ex.

We require some generalisations of these concepts. Let A C V. We say that a function f is rud in A iff f is generated by the

Let p E V. We say that a function f is rud in parameter p iff f is generated schemata for rud functions and the function xA .

by the schemata for rud functions and the constant functions h(;r) = p .

Lemma 3. I f f is nid in A C V, there are rud functions gl, ..., gn such that f is expressible (in a uniform way with respect to the rud definition o f f ) as a composition of gl , ...,g, and the function h (x ) = A n x.

Proof. By induction on the (nid) definition o f f

A set X is said to be rud closed if for all rud functions f , f “ X n C X . A structure M = ( M , A ) is said to be rud closed if for all functions f which

We say a structure M = ( M , A ) is amenable if u E M + A n u E M .

are rud inA, f”M” C M .

Lemma 4. A structure M = ( M , A ) is rud closed i f f the set M is rud closed and M is amenable.

Proof. By Lemma 3.

Lemma 5. Le tA C V. Zf f is rud in A , then f r M n is uniformly X o ( ( M , A n M ) ) for all transitive, rud closed M = ( M , A n M ).

Proof. By Lemmas 2 and 3.

The next result will be of considerable use to us later on.

Lemma 6. Every rud function is a composition of some of the following rud functions


Proof. Let Cdenote the class of all functions obtainable from F,, ..., Fg by composition. We must show thatfrud + f€ C.

For each €-formula cp(xl, ..., xn), set

By induction on cp, we show that for all cp, t, E C. (The required result will then be proved using this fact.)

(a) cp(x)-xiExj, 1 < i < j < n .

Write F,(y) for F3(x,y). Then t,(u) = ui-' X FL-'(F4(E f l u 2 , u"-j)) so t, E c.

(b) Let cpI(x), ..., cp,(x) be such that tvl, ..., t

Let p(x) be any propositional combination of cpl, ..., qp.

E C . PP

Since x - y , x U y (= U {xJ}), x f3 y (= x - (x - y ) ) € C, we clearly have t, E c.

(c) Let &,x) be such that tz E C. Let ~ ( x ) - 3y?(y,x) or VyF(y,x).

Clearly, t=~, ,~(u) = dom (t,(u)) and tvYlp(u) = u n - dom(un - tF(u)). So in either case, t, E C.

(d) v(X) xi = Xj .

132 K.J. DEVLJN

Let O(y,x) =y E x j -y E x i . By (a), (b), t, E e. But look, k=(u,E) q[x] iff (VY E U u ) ~ ~ ~ u " ( " u ) , E ) ~ t Y 7 x l l .

Hence t,(u) = u n n tvy,(u U (Uu)), so t, E C?, by (c).

(e) c p ( x ) z x j € x i , I<j<z'<n.

Let J / (y ,z ,x )=y € z h y = x j ~ z = x i . B y ( a ) , ( b ) , ( d ) , t J I E (?But d x ) - ~ y w ( Y , z , x ) , so by (c), tp E e.

Hence, for any €-formula cp, tV E e. I f f : V n + V , definef* : V + V by f * ( u ) = f n u n . Using our above result,

we prove by induction on the rud definition off, that f rud -+ f * E C . This easily implies the required result.

(b) f ( ~ ) = xj ~ x i .

f * (u )=f"u" = { x - y ~ x , y ~ u } . Letcp(z,x,y)-z EX - y . LetF(u) = t,(u U ( U u ) ) n ( U u X u 2 ) = { ( z , x , y ) ( x , y E u ~ z E x - y } . T h e n f * ( u > = ~ ~ ( ~ ( u > , u ~ ) E e, since t, E e.

Let G(u) = U;=,gf(u), H(u)=h*(Uf=lgT(u)) = h*(G(u)), and K ( u ) = U" U C(u) U H(u). By hypothesis, G, H, K E C . By Lemma 1, let cp(y,x) be an €-formula equivalent to the formula

Clearly, f*(u)=F8(([ t , (K(u))] n [ H ( u ) X u n ] ) , u n ) E e.


LetG(u)= { ( z , y , ~ ) ( ( 3 u E y ) [ z E g ( u , ~ ) ] ~ x E u } . A s a b o v e G E C . But f * ( u ) = F ~ ( G ( ~ ) , u”+’) E e.

Hence f rud +f* E C, for all f .

Finally, let f be rud. We show that f E C?. Setf((z)) = f ( a ) , T(y) = @ i n all other cases. ThusTis rud. So by the above,

7* E C?. Let P ( x ) = {(x)}. Thus P E e. But look, f (x ) = uu {{ f(x)}} =

U U ( ~ { ( X ) } } = U U F ~ ( ~ ( P ( X ) ) , P ( X ) ) E e.

As an immediate corollary of Lemmas 3 and 6 we have:

Lemma 7 . Let A C V and define F9 by F9(x, y ) = A n x. Every function rud in A may be expressed as a composition of some of the (rud in A ) functions Fo, ..., F9.

We shall make immediate use of Lemma 7 in investigating the logical com- z plexity of the predicates kMn far suitable M. We assume, once and for all,

that we have a futed arithmetization of our language.

Lemma 8. bgo is uniformly Ey for transitive, rud closed M = (M, A > .

Proof. Let d: be the language consisting of: (i) variables wi, i E w. (ii) function symbols (binary) f o , ...,f9.

We shall assume we have a futed arithmetization of %. We also assume that the reader understands what is meant by a “term” of L. Henceforth, let M = ( M , A ) be arbitrary, transitive, and rud closed.

We first define precisely how d: is to be interpreted in M. Let Q be the set of functionsp mapping a finite subset of {wil i E a} into

M. We may clearly assume Q is rud. Let C be the (rud) function which to each term r of d: assigns the set of all component terms of r , including variables. Let Vbl, be the rud predicate defining the set {wiI i E w } . Let P be the

134 K. J . DEVLIN

predicate

ThusP is rud in A . We may now define the interpretation of a term T of d: at a “point” p E Q by

y = P [ p ] - “T is an L-term” A p E c A gg[P(C(T), g , p ) A g(7) = y ]

Hence the function

if 7 is an &term and p E Q

otherwise

is (uniformly) ?2? (for transitive, rud closed M). Since M is rud closed we can use the above result to define kE0 as an M-predicate.

Let cp E Frnl‘o. By Lemma 2, pM is rud in A . Hence the function r defined by

I , if ( ~ ~ 1 x 1

i 0, otherwise Iyx) =

is rud in A . So, by Lemma 7, we may assume r = T ~ , where T is a term of L, under the above interpretation (i.e. with Fi interpretingfi for each i). In fact, we may clearly pick a recursive function u mapping FmlEo into the terms of d: so that whenever cp E Fml’o, pM[x] - [ U ~ ) ] ~ [ X ] = 1. But by our above result, this implies that kzo is (uniformly) ?2 (for transitive, rud closed M).

As an immediate consequence of this result, we have

Lemma 9. Let n 2 1. Then &‘ is uniformly ?2: for transitive,’ rud closed M = ( M . A ) .

We conclude this section with a few miscellaneous results of use later. The first two are technical, and will often be used without mention.


Lemma 10. Let M = ( M , A ) be rud closed. If R C M is E,(M), there is a Xo(M) relation P such that R(x) - 3x1Vx23x3 ... Q,x,P(x,xI, ..., x,).

Proof. SupposeR(x) - FM 30~V02303 ... Q,o, (P(u ,o~, ..., v,)[x], where cp is a Co-formula. Using the rud functions (-, ..., -), (-)?, ..., we can easily obtain, via Lemma 1, a Co-formula J / such that R(x) c--f bM 3ulVu2 ... Q,u,$(u,ul , ..., u,)[x]. ThenR(x) - 3x,Vx, ... Qnxn [bM $ [ x , x ~ , ..., x,]], as required.

Lemma 1 1. Let M = (M, A ) be rud closed. If R C M is E,(M), there is a single element p E M such that R is E;( ( p } ) .

Proof. If R is Ey({pl, ..., p , } ) , thenR is also CF(((pl, ..., p , ) } ) .

Let M = (M, A ), n 2 0. Write X < x , M iff X C M and for every En formula cp and every x € X ,

~ ( x , A nX) CP [XI iff b~ CP [XI .

Clearly, i fX, M are transitive and X CM, we always have X < x o M. And for n>O,wehaveX<EnMiffXCMandforeveryPEEF(X),P#O-+ Pnxf0.

Recall that if ( X , E) satisfies the axiom of extensionality, there is a unique isomorphism n : (X,E) 1 (W,E), where W is a unique transitive set. Further- more, if Z C X is transitive, then n 1 Z = id r Z. In fact, n is defined by €-

induction thus: n ( x ) = {n(y) I y E x n X } for each x EX. The next result is of considerable importance.

Lemma 12. Let M be transitive and rud closed. Let X < x : , M. Then ( X , A n X ) satisfies the axiom of extensionality and is rud closed. Let n : ( X , A n X ) S ( W , B ) , where Wis t rans i t i ve .Le t f :Mn+Mberud inA . Then for all z EX, n( f(z)) =f(n ( z ) ) .

Proof. Since M is transitive, M satisfies the axiom of extensionality. Hence as X < q M, so does ( X , A n X ) . Similarly, by Lemma 5, ( X , A n X ) is rud closed. Hence, in particular, z G X +f( z ) E X for f : M n +M rud in A . By induction on the (rud in A ) definition off, ~(f(z)) =f(n(%)) for each I EX.

136 K. J. DEVLIN

$3. Admissible sets

Let M = ( M , A ) be non-empty and transitive. We say M is admissible iff M is rud closed and satisfies the X o-Replacement Axiom: for all Co formulas cp and all u E M , l=M [V~3ycp + V U ~ V ( V X Eu) (3y Eu) cp] [ a ] .

In case A = !J in the above, we call M an admissible set. More generally, M is X,-admissible iff M is rud closed and satisfies the (analogous) C,-Repluce- ment Axiom. Likewise a Enadmissible set. We prove below that M is admissible iff M is C , -admissible. All our results extend trivially from admissibility t o C,-admissibility, with “En” everywhere replacing XI, etc.

Roughly speaking, an admissible set (or structure) behaves like the universe as far as El concepts are concerned. We give a few elementary results which set the tone for the rest of this exposition.

Convention: For the whole of this paper, we shall adopt the following abuse of notation. Suppose M is a structure, cp(v) is a formula, and x EM. We shall write FM ~ ( x ) rather than FM ~ ( V ) [ X ] . Clearly, this is purely a notational convenience.

Firstly, we give the promised “stronger” form of the admissibility definition.

Lemma 13 ( C -Replacement). Let M be admissible, and let cp be a X , -formula, a E M . Then

Proof. Let $ be a E,,-formula such that bM cp(x,y, a) t--, 32$(x ,y , z ,U) . Then

Convention: The essentially superfluous role played by a in the above theorem


leads us to extend our previous convention slightly by allowing formulas to contain members of M as parameters. Again, this is clearly an avoidable convenience.

Lemma 14. Let M be admissible. I fR (x , y ) is Xc,(M), so is (Vy €z)R(x,y).

Proof. Let cp be a Eo-formula with parameters from (“w.p.f.”) M such that W , y ) - kM ~ w P ( ~ , Y ) - Then

So by Z0-Replacement,

which is z 1 ( M ) .

Lemma 15 (A l-Comprehension). Let M be admissible, P E A1(M). Then U E M + P n U E M .

Proof. Let cp, $ be Eo-formulas w.p.f. M such that

So by Xl-Replacement there is u E M such that

138 K . J . DEVLIN

But M is rud closed (SO satisfies what might be called the Eo-Comprehension Axiom), and therefore we conclude t h a t P n u = { z E u I k M ( 3 y E u ) $ ( y , z ) } E M .

The next result has nothing specificalIy to do with admissibility, but is of considerable value. Let f : C M -+ M mean that f : X .+ M for some X C M.

Lemma 16. Let M bearbitray, f : C M - + M b e X,(M).Ifdom(f)is n , ( M ) , then in fact fand dom (f) are A (M).

Proof.

It was necessary to state the above result explicitly because we shall frequently have to deal with functions which, though definable, are not total functions. A particular case of the above theorem would of course occur when dom(f) EM, (whence dom(f) is Z,(M)).

As usual, we shall use the notatiolif(x) =g(x) for partial functions, with its usual meaning(i.e. f(s) is defined iffg(x) is defined, in which case f(x> = 'dx)).

Lemma 17. Let M be admissible, f : C M + M be XI (M). If u E M and u C dom (f), then f " u E M .

Proof. Since M is rud closed andf"u = ran(fr u) , it suffices to prove that fl'uEM.Now,asuEM,fI'uisA1(M)byLemma 16.Letcp(x,y)bea X -formula w.p.f. M such that f(x) = y +-+ kM cp(x,y).

Then kM Vx 3y [(x E u A cp(x,y)) V ( x e u ) ] , so byX I-Replacement there is u E M such that kM (Vx E u ) ( 3 y E u ) cp(x,y). Hence f r u C u X u. So, by A -Cornprehension, f 1 II = (f u ) n (u X u ) E M .

Theorem 18 (Recursion Theorem). Let M be admissible. Let h : Mn+' -+A4 bea L1(M)fiinction such thatforallx E M , { (z ,$) l z E h ( y , x ) } is weN-


founded. Let G = Mn+2 + M be El (M). Then there is a unique X (M) function F such that

(i) ( y , x ) E dom(F) f3 {(z,x)lz E h ( y , x ) } C dom(F) (4 F(y ,x) -G(y ,x , (F(z ,x) Iz Eh(y , x ) ) ) .

Proof. Let @ be the predicate

@(f,x) ~ - - f “f i s a function"^ (Vy E dom(f))(Vz E h ( y , x ) ) ( z E dom(f))

By Lemma 16, h , G are A1(M), so @ is Al(M).

XI (M) predicate F by (using notation which will later be justified) Let cp be a E1-formula w.p.f.M such that @ ( f , x ) - p(f.x). Define a

We verify (i) for this F. Suppose first that ( y , x ) E dom (F) . Then, by definition, 3f[@(f,x) A y E dom (f)] . By definition of @, for such anf we must have(Vz Eh(y,x))(zEdom(f)). HencezEh(y,x)+(z,x)Edom(F). Now suppose that z E h ( y , x ) -+ ( z , x ) E dom (F) . Note that asM is transitive, h ( y , x ) C M . By our supposition,

so by Xl-Replacement,

Pick such a u . As @ is Al(M), by Al-Comprehension we see that w = u n { f l @(’,x)} EM. Hence u w EM. It is easily seen that @(UW,X). Noting that h ( y , x ) C dom(Uw), note that U w /‘h(y,x) EM. Se t f= U w 1 h ( y , x ) U

{(G(y,x, U w f h ( y , x ) ) , y ) } . Clearly, @ ( f , x ) , so ( y , x ) E dom(F). Hence (i) holds for this F.

We now show that F is a function and is unique. By (i), dom (F) is already uniquely determined, so for both of these it suffices to prove the following:

140 K. J. DEVLIN

To this end, suppose not. Then P = { y Iy E dom(f) n dom (f’) A f(y) # f ’ (y )} # 0. Let y o be an h-minimal element ofP . Since y o E P , f ( y o ) # f ‘ ( yo ) . But @ ( f , ~ ) , @ ( f ’,x), so clearlyf(yO) =f’(yO) by the h-minimality o f y o EP. This contradiction suffices (and thus justifies our notation somewhat).

Finally, it is trivial to note that (ii) must hold, virtually by definition.

In view of the many set theoretic concepts defined by a recursion of the above type, it is clear that admissible sets play an important role in set theory.

Say M is strongly admissible iff M is non-empty, transitive, rud closed, and satisfies the Strong CO-Replacement Axiom: for all Co formulas cp w.p.f. M, kM Vu3u(VxEu)[3ycp(x,y)’(3y Eu)cp(x,y)]. (Clearly, such anM will also satisfy the “Strong El-Replacement Axiom”.)

Strongly admissible structures M are (for reasons to be indicated later) also called rion-projectible admissible structures. The difference between admissibility and strong admissibility is closely connected with the difference between En predicates and An predicates, which is in turn closely connected with the difference between a function being partial and total. We shall have more to say on this matter later.

$4. The Jensen hierarchy

Let X be a set. The rudimentary closure of X is the smallest set Y 3 X such that Y is rud closed.

Lemma 19. If U is transitive, so is its rud closure.

Proof. Let W be the rud closure of U . Since rud functions are closed under composition, we clearly have W = {f(x)l x E U A f is rud}. An easy induction on the rud definition of any rudfshows that x E U + TC (f(x)) C W . Hence W is transitive. (TC denotes the transitive closure function.)

For U transitive, let rud(li) = the rud closure of U U {U] . Of crucial importance is:


Lemma 20. Let U be transitive. Then 9(u> n rud(U) = Zu(U).

Proof. Clearly, P(U) n Z;,(UU { U } ) = Xu(U) , so it suffices to show that

Then, exactly as in the proof of Lemma 2 , X E rud ( U ) (by induction on the Zo definit ionofX).NowletX€P(U)nrud(U). T h e n X i s a X (rud(U))

3 ( u ) n z , ( ~ u { ~ ) ) = P ( ~ ) n n t d ( ~ ) . ~ e t ~ ~ P ( ~ ) n z , ( u u {u}).

subset of 0. By Lemma 1, we may in fact assume thatXisE:'gUd( 8 )(UU { U } ) .

transitive, s o x i s

Also very relevant is:

Lemma 21. There is a rud function S such that whenever U is transitive, S(U) is transitive, U U { U } C S(U)and U,,,Sn(U)= rud(U).

Proof. Set S(U) = ( U U { U } ) U (Ui8,,F;(UU {U})2). The result follows by Lemma 6.

Lemma 22. There is a rud function Wo such that whenever r is a well-ordering of u, Wo (r, u ) is an end-extension of r which well-orders S(u).

Proof. Define iu, j f , j ; by:- iu(x)= t h e l e a s t i s 8 such t h a t ( 3 x 1 , x 2 E u ) [ F i ( x 1 , x 2 ) = x ] jy(x) = the r-least x1 E u such that (3x2 E u)[Fiu~,)(xl,x2) = x] jT(x) = the r-least x2 E u such that Fju(&f(x), x2) = x. Clearly, iu, j r , j ; are rud functions of u ,x .

Define Wo(r,u)= { ( x , y ) [ x , y E u ~ x r y } ~ { ( x , y ) l x ~ u A Y @ U }

u { ( x , y ) l x @ u A ~ @ U A [ iu(x)<iu(y) v iu (x )=

The Jensen hierarchy, (J,I a € OR), is defined as follows:

J, = b J,+l = rud(J,)

J, = u,, J,, if lim (A).

142 K.J. DEVLIN

Lemma 23. (i) Each J, is transitive. (ii) a I p + J, C Jp

(iii) rank (J,) = OR n J, = WLY.

Proof. (i) By Lemma 19. (ii) Immediate.

(iii) By induction: rank(J,+l)= rank(rud(J,))= rank(J,) + w (by an earlier remark, this last step is easily verified).

To facilitate our handling of the hierarchy, we “stratify” the J,’s by defining an auxiliary hierarchy <S,\ a E OR) as follows:

Clearly, the J,’s are just the limit points of this sequence. In fact:

Lemma 24. (i) Each S, is transitive (ii) a LO + S, C Sp

(iii) J, = u v < w , S, = sw,.

Proof. (i) By Lemma 21. (ii) Immediate.

(iii) By induction: Jol+l = rud(J,) = u,,, Sn(J,) = u,,, S n ( S , , ) =

“ n E w SW,+H = s,,+w = Sw(a+lY

Lemma 25. (S,I u < m a ) is uniformly X k for all a.

Proof. Set c p ( f ) ~ ‘ ~ i s a f u n c t i o n ” A d o m ( f ) € O R A f ( O ) = O A

(VuEdom(f))[(succ(u)+f(u)=S(f(u-l))) A

b ( v ) + f ( v ) = u,,.f(.>lI .

Clearly, Thus it suffices to show that for any a, u < wa, the existential quantifier here

is uniformly Xk. And by definition, y = S, +-+ 3f(@(f) A y =f(v)).


can be restricted to J,. In other words, we must show that whenever T < oa, then (S,I v < T) E J, . This is proved by induction on a. For a = 0 it is trivial. For limit a the induction step is immediate. So assume a = p + 1 and that T < wp -+ (S,I v < 7) E Jp. Then, by our above remarks, it is clear that (S,I v < 00) is Zip. So by Lemma 20, (S,I v < u p ) E J,. Thus for all n < o, (S,I v < o p + n ) = (S,I v < U P ) U {(Sm(Jp), UP + m)l m < n } E J,, as J, is rud closed.

Lemma 26. (J,I v < a ) is uniformly Ek for all‘a.

Proof. By an easy induction, (ovl v < a ) is uniformly Ep for all a. Since J, = S,,, The result follows by Lemma 25.

Lemma 27. There are well-orderings <, of the S, suck that:

(i) v1 < v2 -+

(ii) <u+l is an end-extension of <,; (iii) <<J v < oa> is uniformly E+ for all a.

C <% ;

Proof. We use Lemma 22. Set <,, = (4, and by induction:

(i) and (ii) are immediate and (iii) is proved like Lemma 25.

Lemma 28. There are well-orderings < J, of the J, suck that :

(i) a1 < “2 -+ <J, <Ja2 ; (ii) <J,+l is an end-extension of<J,;

(iii) ( < J p l (iv) < J, is uniformly x k; (v) the function pr,(x) = { z 1 z < J, x } is uniformly Z k .

J < a ) is uniformly E la ;

(“pr” stands for “predecessors”of course.)

Proof. Set <,, = <w,. (i)-(iii) are immediate by Lemma 27. For (iv), note simply that x 3 v [ x E S, A y = {z I z <, x)] (and that <, E J,), and use Lemma 27.

++ 3 v ( x <, y ) . Finally, for (v), note tha ty = pr,(x) *

144 K . J . DEVLIN

Lemmas 12 and 26 enable us to prove the following extremely powerful result (due in its original form to Godel, the present version being Jensen’s):

Theorem 29 (Condensation Lemma). Let X < c l J,. Then for some 0 I a, X z Jp.

Proof. L e t X < c l J,. Then by Lemma 12, let n : X % W , where W is transitive. We prove by induction on a that W = Jp for /3 = n”(X n a).

Assume, therefore, that whenever v < a and X u < z1 J,, the unique isomorphism nu o f X V onto a transitive set W’ yields W’ = J,uft(xvnv). Note that, as (J,I v < a) is EP, v E X n a ++ J, EX.

Claim 1 : For all v E X n a, ?(J,) = Jn(,].

A E Z k ( X n J,). Since J, EX, A E X p ( X ) . So, if A # 6, then as X<c, J,,, A n X # 6. But A C J,, so A f’ ( X n J,) # 6.1 Hence by induction hypothesis, n‘ : X n J, J n ~ ~ r ~ x , J v n u ) for some unique n’. But look, J, is an E-end extension of J,, so n maps X n J, isomorphically onto a transitive set also. In other

Jntt(Xnv) - JR(,), by the definition on n, as claimed.

To see this, note first that for v E X n a, X f7 J, <z;] J,. [For, let

words, 71’ = 71 1 X n J,, and n”X n J, = J,r9(xnu). SO, n(J,) = r” (X n Jv) = -

For v < a, define rudx(Jv) = the rud closure of X fl (J,, U {J,}).

Claim 2 : X = UvEXn, rudx(J,).

To establish this claim, note that as X < q J,, X is rud closed, so 3 is obvious. For the converse, let x E X . Then x E J, = U,<,rud(J,), so for some rud functionf, F J a ( 3 v ) ( 3 p E J,)(x =f(p,J,)). ButX<z;, J,, so ( 3 v E X n a ) ( 3 p E J,nX)(x=f(p,J,)). In other words,xE Uu6Xn,rudX(Ju). Hence Claim 2.

Claim 3 : For v E X n a, n“ rud, (J,) = rud(Jn(,)).

To see this, let v E X n a. Suppose first that x E rudx(Ju). Then for some rud functionfand some p E J, n X , x = f ( p , J,). By Lemma 12 and Claim 1, n(x) = f ( n ( p ) , JR(,,)). But p E J, c7 X so n ( p ) E Jn(,). Hence n(x) E rud (JnCv)). This proves C. Conversely, suppose y E rud(J,(,)). Then y E rud (n(J,)), by Claim 1, so for some rud functionf and some p E n(J,), y = f(p, n(J,)). Now, n(J,) = 7r”(JunX), so for some q E J, nX, p = n(q) and we havey =f(n(q) , n(J,)) = n( f (q , J,)) E n“ rud, (J,). Hence 3, and Claim 3 is proved.


Note. I t is easily seen that we may regard the following as part of the statement of Theorem 29: If Y C X is transitive, then r 1 Y = id / Y . And for v E X n a, n(v) 2 v, and for all x E X, r(x) 2 J, X.

By an argument well known to all set theorists, it is easily shown that J = UaEOR J, is a model of ZFC. (In fact, setting<J = UaEOR <J,, <J is a J-definable well-ordering of the entire class J , so J satisfies the Axiom of Choice in a strong way.) Using the Condensation Lemma, an equally well- known argument shows that J k GCH. However, in the next section we will prove (and have already indicated this fact in our preamble) that J = L, so all that the above says is that we can use the Jensen hierarchy in place of the L- hierarchy in order to establish the classical results on the constructible universe.

$5. On the fine structure of the Jensen hierarchy

As mentioned in the introduction, a theory similar to the one following can be developed for the usual Lhierarchy, if desired.

Central in our discussion will be the concept of a “uniformising function” for a relation, which is a sort of “choice function” for a given relation. Speci- fically, a function r is said to uniformise a relation R iff dom ( r ) = dom (R) and for a l lx , 3yR(y,x) -R(r(x),x).

Let M = ( M , A ) , n 2 1. We say M is E:,-unifomzisabZe iff every Zn(M) relation on M is uniformised by a Xn(M) function. A few moments reflection will reveal that En-uniformisability is a very strong condition to demand of an arbitrary structure M, since in the more obvious cases, the definition of a uniformising function for a given relation would appear to increase the logical complexity by one or more quantifier switches. However, it will turn out that for all a, all n 2 1, J, i s Z,-uniformisable. For n = 1, this will be easy to prove, but for n > 1 , the corresponding argument will only work when J, is E:,-l-admissible, so a more indirect approach will be necessary. We shall outline the approach required after we dispose of some of the more easy results. First, E l-uniformisability. The E1(J,) well-ordering of each J, gives

146 K.J. DEVLIN

us this with little effort. in fact, we have a much stronger result, of importance in applications of E l-uniformisability.

Let F be a class of structures M = (M, A ) , n 2 1. Say F is uniformly En- uniformisable if, whenever p is a En-formula w.p.f. n {MI M € F } such that (PM is a relation on M for each M E F , there is a En-formula $ (w.p.f. n {MI M E F } ) such that for each M E F, $M is a function uniformising $1.

Theorem 30. (J,, A ) is El-uniformisable. In fact, the class of all (J,, A ) is uniformly X -uniformisable.

Proof. Let p be a X -formula w.p.f. J, such that [cpb, x)]‘ JaA) is a E relation on J,. By contraction of quantifiers, we can, in a uniform way, find a X o formula J / (w.p.f. J,) such that k ( J , , A ) p b , ~ ) c-, 3z$(z ,y ,x ) . Define g by: g(x)= the<J-least w such that k(J,JI)J/((~)o,(~)l,~). Thengis (uniforndy) X l((J,, A ) ) , since

Set r(x) = (g(x))l. Then r is (unifcrmly) E l((J,, A ) ) and clearly uniformises [ d Y , x>l( J,A).

Remark. We call the above construction the canonical X l-uniformisation procedure. Observe that if R(y ,x ) is a E l((J,, A )) predicate, then the canonical Xl-unifomisation o f R is a function whose El((J,,A)) definition involves only those parameters which occur in the definition of R.

Let us take a little time off to examine the above construction more closely. SupposeR(y,x) is a given XI relation, sayR(y,x) * 3zP(z ,y ,x ) , where P is X o . To obtain the Z; uniformisation of R , we first obtain a Z;

uniformisation of the T o relation {(w,x)( P((W)~, ( W ) ~ , X ) } , and then simply pick out the requisite component of the result as our required function. And since < is a X well-order of J, the result is also E 1 . However, returning now to the notation of Theorem 30, we see that, if we try to extend this procedure to the case n > 1, we cannot conclude that the function g is En, the problem being the last conjunct in the explicit definition of g. Let \k(w,x)


denote the predicate [l $ ( ( W ) ~ , ( W ) ~ , X ) ] ‘ ~ “ ~ ’ . Forn = 1, there was no problem, since if \k(w,x) is zo, so is (Vw E t)\k(w,x). However, for n > 1, \k(w,x) is I:,-l, and we can only conclude that (Vw E t)\k(w,x) is I:,-, if (J,,A) is Z,-l-admissible. Otherwise it is merely II, of course, and so the resulting uniformisation of the original X, relation turns out to be En+,. So, in order to establish the general I:,-uniformisation lemma, it is not altogether unreasonable to try and “reduce” all En relations on an arbitrary J, to C, relations on some I:,-l-admissible Jp, for which we have a I:,-uniformisation procedure. In practice, it will turn out that this hint is slightly off target, but in its general tone it is worth bearing in mind.

Closely connected with Z,-uniformisability is the notion of a “2, Skolem function”. Let M = ( M , A ) be transitive and rud closed. By a 2, Skolem function for M we mean a &(M) function h with dom (h) C w X M , such that for some p EM, h is Z:({p}), and wheneverP E Zr({x ,p} ) for some x EM, then 3yP(y) -+ (3 E w)P(h(i,x)). (With h, p as above, we say that p is a good parameter for h.) Note that C, Skolem functions need not be (and in general are not) total! As far as existence of I:, Skolem functions is concerned, we can get away with slightly less than might first appear. In fact:

Lemma 31. Let M = ( M , A ) be transitive and rud closed. Let h be a Z r ( { p } ) function with dom (h) C w X M. Suppose that whenever P E I:,,”({x}) for some x E M, then 3yP(y) -+ (3i E w) P(h (i, x)). Then M has a X , Skolem finction.

Proof. Set i ( i ,x)=h(i , (x,p)). It is easily seen that h“ is a C, Skolem function for M.

Note that in the above, if h is actually I: :, then h” = h. This is used in establishing the following result:

Lemma 32. If (J, , A ) is amenable, then it has a E Skolem function. In fact, there is a C Skolem function for all amenable (J,, A ) .

Proof. Let ( p i [ i < w ) be a recursive enumeration of Fml’l. Let (J,,A) be

canonical I: l-uniformisation of the I::Jae) relation {(y, i ,x)l k $ ~ e ) p i ~ . x ] } .

for (Ja, A ) which is uniformly I: :Jff$”

amenable. By Lemma 9, i=( I; JLJA) is (uniformly) E:J,A’. Let h = haJ be the

148 K. J. DEVLIN

(By Lemma 30 and the ensuing remark, h.is thus uniformly E i J a P ) for amenable (Ja, A ) .) By the remark following Lemma 3 1, it is clear that h is a El Skolem function for (Ja, A ) .

We refer to h,

By a similar argument, we have:

as the canonical E Skolem finction for (amenable) (J,, A ) .

Lemma 33. If (J,, A ) is amenable and X,-uniformisable, it has a X, Skolem function.

The following lemmas indicate our reason for using the word "Skolem" here.

Lemma 34. Let M be transitive and rud closed, and let h be a I;, Skolem function for M. Then whenever x E M , x E h"(o X {x}) < c 'M.

Proof. Set X = h"(o X {x}). Clearly, x EX. Let P E X F ( X ) , P f @. We must show t h a t P n X # $ . Le tp be agoodparameter forh , andpickyI , ...,y, E X with P E E;({yl, ..., y,}). By definition of X, there are j l , ..., j , E w such that y1 = h(jl ,x), ...,y, = h(j,,x). Since h is Z F ( { p } ) , it follows that PEXF({p,x}). Hence, P# @-+ 3yP(y ) -+(3 iEo)P(h ( i , x ) )+(3yEX)P(y ) .

n

Lemma 35. Let M be a transitive and rud closed, and let h be a E n Skolem function for M. I f X C M is closed under ordered pairs, then X C

h"(U XX)<x M. n

Proof. Set Y = h"(w X X ) . By Lemma 34, X C Y . Let P E E,"( Y ) , P # @. We must show that P n Y # 0. Let p be a good parameter for h , and pick yl , ..., y m E Y w i t h P E X F ( { y l , ..., ym}).Pickjl ,..., j,Ew andxl , ..., x ,EX such t h a t y l = h ( j l , x l ) ,..., y, =h(j , ,x,) .Letx=(xl ,..., x,).Byassump- tion, x EX. But clearly, ash is ZF({p}), P i s then Ey({p,x}), so P f @ +

3yP(y) + (3iE o)P(h(i ,x)) + (3y E Y)P(y ) .

Corollary 36. Let M, h be as above. Let X C M and suppose h"(w X X ) is closed under ordered pairs. Then X C h"(w X X ) i M.

n


Proof. Let Y = h”(w X X ) . Clearly, Y = h”(w X Y ) , so the result follows by the lemma.

Lemma 37 (Godel). There is a bijection @J : OR2 * OR such that @(a,P) 2 a, P for all a,P, and @-’ I wa is uniformly C for all a.

Proof. Define a well-order <* of OR2 by

Let @ : <*

Lemma 38. There is a I: l(J,) map of wa onto (oa)* for all a.

OR. By induction on a, @-’ 1 wa is C p (uniformly).

Proof. Let Q = (a1 @(O, a) = a}. Then Q is closed and unbounded in OR. In fact, Q = (a1 @ : a2 ++a}, so wa E Q + wa = a. We prove the lemma by induction on a. Assume it is true for all v < a. Case 1 : wa E Q. Then W’ 1 a suffices.

Case2: a=P + 1. I f P = 0, then w a = w E Q , so we are done by Case 1. Hence we may assume 0 2 1. Then clearly, there is a X1(J,) map j : wa ++ u p . By hypothesis, there is a E l(Jp) map of wp onto I: l(Jp) map g of v, y E wa, define

so there is certainly a one-one into wb. Theng E rud (Jp) = J,, so for

Then f is E1(Ja) and f maps ( ~ a ) ~ one-one into 06. Now r a n 0 = ran(g) E J,, so if we define h by (for v E wa)

f-’(v), if v E ran (f)

(0,O) , otherwise

we see that h is C1(J,) and h : wa-

i h(v) =

150 K. J. DEVLIN

Case 3 : lim(a) A oa 4 Q. In this case let ( u , T ) = @-'(ma).

onto wa, and is I: l(J,). Pick y < a with v, T < oy. Then @-I 1 wa is a C '(J,) map of o a one-one into oy. And by assumption, there is a map g E J, mapping (or12 one-one into wy. SO, settingf((L, 6 )) = g((g(@-'(L)), g(W1(0)))), I , 0 < wa, we see that f i s a X l(J,) map of ( 0 ~ ) ~ one-one into d = g " ( g " ~ ) ~ . But d E J,, so we can define a X1(J,) map h on wa by

T h u s v , T < w a . S e t c = {z l z<*(v ,~ )} (€J , ) .Thus@} c m a p s c one-one

f - ' ( ~ ) , if e ~d

(0, O), otherwise. I h ( 0 ) =

onto Clearly, h = o a --+ ( ~ a ) ~ . The lemma is proved.

Using this lemma, we may now establish the following important result:

Theorem 39. There is a E l(J,) map of oa onto J, for all a.

Proof. Let f : oa% ment of J, for which such an fexists. Define f o , f 1 by demanding that f ( v ) = (fo(v),fl(v)) for all v E oa. By induction, definef, : oa % (ma)" thus: fo = id 1 ~ a ; f , + ~ ( v ) = ( f o ( v ) , f , .fl(v)). Hence eachf, is Z k ( { p } ) . Let h = h,, the canonical C l Skolem function for J,. Set X=h"(wX

be Ep({p}), wherep E J, is the <J-least ele-

(wax { P I ) ) .

Claim 1. X is closed under ordered pairs.

To see th i s , le ty l ,y2 EX, sayyl =h( j l , (v l , p ) ) , y2 =h( I2 , ( v2 ,p ) ) . Let ( v l , v 2 ) = f2(7). Then { ( y 1 , y 2 ) } is a E k ( { ~ , p } ) predicate, so by definition of h , ( y 1 , y 2 ) E X , asclaimed.

So by Corollary 36, X<z, J,. By the condensation lemma, let n : X z Jp, 8 5 a. Since wa c X , we clearly have 8 = a here.

Claim 2. For all i E w , x E X , n(h(i,x)) = h(i, n(x)) .


To see this, observe first that as h is X p , there is a rud function H such that y = h( i ,x ) * ( 3 t E J , ) [H( t , i , x , y ) = 11. Now let i € w , x EX. Since X < x J,, y = h( i ,x ) E X (if defined). Thus, by the above, since x,y E X <z], J,, (3t EX)[H( t , i , x ,y ) = 11. By Lemma 12, therefore, (3tEX)[H(n(t) , i ,n(x) , n(y))= l].Sincen”X= J,, thiscan be rewritten as(3tEJa)[H(t,i,n(x),n(y)) = 11. Thus n(y ) = h(i,n(x)), as claimed.

1

Now, f C ( ~ a ) ~ , so as n 1 o a = id 1 oa, n’y= f. And by isomorphism, n‘yis Ep({n(p)} ) . So as n(p) IJ p , the choice of p shows that n(p) = p . So, by Claim 2, if i E a, v E wa, n(h(i, (v , p)) ) = h( i , (u, p) ) , which is to say n t X = i d r X . T h u s X = J , .

Now define i : ( w ~ r ) ~ + J, by setting

y , if (3 t E S,) [H(t , i , (v, p ) , y ) 1 1

8, otherwise. I i ( i , U , 7 ) =

Thenzis El(J,), andclearlyi”(oa)3 =h“(oX(oaX{p}))=X=J, . Therefore, f3 is as required by the theorem.

Observe that in Lemmas 38,39, the maps constructed generally have parameters in their definitions. Note also that, being total, these maps are in fact A,(J,).

J, is an admissible set. Recalling the results of $3 , we now investigate those ordinals cy for which

Let us call an ordinal a admissible iff a = o p and Jp is an admissible set.

Theorem 40. wa is admissible iff there is no E l(J,) map of any y < wa cofinally into we. (Note that such a map, having domain y E J,, would in fact

be Al(J,).)

Proof. (+). Let y < wa and suppose f : y + wa is El(J,). Then ( V i E y)(3( E oa)(f(.$) = (). If J, is admissible, then by X l-Replacement, ( 3 ~ € w a ) ( ~ i E ~ ) ( 3 ( € ~ ) ( f ( ~ ) = ( ) , sofisnotcofinal inoa.

(+). Assume wa is not admissible. If cy = f l + 1, then the Z1(J,) map ((wfl t n , n > l n Eo} maps o cofinally into wa, so we are done. Assume then that lim (a). Since J, is not admissible, there must be a E I(J,) relation R and a ~1 E J, such that (Vx E u) (3y)R(x,y) but for all z E J,,

152 K. J . DEVLIN

l ( V x E u ) (3y E z ) R ( x , y ) . Take y < a with u E J,. By Theorem 39, let f be a C1(J,) map of wy onto J,. Thus f E J,, and u Cf"oy. Define g : o y -+ oa

the least T such that ( 3 y E S,)R( f ( v ) , y ) , if f(v) E u

0, if f ( v ) $ u .

Theng is a C1(J,) map of o y cofinally into wa.

Recalling our discussion at the end of 8 3 , let us call an ordinal a strongly admissible (or non-projectible admissible) iff a = 00 and Jp is strongly admissible. Imitating the proof of Theorem 40, we have:

Theorem 41. wa is strongly admissible iff there is no Z ,(Jay> map of a bounded subset of wa cofinally into war.

The above two results illustrate our earlier remark concerning the difference between a function being partial and being total, and the corresponding difference between a predicate being En and being A n . The next two results, which strengthen the last two, and are also due to Kripke and Platek, also highlight this distinction.

Theorem 42. The following are equivalent: (i) wci is admissible.

(ii) (J,,A)isamenable forallA E A1(J,). (iii) There is no (J,) function mapping a y < oa onto J, ~

(Of course, any such function would in fact be A I(J,).)

Proof. (i) + (ii). By Lemma 15 (A , -Comprehension). (ii) -+ (iii). Assume (ii) h l(iii). Let y < wa, and let f : y - J, be C1(J,). Thenfis A,(J,),sod= {v l v$ f(v))is A,(J,).Thus by ( i i ) , d=d nyEJ,. So, d = f ( v ) for some v < y, so v E f ( v ) ++ v E d - v 4 f (v ) , a contradiction. (iii) -+ (i). Assume (iii) A l ( i ) . If a = 0 + 1, we can easily construct a Zl(J,) map of wp onto wa, so Theorem 39 yields the required contradiction. Assume lim (a). By Theorem 40, there must be a T < wa! and a Z1(Ja) mapfof T cofinally into ma. Let f be Ek((p}). Pick y < a with T, p E J,. Let h = h, be the canonical El Skolem function for J,. Set X = h"(w X J,). As J, is closed

onto


under ordered pairs, Lemma 35 tells us that X < z1 J,. Let n : X Jp. Thus IT r J, = id F J,. By an argument as in Theorem 39, n X = id X, so X = Jp. Now, f isXF({p}) a n d p € X - ( q J,, soXisclosedunderf. B u t T C X a n d so f ’’7 C X , which means, since f ”7 is cofinal in wa and X = Jp is transitive, that wa C J p . Thus@ = a, a n d X = J,. Define a Zl(J,) map i : w X T X J , + J, as follows. Let H be a X k relation such tha ty = k( i ,x ) - (3t€J,)H(t , i ,x ,y) . Set

y , if (3t E Sf , v ) )H( t , i , x , y )

0, otherwise. i K(i, v ,x) =

Then h is total on w X T X J,, and i ” (w X 7 X {x}) = kf’(w X {x}) for any x , as f “T is cofinal in we. Hence h”(o X T X J,) = X = J,. By Theorem 39 there isg E J,, g : o y w X 7 X J,. Then K e g is a Z1(J,) map of wy onto J,, contrary to (iii).

Theorem 43. The following are equivalent: (i) wa i s strongly admissible.

(ii) (J, , A ) is amenable for all A E C,(J,). (iii) Tkere is no X l(J,) function mapping a bounded subset of wa onto J,.

Proof. (i) + (ii) + (iii). Similar to the above. (iii) + (i). Assume (iii) n l ( i ) and proceed much as before. So, we assume lim (a), f i s (by Theorem 41) a E l(J,) map of some a C r < wa cofinally into ma, f E Zp({p}), and T < oy, p E J,, y < a. As before, if k = k , and X = k”(o X J,), then X= J,. Now, since we do not need to bother about functions being total, we can easily contradict (iii). By Theorem 39, let g E J,, g : wy- w X J,. Setf(v) =h(g(v)). ThenSis a Z1(Ja) map of a subset of wy onto J,.

Note that an immediate corollary of Theorem 42 is:

Theorem 44. If K is a cardinal, then K is an admissible ordinal.

Using admissibility theory, we can give a quick proof that L = UaEOR J,.

Theorem 45. I f MY is admissible, then J, = L, ,.

154 K.J . DEVLIN

Proof. If a = 1, then J , = L, = the hereditarily finite sets. Assume a> 1. Thus o E J,. Since J, is admissible, the recursion theorem tells us that rud(x) =

Un<,Sn(x) is C1(J,). But if u is transitive, then C,(u) = P ( u ) n rud(u). Hence the map Ly * X,(Ly) = L,+l is E1(J , ) (y< war). So, by the recur-’ sion theorem again, we see that (L,l v < war) is uv.,w, L, C J,. For the converse inclusion, it suffices to show that L,, is admissible. (For then, by the recursion theorem, (S, I v < oar) is Cl(L,J, so J,=U,<,,S,CL,,.)LetR beXO(L,,),xEL,,,andassume (Vz Ex) 3yR(y,z). Since (L,I v < war) is El(Ja), we may define a X1(J,) predicate R’ by R’(v , z ) - z E x A (3y E L,)R(y,z). Since J, is admissible, there i sT<wa with(Vz E:x) (3v<~)R’(v ,z ) . Hence(Vz€x)(3yEL7)R(y,z). So as L, E L,,, L,, satisfies the X O-Replacement axiom. Since lim (a), it follows easily that L,, is admissible.

- .- l(Ju). Hence L,, =

Let a,n 2 0. The E,-projectum of a, p:, is the largest p <_a such that (J,,A) is amenable for allA E C,(J,) fl 3 ( J )

Roughly speaking, our reason for introducing the X, projectum is this. We have seen that, for example, we can reasonably handle C,(J,) predicates when J, is C,-admissible. This is because Cn-admissibility is a sort of “hardness” condition on J, for C, predicates. For, if we take an arbitrary J,, it may be “soft” for Xn(J,) predicates; we may, for instance, find Zn(J,) subsets of members of J, which are not themselves members of J,, or even E,(J,) functions which project a subset of a member of J, onto all of J,. But if J, is C,-admissible, none of these situations can arise. Thus, we try to isolate that part of J, which is “hard” for E,(J,) predicates, a sort of “En- admissible core” of J,. One natural way of formalising these ideas is provided by the Z,l projectum. Clearly, J is a reasonable interpretation of the notion of a “C,-hard core” of J,. We sh%l eventually give two characterisations of the x,-projectuni which make it appear even more reasonable - if not inevitable. One of these is that p z is the smallest p < a for which there is a X,(J,) map of a subset of u p onto J,. Then, since we clearly have, for wa admissible, that oa is strongly admissible iff p: = a, we obtain some justification for our alternative name of “non-projectible admissible” for strong admissibility.

the notion of the C , -projecturn of an ordinal.

P ’

P”

I t is convenient, at this point, for us to define an obvious generalisation of

Let (J , ,A) be amenable. The C,-projectum of (J, ,A), p z p , is the largest


p 5 a such that (J,,B) is amenable for all B E C,((J,,A)). Note that by Theorem 43, We shall make strong use of the X,-projectum in proving that every J, is

Z,-unifomisable, all n 2 1. Since most of the following lemmas are directed towards this goal, it is worth indicating briefly our strategy.

We already know that (J,,A) is Zl-uniformisable for all (J,,A). What we shall do is attempt to “reduce” Zn(J,) predicates to C1((Jpn,-4)) predicates for someA C J need to have at our disposal a Cn(J,) map of a subset (at least) of J onto J,. Thus, what we shall do is to simultaneously prove, by induction on n , a, the following two propositions:

(P 1) J, is Z,+l-uniformisable

(P 2) There is a Cn(J,) map of a subset of o p : onto J,.

is always strongly admissible.

which is itself En(J,). To carry out this re%uction, we G 4

The proof of (P 1) goes roughly as follows. Let R be a Cn+l(Ja) predicate J, be C (J ). Now, f -’ is a T,(J,) relation, so by on J,. Let f : C u p :

assuming x,-uniformisability, f-’cg be ‘‘shrunk” to a X,(J,) map of J, into up: . This reducesR to a X‘n+l(J,) predicate R’ on J p n . NOW find a C,(J,) predicateA C J by a C1((Jpn,A)) function, and then reverse the proced&e to recover a Cn+l(J,) uiformising function for R. There is one doubtful point in the above outlbe. Can we in fact find a setA as required. That we can has to be proved as we proceed, so we shall in fact simultaneously prove three propositions, (P I), (P 2 ) , and a proposition (P 3 ) to be formulated precisely later.

such that R’ is in fact XI((Jpn,&). UnifomiseR’ P:

Lemma 46. Let n 2 1, and assume J, is X,-unijiormisable. Let y 5 (Y be the least ordinal such that ;P(wy) (I Xn(J,) a subset of o y onto J,.

Proof. By Lemma 33, J, has a C n Skolem function, h. Let h be Z::((p)) . We may assume p is the <J-least element of J, for which such an h exists. Let a C oy, a E z,(Jg), a $ J,. Let q be the <J-least element of J, such that a E z+({q}). Define h by z ( i , x ) = h(i , (x ,p ,q)) . I t is easily seen that h is a C , Skolem function for J, and that ( p , q ) is a good parameter for x. Set X = h “ ( o XJy).Now, the re i saXI ( Jy )mapg : wy*J,, s o h - g i s a Z,(J,) map of a subset of wy onto X . Hence it suffices to show that X = J,.

J,. Then there is a X,(J,) map of

-

156 K.J. DEVLIN

Clearly,X<zn J,. Let n : X E J,,p <a. Then n 1 J, = i d 1 J,, so in particular, n"4 = a . Also n"a is En0({n(q)}) . But look, this implies that a = a"a E

Jp+l. Hence we must have /3 = a (and here we have used our hypothesis that P(wy) n Zn(J,) Q J,!). Thus, in particular, a = n"a is Z?({n(q)]), so by the choice of 4 , we see that r(q) = 4 . Again, it is easy to see that h' = a .h - n-' is a Z>({n(p)} ) En Skolem function for J,, so by choice o fp , n ( p ) = p . But then h , h' are both defined by the same X n formula (with parameter p ) in J,, so h = h'. It follows immediately that a * h = i , of course. So for i E a, x E J,, n-h ( i , x ) =-h *n(i ,x) =h(i,x). Thus n 1 X = id 1 X , a n d X = J,.

J

Lemma 46 plays a direct part in the proof of (P 1)-(P 3). The next lemma, however, is only used during the proof of the lemma which follows it, and may, on first sight, appear somewhat uninspiring.

Lemma47.Let(J,,A)beamenable,p=p~JI. I f B C J , isXl((Ja,A)), then E,((Jp,B)) C Z2((Ja,A)).

Proof. Case 1. There is a Zl((J , A ) ) map of some y < wp cofinally into wa.

for eachx E J x E J p . ThusB IS A,(<J,,A)). And since B(x )o (3vEy)B ' ( (v ,x ) ) , E l ( ( Jp ,B)) C E,((Jp,B')). Thus, we need only prove that Xl((Jp,B') C

Z2((Ja,A)). It clearly suffices to prove that EO((Jp,B')) C E2((J,,A)). Let R be Zo((J ,B')). Thus R is rud in B' and some parameterp E J,. By choice of p , ( Jp ,B ) is amenable, so by Lemmas 3 and 4, there is a Eo(Jp) predicate P and functions fl, ...,fm+k, rud in parameter p , such that R(x) - P(x,fl(x), ...,f,(X),B'nf,+,(x), ..., B'nf,+,(x)). HenceR(x)- 3 ~ 1 , ..., 3ykIv1=B'nfm+l(x)A . . . A yk =B'nf,+,(x)AP(x,f,(x),... ..., f , (x ) , y parameterp, so it suffices to show that the function b(u) = B' n u is E2((J,,A)). It is in fact n,((J,,A)), because: y = b(u) t--f Vx [x Ey * x E u A B'(x)] , aud B' is Al((J,, A ) ) . Case 2. Otherwise.

this reduces, by the amenability of (Jp,B), to proving that the function b(u) = B n u is Z2((Ja, A ) ) on J,. Now, we clearly have

or_ Let g be such a map, and let B be Eo((J,, A ) ) such that B(x) t--, 3zB(z,x)

Define B ' by B'((v,x)) e ( 3 z ESg(v))B(~ ,x) , for v €7, p ;

e

..., y k ) ] . Now P is certainly L o(Ja), and f l , ..., fm+k are rud in

As before, we must show that Zo((Jp,B)) C Z2((J,,A)). Again as before,


y = b(u) * (Vx E y ) (x E u A B(x)) A (vx E u) (B(x) + X E y )

Now, the second conjunct here is Ill((J,,A)). We show that the first conjunct is Zl((J,,A)), which is sufficient. I t reduces to showing that (Vx Ey)B(x) is Xl((J,,A)). But look, we know that Case 1 fails to hold, so this is proved just as in Lemma 14.

The next lemma is the key step involved in proving, by induction, the as yet unformulated (P 3).

Lemma 48. Let (J,,A) be amenable, p = pAA. Suppose there is a Z,((J,,A)) map of a subset of u p onto J,. Then there is a B C J,, B E Zl((J,,A)), suck that Z ,( (J,, B)) = P (J, ) n Z

Proof. L e t u C u p , a n d l e t f : u * J , be Z,((J,,A)).PickpEJ, such that f i s C:Ja’A’({p}) . Let (ql i < w) be a recursive enumeration of Fml’I. Set

(( J, , A ) ) for all n 2 1.

C B = ( ( i , x ) l i € ~ A x E J , A k(Jk,A)pi[x,P]}.

Now, (J,,A) is amenable, and hence rud closed, so by Lemma 9, B E Cl((J,,A)). And of courseB C J,.

Commencing with Lemma 47, an easy induction shows that for all n 2 1,

For the converse, letR(x) be a X,+l((J,,A)) relation on J,, n 2 1. ~ , ( (J , , fw c Z,+I((J,>A)).

Assume, for the sake of argument, that n is even. L e t P be a Cl((J,,A)) relation such that, f o r x E J,, R(x) +-+ 3y1Vy2 ... Vy,P(y,x). Definepby p(z,x) * [z,x E J, A P ( f ( z ) , x)] . By choice off, any x E J, is XiJe’A’( {p , v}) for some v < u p , so by definition of B, pis rud in B and some parameter v < u p . In particular,pis Al((J,,B)). Again, D = dom(f) is rud in B and some parameter T < u p , so D is also A,((J,, B)). But for x E J,, R ( x ) +-+ (32, E D) (Vz2 E D ) ... (Vz, E D) F(z,x), which is thus q(J,m.

We are now ready to formulate (P 3) and prove our promised uniformisa-

Let a, II 2 0. A C , master code for J, is a setA C J ,, A E Z,(J,), such tion theorem.

that whenever m 2 1, C,((JpB,A)) = 9(J ,) n Z,+,,&). Qff

158 K.J . DEVLIN

Theorem 49. Let a, n 2 0. m e n : (P I ) J, is En+l-uniformisable. (P 2 ) There is a Xn(J,) map of a subset of u p : onto J,. (Unless n = 0, when

(P 3 ) J, has a xn master code. it is X l(J,).)

Proof. We prove the theorem (for all n ) by induction on a. For (Y = 0, it is trivial. So assume (Y > 0 and that (P 1)-(P 3) hold (for all n ) for all 0 <a. We prove (P 1)-(P 3) at a by induction on n.

Case I : n = 0. (P 1) is already proved (Theorem 30). (P 2 ) p : = a, so (P 2 ) is already proved (Theorem 39) (P 3) since p: = a , A = 6 is a Eo master code for J,.

Case 2 : n = m t 1, m 2 0. Let p = p: for convenience.

We first prove that p is the least ordinal such that some Xn(Ja) function maps a subset of w p onto J,. To this end, let 6 be the least such ordinal. Suppose first that 6 < p . Then B = {t E w61 g 4 f(g)} is a En(J,) subset of J,, so by definition of p , (J,,B) is amenable. Thus, as 6 < p , B = B n w6 E

J, CI J,. S O B = f(g) for some E w 6 , whence g Ef([) ++ [ E B * $f(E), which is absurd. Hence p 5 6. Suppose p <6. By definition of p , this means that for some En(J,) set B C J , , (J,,B)is not amenable. Since (Jl ,B>must be amenable. 6 > 1. I f 6 = y + 1, then since there is a E l(J,) map of o y onto w6, there i s a Xn(J,) map of a subset of wy onto J,, contrary to the choice of 6. Hence lim (6). I t follows, since (J,,B) is not amenable, that there is T < 6 with B f3 J, @ J,. By induction hypothesis, J, is En-uniformisable. So as 7 < 6. Lemma 46 implies that Y(w7) n En(Ja) C J,. But there is 11 E J, , h : R 0 J, E J,. Hence for some 0 < a, B n J, is Jp-definable. Let p be the least such, and let r be least such that B n J, is Zr(JP). By definition, ( J * , B n J,) is amenable, so if T _ p‘p. By induction hypothesis, there is a X,(Jp) mapg from a subset of wpi onto JP. And since B n J, E JP+l and B n J, 4 J , , t 1 > 6 , or p >_ 6. Hence there is a E,.(Jp) map g’ from a subset of u p ; onto U S . Then f -g’ is a Xn(J,) map of a subset of u p ; onto J,. But we have established that p i 5 T < 6, so this contradicts the choice of 6. Hence

3 J ~ , so this implies Y(J,) n E ~ ( J , ) c J,. In particular,

P P

PP

6 = p . (P 2) follows immediately from the above result of course.


We turn now to (P 3). By induction hypothesis, let A be a Ern master code for J,. Set 77 = p r for convenience. By the above, let f be a ZJJ,) map of a subsetofup ontoJ,.Bychoice ofA, f ’ = f t(f-l”J,)isaC1((Jq,A))map of a subset of wp onto J,. By choice of A , it is clear that p = p t = P , , ~ . Finally, of course, (J,,A) is amenable. So, we may apply Lemma 48 to (J,,A) to obtain a Zl((Jv,A)) setB C J, such that C,((J,,B)) =3 (J,) n Z,+l((Jq,A)) for all r 2 1. By choice ofA, B E Zn(Ja) and Z,((J,,B)) = 3(J,) n X,+JJ,) for all 1-2 1. Hence B is a Enmaster code for J,.

Finally we prove (P 1). Let B be, as above, a C, master code for J,. Let R ( y , x ) be a Zn+l(Ja) relation on J,. Define, with f as above, a relation R on

hence Zl((J,,B)). Let? be a Zl((J,,B)) function uniformising R. Sincefis Z,(J,), so is f -’. But J, is C,-uniformisable, by induction hypothesis, so we can 1etf”be a C,(J,) function uniformising f-l. Set r = f a? f -’. It is clear that r is a Zn+’(Ja) function which uniformises R. The proof is complete.

1

N

J, by a r , x > - [ Y , X E J, A R ( f ( Y ) , f ( % ) ) l - ThenE is Z*+l(JJ and

The above results give us two (intuitive) equivalent formulations of the En -projecturn:

Theorem 50. Let a , n 2 0. Let 6 be the least ordinal such that some Z,(J,) function maps a subset of w6 onto J,. Let y be the least ordinal such that 3 (wy) n C,(J,) Q J,. Then 6 = y = pz .

Proof. That 6 = p i was actually proved during the proof of Theorem 49. Since we now know that J, is Z,-unifonnisable, Lemma 46 tells us that 6 5 y. Assume 6 < y. Now by definition, let u C w6, and let f : u Zn(Ja). Let Z = (51 [ ($ f(l)}. Then 2 C 06 and Z E Z,(J,), so by definition of y, 2 E J,. Thus Z = f([) for some 5, so [ E f([) * ( 4 f([), which is absurd. Hence 6 = y.

J, be

There is, of course, a concept which, for A, predicates, plays the role that the En projectum plays for X, predicates. And, as might be expected, there is a corresponding ‘‘total function” or A, equivalent of Theorem 50 for this concept .

Let a, n 2 0. The A,-projectum of a (sometimes called the weak En- projectum), Q:, is the largest Q < a such that (J,,A) is amenable for all An(J,) setsA C J,.

160 K.J. DEVLIN

Thus the A,-projectum of a represents the "hard core" of J, with regards to A, predicates on J,. Clearly, 77; 2 p t . We do not, however, necessarily have equality here. For example, let Q be the first admissible ordinal > w. Then it is easily seen that 77; = a, whereas p; = w.

Corresponding to Lemma 46, we have:

Lemma 51. Let n 2 1, and let y be the least ordinal such that 3(oy) n An(Ja) J,. Then there is a Xn(J,) (and hence A,(J,)) map ofwr onto J,.

Proof. Let n = rn + 1, n 2 0. Since Xm(J,) C A,(J,) C Z,(J,), Theorem 50 implies that p: 6 y 5 p';y". Theorem 50 also implies that there is a X m ( J a ) map of a subset of u p : onto J,. So, we can clearly define a C,(J,) map of wpz itself onto J,. This reduces our problem to showing that there is a Z,(J,J map of oy onto u p : . As a first step, we have the:

Claim. There is a X n ( J a ) map g from wy cofinaliy into u p ? .

LetA be a Em master code for J,. By hypothesis, let b C wy, b E A,(J,), b @ J,. By choice o fA , b is A ~ ( ( J ~ ~ , A ) ) . Suppose b is in fact defined by:

whereBo,B1 areXO((J m , A ) ) . Then pol

But ( JpE,A) is amenable, and hence rud closed, so as b 4 J no r < up: such that

there can be PF'

Define g : oy + u p : by

Clearly, g is Zn(Ja) and cofinal in u p : , proving the claim.


We now prove the lemma. Since p : I y, there must be a Xn(J,) map f from a subset of o y onto wp; (I wa), and of course such anfwill then be E1((Jpm,A)). Definey: ( w ~ ) --.up! as follows. Le t fbe given by f(v) = 7 , - 3yF0,,7,v), where F is E0((Jp,,A)). Set

2 onto

ff

8 , if (3Y ESy( , ) )F(y,8,4

0, otherwise.

- f (v, 7) =

Then Sis X1((JpmlA)), and hence Xn(J,). AndSclearly maps ( ~ 7 ) ~ onto u p : , as g cofina? m u p ; .

lemma follows. Since we have (by Lemma 38) a X1(Ja) map of wy onto the

Corresponding to Theorem 50, we have:

Theorem 52. Let a, n'> 0. Let 6 be the least ordinal such that some X,(J,) (and hence A,(J,)) function maps 06 onto J,. Let y be the least ordinal such that P ( o y ) n A,(J,) Q J,. Then 6 = y = Q:,

Proof. Suppose y < q:. Let B C oy, B E A,(J,),B 4 J,. Then wy n B = B 4 Jqn, contrary to (Jgz,B) being amenable.

Suptose now that Q: < y. Then there isA C J,, A E A,(J,), such that (J,,A) is not amenable. In particular, y > 1. Suppose y = 5' + 1. There is then a Xl(J,) map of w( onto wy, so by Lemma 51 there is a Zn(Ja) map,f, of 05' onto J,. Then Z = { L E wE I L 4 f(~)} is a A,(J,) subset of w [ . Clearly, Z 4 J,, so this contradicts the choice of y. Hence lim(y). Thus as (J,,A) is not amenable, there must be T < y withA n J, 4 J,. But by choice o f y , T < y -+A n J, E J,, so for some 0 <a, A n J, is J,-definable. Let 0 be the least such.ThenA n J , E P ( J , ) n A m ( J e ) f o r s o m e m E w , a n d A nJ,$Je. Thus by Lemma 5 1 there is a E,(J,) mapf of WT onto J , . (Actually the hypotheses of Lemma 5 1 require that we have a A,(Jo) subset of OT not in Je , whereas we have only exhibited a subset of J, with these properties. However, since there is available a X l(J,) map of WT onto J ,, this point causes no problem.) Since 8 <a, f E J,. But A n J, 4 J,, andA n J,E Je+l, so 19 2 y, and there is thus a map f ' E J, of OT onto wy. By Lemma 5 1, again, this gives us a Xn(Ja) map k of 07 onto J,. Then, clearly, K =

162 K . J . DEVLIN

{ 1 I i @ K(1)) is a A,(J,) subset of WT not lying in J,, contrary to T < y.

f : w6%J, beZ,(J,). LetZ={vlv$f(v)}.ThenZE3(w6)n An(J,) ~ J,. But this contradicts the choice of y. Hence 6 = y.

Hence y = 77:. Now, by Lemma 5 I , we have 6 5 y. Suppose 6 < y. Let

Remark. Lemmas 46 and 5 1 can be regarded as much sharper versions of the following, much earlier theorem of Putman:

Suppose 9 ( y ) n L,. Then contains a well-ordering of y of order type a. (For y >_ 0.)

Putman actually proved this result for the case p = w, but his proof works in the general case.

The methods described above have, of course, many uses, We give just one, very general, example, showing that (in certain circumstances) it is possible to carry out Lowenhein-Skolem arguments for non-regular ordinals a which can generally only be done when a is actually a regular cardinal.

More precisely, the following theorem, is well-known:

Theorem 53. Le t K be a regular cardinal. Le t y < K 2 u p , and suppose that Y C Jp, I Y I < K . Then there is X < Jp such that Y U y C X a n d K n X E K .

To prove this, one simply forms an a-chain Xo< XI< ... 4 X , 4 ... 4 Jp of elementary submodels of Jp , takingXo as the Skolem hull of Y U y in Jp, and Xn+l as the Skolem hull of X , U sup ( K f' X , ) in Jp, and then X = U n < w X n is the required submodel of Jp. By construction, K n X is transitive, and hence an ordinal, and since K is regular, I XI < K , so K n X E K .

It should be observed that K being regular is a necessary condition for the above procedure to work (in general). However, providing we can, in some way, ensure that for each n , sup ( K n X,) < K , then we can, of course, get by with just cf(K) > w . The theorem below shows that, in certain cases we can do just this, providing we relax our demands somewhat.

Let n 2 1, a 5 wp. We say that a is E,-reguZar at p iff there is no X ,(Jp)

For example, by Theorem 43, wa is strongly admissible iff wa is X 1 - map of a bounded subset of a cofinally into a.

regular a t a.


Theorem 54. Let n 2 1, wp 2 a 2 1. Suppose a is X,-regular at p. Let Y C Jp, w 2 1 Y 1 < cf (a), and let y < a- Then there is an X <zn Jp such that Y U y C Xand a n X E a.

Proof. Since cf (a) > w , it clearly suffices to prove that, under the stated hypotheses, there is X i r , Jp such that Y U y C X and sup (a n X ) < a.

Let h be a E n Skolemfunction for Jp (by Theorem 49 and Lemma 33). Since w 5 I Y 1 < cf(a), we may, without loss of generality, assume that Y is closed under ordered pairs. Furthermore, let @ be the function defined in Lemma 37. Since a is X:,-regular at 0, a is certainly strongly admissible. Hence, by Lemma 37, { .$ E a I @ ' I t 2 C .$} is unbounded in a. It follows that we may also, without loss of generality, assume that arty2 C y. Recall that y2 is zip.

Let X = h t t ( o X (Y X 7)). Then we claim that Xis closed under ordered pairs. To see this,letxl,x2 EX, sayxl =h(il,(yl,vl)),x2=h(i2,(y2,v2)). Lety = (y1,y2)E Y andv= @((vl,v2))Ey. Clearly {(xI,x2)} is t :P({p,(y , v ) } ) , where p is a good parameter for h. Thus for some i E w, (x1,x2)= h(i , (y ,v)) E X , as required. So, by Corollary 36, X < z H Jp. And of course, we clearly have Y U y C X . We show that sup (a n X) < a.

F o r y E Y , iEw,definehi,y: C y + a b y h . (v)l .h(i ,(y,v)) .Thushi, , is Xn(Jp), and so as a is En-regular at 0, sup (hi,y y) - y( i ,y) < a. Since I Y I < cf(a), it follows that supyEy y( i ,y ) N- y( i ) < a . Since cf(a) > w, we conclude finally that supjEw y(i) < a. But clearly, supiEw y(i) = sup(& n X ) , so we are done.

'Jn

The above lemma may be used to prove that if V = L and K is a regular uncountable, non-weakly compact cardinal, then there is a Souslin K-tree. (Jensen.)

J.E.Fenstad, P. G.Hinman (eds.J, Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

DEGREE THEORY ON ADMISSIBLE OKDINALS

Stephen G . SIMPSON Yale University and The University of California, Berkeley

0. Introduction

The study of recursive functions on the ordinal numbers was initiated by Takeuti in the late 1950’s. Takeuti’s concept was generalized by several authors to that of a-recursive functions on admissible initial segments a of the ordinals. An intensive study of the generalized concept was begun by Sacks in 1964 [ 111.

Sacks’ unstated program was to suitably generalize all the theorems of ordinary recursion theory to &-recursion theory. A typical theorem of ordinary recursion theory is the Friedberg-Muchnlk solution to Post’s problem: there are two recursively enumerable subsets of w such that neither is recursive in the other. A recent paper of Sacks and Simpson [ 131 generalizes the Friedberg- Muchnik theorem as follows: for any admissible ordinal a, there are two a- recursively enumerable subsets of a such that neither is a-recursive in the other. (The Sacks-Simpson proof is of some methodological interest in that it involves a rather delicate downward Lowenheim-Skolem argument within La, the a fh constructible level.)

Our object in the present paper is to survey the recent work in a-degree theory for arbitrary admissible ordinals a. (An a-degree is an equivalence class under the relation: A is &-recursive in B andB is wecursive in A . ) The flavor of a-degree theory can be savored only in the proofs; we therefore give detailed proofs of a few theorems. The reader must understand that in this survey

The preparation of this paper was partially supported by NSF Contracts GP-34088X and GP-24352. The main result of Section 4, due to Sacks and the author, was presented by the author in an invited address to the 1971 Summer School in Mathe- matical Logic at Cambridge, England, sponsored in part by the North Atlantic Treaty Organization.

165

166 S.G. SIMPSON

paper we are leaving many important matters out of account, e.g. (in decreasing order of importance):

1. reasons for generalizing recursion theory ; 2. the lattice of a-r.e. sets ’ ; 3 . alternative notions of relative a-recursiveness ; 4. non-regular and non-hyperregular a-r.e. sets 5 .

A detailed outline of the paper follows. In Section 1 we review the basic notions of a-recursion theory such as a-

degree, a-r.e. sets, and the a-jump of a set. We also discuss two important auxiliary notions, regularity and hyperregularity, which have no counterpart in ordinary recursion theory.

ness Theorem: an a-degree b > 0‘ is regular if and only if it is the jump of a regular, hyperregular a-degree. We then give a necessary and sufficient condition on a that there exist an a-degree d such that every b 2 d is regular. We show in particular that d exists if a is countable.

In Section 4 we reprove the main theorem of Sacks-Simpson [ 131 in the following strong form: for any admissible ordinal a there are a-r.e. degrees a , b such that a 4 b, b $ a , a U b is regular and hyperregular, and (a u b)‘ = 0’.

In Section 5 we sutvey recent work of Manny Lerman, Richard Shore, and others. Their work is further evidence that most if not all theorems of ordinary degree theory can be generalized (not straightforwardly!) to ar-degree theory.

In Section 6 we discuss briefly the connection between a-recursion theory and Jensen’s “fine structure” theory.

In Section 2 we prove the following generalization of Friedberg’s Complete-

1. Fundamental notions

We employ von Neumann’s notion of ordinality; thus an ordinal is identi- fied with the set of smaller ordinals. Our set theoretical notation is standard.

* See Kreisel [ 2) especially Section 3. See Machtey (81;Chapter 1 of Simpson [ 171; Lerman-Simpson [7] ; and Lerman

See pp. 157-158 of Kreisel [ 21 and pp. 39-44, 92-94 of Simpson [ 171. See Chapters 2 and 3 of Simpson [ 171.

3

[4 i i

DEGREE THEORY ON ADMISSIBLE ORDINALS 167

In particular Ux (union of x ) , x ny (intersection of x andy), x X y (Cartesian product of x andy), x r y (x restricted toy), x”y (the range of x r y ) , x - y (set theoretical difference o f x andy), x C y ( x is a subset ofy), x E y ( x is an element ofy), 8 (empty set), (x,y) (ordered pair), dom ( x ) , rng(x) (domain and range of x ) , x : y + z ( x is a function from y into z ) have their usual meanings.

Throughout this paper a is a fixed but arbitrary admissible ordinal. Lower case Greek letters denote ordinals less than a, i.e. elements of a; except for /3 and A which may denote limit ordinals less than or equal to a. Upper case Roman letters denote subsets of a. A set X 5 /3 is unbounded in /3 if U X = 0; otherwise it is bounded below p. We sometimes write unbounded for unbounded in a and bounded for bounded below a.

A partial function from a into a (i.e. a function whose domain and range are subsets of a) is a-recursive if its graph can be enumerated via an equation calculus resembling Kleene’s but allowing infinitary bounded quantifications such as (3x < 6) where 6 < a. (For details see [ 121 .) A subset of a is a- recursive if its characteristic function is a-recursive. A subset of a is a- recursively enumerable (abbreviated a-r.e.) if it is the domain or range of an a-recursive partial function. Every a-r.e. set is the range of a one-one a- recursive function whose domain is an ordinal less than or equal to a. .

A subset of a is &-finite if it is a-recursive and bounded. A basic of a-recursion theory is:

principle

i f f is an a-recursive partial function and K C dom (f) is a-finite then f ”K is a-finite.

In the usual way one defines an a-recursive function k : a X a+ (Y such that: (i) k(y ,q) = 0 implies y < q ; and (ii) {y I k ( y , q) = 0) ranges over the a-finite sets as q ranges over a. We fuc the notation K , = {y 1 k (y ,q ) = 0) and call q a canonical index for K if K = K , .

a-recursive function f : a + /3. It is easy to see that a* is equal to the least /3 The projectum of a, denoted a*, is the least /3 such that there is a one-one

For any limit ordinal a the notion of a-recursive partial function can be defined. One can then define a-finiteness as above and prove: a is admissible if and only if principle (*)holds. Note that iff is a-recursive and 7 < 01 then h [ f ( ( g , y ) ) is a-recursive.

168 S.G. SIMPSON

such that not every a-r.e. subset of is a-finite. I f K is an a-finite set, the a- cardinality of K is the least /3 such that there is an a-finite one-one correspondence between K and p. An ordinal less than a is an a-cardinal if it is equal to its own a-cardinality. An a-cardinal 0 is regular if every a-finite subset of 0 of order type less than is bounded below p. An a-cardinal is singular if it is not regular. I t is easy to see that if a* < a then a* is the largest a-cardinal.

As in ordinary recursion theory one defines an a-recursive function r : a X a* +a such that: (i) K,,(,,,) C Kr(7,,) E T whenever u < T < a; (ii) u{Kr(u,E)I u < a} ranges over the a-r.e. sets as E ranges over a*. We fix the notations Rz = Kr(u,E) and R E = U{R: /u < a}. We call E < a* an index for R i fR = R,. As in ordinary recursion theory, one can a-effectively compute an index for an a-finite set K from a canonical index for K but not conversely.

We now discuss relative a-recursiveness. For A , B 5 a we say A is a-many- one reducible to B (abbreviated A 5 &I) if there is an a-recursive function f : a + a such that for ally

A is a-recursive in B (abbreviated A <a B) if there is an index E <a* such that for all y, 6

Clearly A 5 maB implies A 5, B but not conversely. The &-jump o f B is defined by

B ' = { E < a* I ( 3 g)( 371) [ ( t ; , 7 7 ) E R E & KE 2 B & K, n B = (31) .

I f f is a partial function f roma into a, we say f is weakly &-recursive in B (abbreviated f 5 w&?) if there is an index E < a* such that for all y, 6

f ( r ) = 6 c--, (W( 377) [(Y, 6 , t;, 77) E R E KE C_ B K , n B = 61 .

For such an e we write f = [ E ] and call E an index for f from B. We say A is a-recursively enumerable in B ifA i s the domain or range of a partial function weakly a-recursive in B.


IfA C a we denote by cA the characteristic function ofA. Caution: A I, B implies cA I ,$ but not conversely. However, the following facts are easily verified.

1.1 5, is transitive, and f i,, A <, B implies f 1.2 B I,, B' but not B' 2, B. 1.3 A 5, B if and only ifA' 1.4 A is&-r.e. in B if and only ifA I,, B'.

B.

B' .

Two sets A and B are said to have the same a-degree if A -, B i.e. A 5, B and B 5 A. Lower case boldface lettersa, b, ... are used to denote a-degrees. If a, b are the a-degrees of A, B respectively then we write: a 5 b for A 5, B; a U b for the a-degree of A @ B where

Fy

0 for the a-degree of 0 = 0; a' or jump (a) for the a-degree of A ' . Thus the a- degrees are an upper semilattice under <, U; moreover the jump operator is well defined, and increasing on a-degrees.

We now discuss two important auxiliary notions due to Sacks.

Definition. A C a is a-regular ifA n 0 is a-finite for all p < a.

Definition. A 2 a is a-hyperregular iff "0 is bounded whenever 0 < a,

f : p + a , f 5 , , A .

Definition. An a-degree is regular (resp. hyperregular) if it contains an a- regular (resp. a-hyperregular) set.

Clearly every subset of o is w-regular and w-hyperregular. The existence of non-a-hyperregular sets is what makes a-recursion theory more intricate than ordinary recursion theory. For instance, if a* < a then not even every a-r.e. set is a-regular (and conversely). This means that the difficulties associated with non-a-regularity can arise even in arguments concerning a-r.e. sets. Such difficulties are alleviated somewhat by the following theorem.

1.5 (Sacks [ 111) . To every a-r.e. set A there is an a-regular a-r.e. set B such that B 5, A.

170 S.G. SIMPSON

A short proof of 1.5 can be found in Chapter 2 of [ 171.

1.6 (Sacks [ 111). For A E a the following are equivalent :

(i) A is a-regular and a-hyperregular. (ii) f "K is a-finite whenever f <wa A , K C: dom ( f ) , K a-finite.

Proof. That (ii) implies (i) is immediate from the definitions. To prove (ii) assuming (i), suppose f i,, A , K E dom(f), K a-finite. Let E <a* be an index for f from A . Let /3 < a be the a-cardinality of K , and let i be an a- finite function from P onto K. For each y </3 letg(y) be the least u such that K t C A andK,nA = (Ofor some6,[,q such that(i(y),F,t,q)ER:.Then g swa A since A is a-regular. Hence g"0 is bounded since A is a-hyperregular. Let 7 < a be such that g"P C 7. Let H = A n 7. Then H is a-finite since A is a-regular. Note that ( i (y ) , 6, t , q ) E R: implies 6, t , q < 7 which implies K t U K , C r . Thus 6 E f "K if and only if

Thus f " K is a-finite.

The end of a proof is indicated by m.

We close this section with some useful technical lemmas concerning E2 functions. Following Rogers [ 10: pp. 301-3071 we define the aurithmetical hierarchy. Thus a relation on a is C, if it is a-recursive, ",I if its complement is C,, C,l+l if it is the projection of a H n relation, and A n if it is both En and n,. In particular a relation is C, if and only if it is ct-r.e. A partial function f is said to be En if its graph is En. In particular f is C, if and only if it is a-recursive. Caution: the bounded quantifier manipulations described in Rogers [ 10: p. 31 11 do not generalize to a-recursion theory except in very special circumstances. However, as in ordinary recursion theory, one has:

1.7. Lemma. Let f be a partial function from a into a. Then the following assertions are painvise equivalent:

I t can be shown thatA C a is a-regular and a-hyperregular if and only if the structure ( L , , t , A ) i s admissible.

DEGREE THEORY ON ADMISSIBLE ORDINALS 1 7 1

(i) f is 2,; (ii) f is weakly a-recursive in 0';

(iii) there is an a-recursive function f ' : a X a -+ a such that for all y

In (iii) the limit is taken in the discrete topology as u approaches a. As usual, x =y means that x is defined iffy is defined in which case x = y . Thus in (iii) f ( r ) = 6 iff(3a)(VT>_o)f'(T,y) = 6 .

For each limit ordinal /3 5 a we define A 2 - cf(0) to be the least h such that there is a A2 function with domain h and range unbounded in 0.

I .8 Lemma. A2 - cf (a) = A2 - cf (a*).

Proof. Let f : a -+ a* be a one-one a-recursive function. Note that y n rng(f) is a-finite for each y < a* since rng(f) is a-r.e. Hence {o If(a) < y } is bounded for each y <a*. Hence f "X is unbounded in a* whenever X is unbounded in a.

Suppose h 5 a and g : A + a is A, and rng(g) is unbounded in a. Then fg : h -+ a* is A, and by the previous paragraph rng(f) is unbounded in a*. So A, - cf(a) 2 A, - cf(a*).

For the converse suppose h : h +a* is A, and rng(h) unbounded in a*. Define k : h -+ a by

Using 1.7 it is easy to see that k is A2. Furthermore rng(k) is obviously unbounded in a. So A, - cf(a) 5 A, - cf(a*). rn

If (Ip[ p < v ) is an a-finite sequence of (canonical indices for) a-finite sets, then of course u {Zpl p < v) is a-finite. The following trivial lemma provides another useful sufficient condition for the union of a sequence of a-finite sets to be a-finite.

1.9 Lemma. Suppose v < A, - cf (a) and let (I,, 1 p < v ) be a simultaneously a-r.e. sequence of a-finite sets. Then U {I,, 1 p < v } is a-finite.

172 S.G. SIMPSON

Corollary. Put X = A -- cf (a) and suppose f : A + a is A2. Then (i) ( f ( p ) I p < v ) is a-finite for each v < A;

(ii) the sequence (Cf(p)I p < u)l v < A) i s A,.

Proof. Let f : a X X -+ a be an a-recursive function such that f ( v ) = limo f (u , v) for all u < A. Such anfexists by 1.7. Let us say that f (v) changesvalueatstage u i f ( V u ‘ < u ) ( ~ ~ ) ( u ’ ~ ~ < ~ & f ( ~ , ~ ) # f ( u , v ) ) . Let I , be the set of all u such that f(v) changes value at stage u. Then each I,, is a- finite, and the sequence U,Iv < A) is simultaneously a-recursive. Conclusions (i) and (ii) are now immediate from 1.9.

2. The a-jump operator ’ Friedberg determined the range of the jump operator in ordinary degree

theory by proving the following theorem: for every a-degree b 2 0’ there is an w-degree a such that a’ = a U 0’ = 6 . (For a proof see Rogers [ 10: p. 2651 .) We now generalize Friedberg’s theorem and its proof to a-degree theory as foliows:

2.1 Theorem. Let b be an a-degree 2 0’. Then b is regular if and only if there is a regular, hyperregular a-degree a such that

a ’ = a u O’= b .

Proof. Theorem 1.5 implies that the a-degree 0’ is regular. (In fact, it can be shown that a‘ is regular whenever a is regular and hyperregular.) The “if’ part of 2.1 follows immediately.

For the “only i f” part we shall employ a forcing construction. This in itself is not surprising since Friedberg’s original proof may be viewed as a forcing argument. However, unlike Friedberg or Cohen, we shall do our forcing over a possibly uncountable ground model, L,.

ranging over conditions. Thus p is an a-finite function from an ordinal lh ( p ) , A condition is an a-finite sequence of 0’s and 1’s. We use p , q , ... as variables

The main result of Section 2 was first proved in August 1971. It was presented in an invited address to this Symposium.


the length of p , into (0,l). We say q extends p if p C q . Let Cond be the set of conditions. A set D C Cond is dense if every p E Cond is extended by some q E D .

For X C a we denote by cx the characteristic function of X . Conditions are thought of as a-finite approximations to cx. If D is a set of conditions, we say X meets D if cx y E D for some y < a.

Our notion of forcing is defined as follows :

(9 P I t [ E l (r) = 6 iff ( 3 ~ ) ( 3 q ) [ ( y , ~ , E , Q ) E R ' , ~ ( P ) & p ' ' ~ ~ c (1) & p " ~ , c (011; (ii) p decides [el (y) iff p H- [ E ] (y) = 6 for some 6 ; (iii) p H- ( [E] (y) is undefined) iff no q 2 p decides [ E ] (7); (iv) p determines ( [ E ] (y) I y < 0) iff either p decides [ E ] (y) for all y < where yo is the least y such that p does not decide [el (7).

or p ( [E] (yo) is undefined)

Note that "p decides [ E ] (y)" is a-recursive as a relation of p , E , y. We now define certain important sets of conditions. The definition splits into cases depending on the nature of a.

Definition. Case I. a* is a regular &-cardinal. Then for each 0 < a* put

Case I f . Otherwise, i.e. a* = a or a* is a singular a-cardinal. Then for each p<a* put

D p = ( p E C o n d l p deterrnines([E)(y)\y<p)foreachE<fl).

2.2 Lemma. For each < a*, Dp is dense.

Proof. The proof splits into cases following the definition of Do.

B A more conventional way of writing CB 1 u €1 (7) = 6 is: [ e J (7) = 6.

174 S.G. SIMPSON

Case I . Define an a-recursive function

m : C o n d X a * + a * + l

by m ( p , E ) = the least y < a* such that p does not decide [ E ] (7); m ( p , ~ ) = a* if n o such y exists. Fix p < a* and define an a-recursive partial function O from Cond into Cond by B(p) = the least q (in the canonical a-recursive well ordering of Cond) such that p 5 q and m(p, E ) < m(q, E ) for some E < 0. Then, given a condition p , define an a-recursive increasing sequence of conditions by qo = p ; q, = U {qEl [ < q } if q is a limit ordinal; 4[+1= O(q5) if 8(qE) is defined; to = the least

define

such that O(qE) is undefined. We claim that to < a. Suppose t o the contrary that to = a. For each E < 0

Then the It’s are simultaneously a-recursive and a = u { I , / € <@}. Further- more m(qo, E ) < ... < m(q[ , €) < m(q[+,) < ... < a* so each I , has order type less than or equal t o a*. Let H be the set of E < such that I E has order type a*. Then H i s an a-r.e. subset of /3 < a*. Hence H is a-finite. I t follows at once that U { I E ( E E H } is a-finite. Let y < a be an upper bound for u {I,( E EH}. Let J , = I , n { 6 I y 5 6 < y + a*}. Then (J,I E < 0) is an a-finite sequence of a-finite sets each having order type less than a*. Furthermore u {J,I E <0} = ( 6 I y 5 6 < y + a*}. This contradicts the Case assumption that a* is a regular a-cardinal. The claim is proved.

Thus qEo is a condition extendingp. For each E < p it is clear from the construction that qt0 decides [ E ] (y) for each y < m(qto, E ) ; furthermore if m(qEo, E ) < a* then qto + ( [ E ] (m(qEo, E ) ) undefined). Thus qtO E Do. Since p is arbitrary, Do is dense.

Case II . Fix p < a* and define an a-recursive function

m : C o n d X a * + D + l

by m(p , E ) = the least y < O such that p does not decide [ E ] (y); m(p, E ) = p if n o such y exists. Now define B and the sequence (qtl $ 5 to) as in Case I .

Again we claim to < a. Suppose t o the contrary that to = a. Define16, E < 0, as in Case I . Again the I,’s are simultaneously a-recursive and


a = U {I,l e < 0). Furthermore m(qo, E ) 5 ... 5 m(q5, e) 5 m(qE+l, E ) I ... 5 (3 so each I , has order type 5 0. Hence there is an a-recursive partial function i from a subset of (3 X 0 onto a (namely i ( t , E ) = the tth member of ZE). This is impossible since fl < a*.

Thus qt0 is a condition extending p . As in Case I we note that qEo E Dp hence Dp is dense. =

Remark. From the above proof for Case I we can extract the following combinatorial lemma:

Assume that (3 < a* and that a* is a regular a-cardinal. Let (Z,I v < (3 ) be a simultaneously a-recursive sequence of a-finite sets each of order type less than or equal to a*. Then U {Z,l v < (3) is a-finite.

One can extract a similar but weaker combinatorial lemma from the proof for Case 11.

Corollary. There is a A, function q : a* X Cond -+ Cond such that for all p E Cond and 0 < a*,

Proof. Define q((3,p) = qEo where qto is as in the proof of Lemma 2.2. Using Lemma 1.7 it is easy to see that this q is a A z function.

A set A 5 a is said to be generic if A meets Dp for each (3 < a*. The following Lemma and its proof show that our notion of genericity is not so special as it appears. (It can be shown thatA E a is generic if and only if (L,,A) is admissible and every X 2 sentence true in (L, ,A) is forced by some condition p C cA . However, this remark does not simplify the proof of Theorem 2.1 .)

2.3 Lemma. Let A 5 a be a generic set. Then A is a-regular and a-hyperregular, and

A ’ - , A @ 0 ’ .

Proof. For each y < a one can find E <a* such that p decides [c] (0) if and only if l h ( p ) >_ y. Hence for each y < a there is (3 < a* such that lh(p) 2 y for all p ED,. It follows that A is a-regular.

116 S.G. SIMPSON

SupposeA is not a-hyperregular. Let K be the least ordinal such that for some e < a*, K E dom([eIA) and { [eIA(y)I y < K } is unbounded. Obviously K < a since A is not a-hyperregular. I t is also clear that I( is a limit ordinal.

We claim that K is a regular a-cardinal. Suppose not and let ( K ~ I p < n) be an a-finite sequence such that n < K = U {K, , ] y < n}. For each y < 7~ let h(y) be the least u such that for all y < K , , there exists ( g , ~ , y , 6 ) ER: with K g E A and K7, n A = @. Clearly h ( p ) is less than a since K~ < K . Also h Sw, A since A is a-regular. Hence {h(p)i y < n} is bounded since n < K . It follows that { [elA (y) I y < K } is bounded, a contradiction. The claim is proved.

We now split into Cases as in the definition of the Do’s. In Case I we have K <a* and we set = E i- I . In Case 11 we haveK <a* and we set0 = max { K , E i- 1). In either case A is generic so let p E Dp be such that p C cA . Then for each y < K it is clear that [e lA(?) is the unique 6 such that p tt [~](y) = 6 . Hence { [eIA(y)I y < K } is a-finite, a contradiction. SoA is a-hyperregular.

a-recursive functions f ,g : a + a* such that for all y, 6 Obviously A @ 0’ 5, A’ . That A’ <, A t~ 0’ follows from the existence of

and

The existence off is a consequence (via 1.6) of the fact that A is a-regular and a-hyperregular. The existence of g follows directly from the genericity of A .

We are now ready to complete the proof of Theorem 2.1. Let B be an a- regular set. Thus, for each y < a, cs r y is a condition. If p and q are conditions we write

i.e. p * q is the concatenation of p and q . Let X = A2 - cf(a). Recall Lemmas 1.7, 1.8, and 1.9 which state several useful properties of A. Let F(resp. G) be a A2 function from h onto an unbounded subset of a (resp. a*). Define

p , = U { p , I p < v } if v = U v ;


where q is the A, function mentioned in the Corollary to Lemma 2.2. For each v < A, {F(p) I p < v} is bounded below a. Hence only a-finitely

much of B is used in the definition of (pp I p < 2v). Hence, by 1.7 and 1.9, (p,l p < 2 v ) is a-finite. Thus (pvI v < A) is an increasing sequence of conditions. Moreover U {Ih(p,)I v < A} = a since F(v) 5 l h ( ~ ~ . + ~ ) . Hence there is a unique set A C a such that cA = u {pv( v < A}. For each v < A we have p2v+2 E D G ( u ) ; henceA is generic. I t follows by Lemma 2.3 thatA is a- regular and a-hyperregular, and that

I t remains to show that

This is an immediate consequence of the following claim. For each v < A let P(v) be (a canonical index for) (p,I 1.1 < 2v).

Claim: P is weakly a-recursive in B @ 0’ and in A @ 0’

We shall give an informal, “Church’s thesis” type of argument for the claim. To compute P(v) from B CB 0’. First compute y = U { F ( p ) I p < v}. By

Lemmas 1.7 and 1.9 this computation uses only an a-finite amount of information about 0‘. Then use 0’ and the a-finite sequence cB ry to compute P O , p 1 , ..., p p , ._. ( p < 2v) in order. Again by 1.7 and 1.9 this uses only a- finitely much information about 0’.

To compute P(v) from A @ 0’. First, compute (F(,u)l,u < v ) and (C (p ) I p < v ) as above. Next, start generating nu, the set of all (canonical indices for) a-finite sequences of conditions ( p l l p < 2v) such that U { p ; I 1-1 < 2 v 1 C cA and

178 S.G. SIMPSON

for all p < v. Clearly n, is a-recursive in A . As you generate n,, examine simultaneously all the members of n, trying to find one with the property that

for all p < v. This involves computing pieces of the A , function q . Do this a- recursively in 0', taking care to dovetail the examination of the various members of n, so that you don't spend too much time on any a-finite subset of Il,. (By Lemma 1.9 the examination of any single member of n, will eventually terminate having used only a-finitely such information about 0'; however, there is no guarantee that the examination of an a-finite subset o f n , will so terminate.) Eventually you find a sequence ( p l l p < 2 v ) in n, with the desired property. Then P(v) = (pL1 p < 2v). More importantly, you have used only &-finitely such information aboutA @ 0'.

We have informally described computation procedures showing that PI,, B @ 0' and P<,, A @ 0'. The proof of Theorem 2.1 is complete. m

One can use the same method to prove the following generalization of a theorem of Spector (see Rogers [ 10 : p. 2671):

2.4 Theorem. There exist regular, hyperregular a-degrees a, b such that a u b = a ' = b ' = 0'.

3. Upper segments of regular a-degrees

An interesting corollary of Friedberg's jump theorem is this: there is an w- degree d such that every w-degree 2 d is the jump of some w-degree (namely d = 0'). We now ask: for which admissible a does there exist an a-degree d such that every a-degree 2 d is the jump of some regular, hyperregular a- degree'? By Theorem 2.1 this is equivalent to asking: for which admissible a does there exist an &-degree d such that every a-degree 2 d is regular?

We write I a 1 for the (von Neumann) cardinality of a i.e. the least /3 such that there exists a one-one correspondence between a and /3. We write cf(a)


for the cofinality of a, i.e. the least h such that there is a function from h onto an unbounded subset of a.

3.1 Theorem. Assume the Generalized Continuum Hypothesis. Let a be an admissible ordinal such that I a I is a regular cardinal. Then the following assertions about a are painvise equivalent.

(i) There is an a-degree d such that every a-degree 2 d is regular. (ii) To every a-degree a there is a regular a-degree b 2 a.

(iii) Every X C a of cardinality less than I a1 is a-finite.

Before proving 3.1 we give two lemmas in which 1 a 1 is not assumed to be regular.

3.2 Lemma. Let a be an admissible ordinal such that Assertion 3.1 (ii) holds. Then every X a of cardinality less than cf(a) is a-finite.

Proof. Let X 5 a have cardinality < cf(a). By 3.1 (ii) there is an a-regular set B such that cx Swa: B. Let E <a* be such that cx = [ e l B . For each y E X let h(y ) be the least u such that

Since I XI < cf (a), {h(y) I y E X } must be bounded below a, say by 7. Then for all y we must have y E X iff cB r T k [ E ] (y) = 1. Hence X is a-finite.

3.3 Lemma. Assume the Generalized Continuum Hypothesis. Let a be an admissible ordinal such that Assertion 3.1 (ii) holds. Then cf (a) = cf ( I a I ).

Proof. The hypothesis 3.1 (ii) clearly implies that the set of regular a-degrees has cardinality 2'"'. On the other hand, the set of regular a-degrees clearly has cardinality at most I a I cf (&). If cf (a) < cf (la I) then by the GCH we would have 2 ' O ' <_ lalcf(") = la1 contradicting Cantor's theorem of cardinal arithmetic. So cf (a) 2 cf ( I a I).

For the other direction recall Konig's theorem of cardinal arithmetic which

180 S.G. SIMPSON

says I a I C f ( l o l ' ) > IaI. Hence there is a non-a-finite set X C 1 a1 of cardinality cf ( 1 a I). By Lemma 3.2 X cannot have cardinality less than cf (a). So cf( 1 @I) 2 cf(a).

Proof of 3.1. The implication (i) 9 (ii) is trivial since any two a-degrees have an upper bound. The implication (ii) * (iii) is a special case of Lemma 3.2 in view of Lemma 3.3.

To prove (iii) * (i) first note that (iii) implies cf (a) = I al. Let (?,I u < I a t ) be an increasing sequence of ordinals such that a = u{y,I I.' < 1 a ( } . Let g be a function from 1 a I onto a. Define

and let d be the a-degree of D. Given B 5 a define

Then D and B* are a-regular in view of (iii). Furthermore it is easy to see that

Thus every a-degree 2 d is regular. 9

Corollary. Let a be a countable admissible ordinal. Then there is an a-degree d such that every a-degree 2 d is regular.

Remarks. such that la1 = cf(a)= w1 but not everyconstructibly c o u n t a b l e X C a i s a - finite. Such an a falsifies some attractive conjectures.

1. I t is not hard to construct (in ZFC) an admissible ordinal (Y

2. I f I a1 is not regular then we conjecture that the equivalence 3.1 (i) - 3.1 (ii) still holds. We can prove in ZF + V = L l o that 3. I (ii) is equivalent to

lo The set-theoretical hypothesis V = L can be expressed in mecursion theoretic language as follows: for every infinite cardinal K , every subset of K bounded below K is K-finite.


the assertion that there is a function f : I a1 '%' a such that f r y is a-finite for each y < I aI.

4. Incomparable a-r.e. degrees

In this section we generalize to a-recursion theory a strong form of the Friedberg-Muchnik theorem.

4.1 Theorem. Let a be an admissible ordinal. Then there are a-r.e. degrees a, b such that

(i) a I b and b 4 a ; (ii) a U b is hyperregular; (iii) (a u b)' = 0'.

(By Theorem 1.5 a U b is necessarily regular.)

In the proof we shall define a-recursive functionsA', B', f (u , E ) , and g(u,E) ( E <a*). Here ( A u [ u < a ) and (B"I u < a ) will be nondecreasing sequences of a-finite sets. We shall then setA = U {A'I u < a } and B = u {B'I u < a} . ThusA and B will be a-r.e. For each E < a* f (u , E ) and g(u, E ) will be nondecreasing as functions of u. I t will turn out that f ( ~ ) = lim, f (u , E ) and g(E) = lim, g(u, E ) are a-finite. Thus by 1.7 f,g :a* + a will be A 2 functions. The construction will be designed to insure that

( f ( ~ ) , E ) €A+--+ 3 q [ ( f ( ~ ) , E , Q ) E R , & K , n B = @]

and

for each E < a*. From this and the a-recursive enumerability of A and B it will follow that A 4, B and B 4, A .

At the same time we shall define a certain auxiliary a-recursive function P ( U , E ) whose purpose will be to preserve certain computations so as to make A @ B a-regular and a-hyperregular and (A @ B)' =, 0'. The definition of p(u , e) will closely parallel the key definitions in Section 2 whose purpose was llkewise to make a certain set a-regular and a-hyperregular. (Thus for instance there will be a split into Cases I and I1 exactly as in Section 2.)

182 S.G. SIMPSON

Remark. The construction and proof below are arranged so that the reader not interested in conclusions 4.1 (ii) and 4.1 (iii) can skip certain parts. Specifically, such a reader should (a) ignore the functionsy(u, y, E ) , m(u, E ) ,

and p ( a , f); (b) replace the below definitions off(< u, E ) andg(< u, E ) by

and

(c) regard Lemma 4.6 as an immediate consequence of Lemma 4.2; (d) disre- gard entirely Lemmas 4.3, 4.4, and 4.5.

Before describing the construction we must define some important functions and parameters which do not depend on the construction.

(1) Let h be the A 2 cofinality of a. By Lemma 1.8 h is equal to the A , cofinality of a* so let G : h -+ a* be a A 2 function whose range is unbounded in a*. For each v < X define H(v) = u {G(p)I p < v}. Thus we have

O=H(O)< ... I H ( v ) < H ( v + I ) < ...<a*

and each E < a* satisfiesH(v) 5 t < H ( v + 1) for a unique v < A.

Remark. Thus H partitions a* into A “blocks” each of size less than a*. The use of H was suggested by R. Shore and has the effect of making the present proof somewhat simpler than that of Sacks-Simpson [ 131. Shore has also put this “blocking” idea to good use in his own work (see Section 5).

(2) Let G : a X X -+ a* be an a-recursive function such that C(v) = lim, G(.u, v) for all v < h. (Such a G exists by Lemma 1.7.) Define H(o,v)= U{G(u,1.1)Ip<v}.ThenH(u,v) isa-recursive and H(v)=lini,H(a,v); in fact, by Lemma 1.9, we have

Furthermore, for each u < a we have

O = H(u, 0) < ... i H ( u , v ) I H ( u , v + I ) < ... <a*


(3) Let L : a +. h be an a-recursive function such that L-'({v}) is unbounded

We can now present the construction. As a preliminary to stage u of the in a for each v < A.

construction we set

(l-Y)y<cl* Y ( 0 9 Y 3 f ) = 0

(l-Y),<,Y(O>Y>f) =

if a* is a regular a-cardinal,

otherwise ;

m(u,E) =

if ( c p , ~ ) € A < ~ ,

f (< 0, E ) = max {p, p ( u , E ) } otherwise

where

if ( J / , E ) E B < '

otherwise g(< 'J, 4 =

max { 9, p ( u , E ) }

where

Note: In the definitions of y(o, y ,~ ) and m ( u , ~ ) the usual convention concerning the bounded p-operator is observed. Namely, ( ~ L X ) ~ < ~ ... is defined

184 S.G. SIMPSON

to be the least x less thany such that ... if such anx exists andy otherwise.

L is as in (3) above. Define Stage u of the construction is now described as follows. Put u =L(u> where

AO=A<OU {c f ( _ H ( U , U ) & A ~ Z A < ~ ,

u + 1 if E' 2 H(u , p ) for some p such that H ( p ) changes value at stage u ,

g(< u, E ' ) otherwise.

d o , €7 =

Define

Bu = B<"U {(g(u,E),E) IH(u ,v) < e <H(u ,u+l ) & (5)}

where

(5) 3 7 7 [ ( g ( u , E ) , E , 7 ) ) E R ~ & K ~ nAu= @] .

Define

u + 1 if E ' > H ( u , u + I ) & B ~ Z B < O ,

u + 1 if E' 2 H ( u , p ) for some p such that H(p) changes value at stage u ,

f ( < u, E') otherwise.

f ( 0 , El) =

This concludes stage u and completes our description of the construction.

We now prove a series of lemmas leading to the conclusion that the construction works. For each v < X put

Obviously the 1,'s are simultaneously a-recursive. Our first lemma corresponds to the Friedberg-Muchnik observation that in their construction each require-


ment is injured only finitely often (see Rogers [ 10: p. 1661).

4.2 Lemma. Each Z, is a-finite.

hoof. By induction on v. The induction hypothesis tells us that each Z,,, p < v, is a-finite. Since v < h = A 2 - cf(a) it follows by Lemma 1.9 that U<Z,l p < v} is a-finite. Let uo be such that {Z,, I p < v} C uo. Let u1 2 uo be such that no H ( p ) , p I v + 1, changes value at any stage u 2 u l . For each E < H ( v ) put

V ( E ) = the least u 2 u1 such that (g(u,E),E) E B u

We claim: (6) if E < H ( v ) and c p ( ~ ) is defined theng(u, E ) = g(p(f), E ) for

Claim (6) is proved by induction on u > ( P ( E ) . The induction hypothesis all u > l p ( E ) .

tells us that g(< u, E ) = g(cp(e), E ) since (g(cp(E), E ) , E ) E B<'. But g(a, E ) = g(< u, E ) since H(v) = H(u, v) and u 2 uo.

is a-finite. Let u2 >_ u1 be such that rng(cp) s 02. Our cp is an a-recursive partial function and dom (9) 5 H(v) < a* ~ Hence cp

We now claim: (7) for each E <H(u , v+l) and u 2 u2, f(u, E ) = f(< u, E ) .

To prove claim (7), suppose u 2 u2 and E' <H(u,v+l) and f(u,~') f f(< u, E ' ) . Since u 2 u1 we must have E' >H(u,L(o) + 1) and B' # B". Hence L(u) 5 v and there must be an E such that H(u,L(u)) 5 E < H(u,L(u) + 1) and (g(u, E ) , e ) E B u - B<". Then E <H(u, v) = H(v) . Hence P ( E ) is defined and p(~ ) 5 u. But then g(u, E ) = g(cp(e),E) by claim (6) so in fact cp(e) = u. Hence u < u2 a contradiction.

Now for each E < H(v + 1) put

$ ( E ) = the least D 2 u2 such that Cf(< u, E ) , E ) € A u

Again $ is an a-finite function, and we let u3 >_ u2 be such that rng($) C 03. Again we make two claims: (6!) if E < H(v + 1) and $ ( E ) is defined then f ( u , ~ ) = f ( < $ ( ~ ) , ~ ) f o r a l l u > $ ( ~ ) ; ( 7 ' ) fo reache<H(u ,v+ l ) a n d u > _ u 3 , g(o, E ) = g(< u, E ) . The proofs of (6') and (7') are similar to the proofs of (6) and (7).

It follows from (7) and (7') that I,, E u3 hence Z,, is a-finite.

186 S.G. SIMPSON

4.3 Lemma. Let v and u3 be as in the proof of Lemma 4.2. Suppose E<H(v+l)andy<rn(u ,E) , u 2 u 3 . Then y ( u , y , ~ ) = y ( ~ , y , ~ ) a n d y < rn(~, E ) for at1 T 2 u.

Proof. Let us write C7 = A T @ BT. It suffices to show that CT ny(u , y , e ) = 6“ n y(u , y, E ) for all T 2 u. Suppose not and consider the least T 2 u such that this fails. We havey(T,y,E)= y ( a , y , ~ ) hencey(u,y,E)<p(T,E“)I min {f(< T , E ” ) , g(< T , E ” ) } for all E” > E . Hence there must be an E‘ 5 E such that (f(< T , E ‘ ) , E ’ ) EAT - A<‘ or (g(T, E ’ ) , E ’ ) E BT - B“. This is impossible s i n c e ~ 2 u ~ . m

4.4 Lemma. For each E < a*

{y(u ,y ,~ ’ ) I T < a & y < m(u,e’) & E’ < E }

is a-finite.

Proof. Let v and u3 be as in Lemma 4.3 where E <H(v + 1). Thus, for each E‘ < E , rn(u ,~’ ) is nondecreasing as a function of u 2 u3. Let m(E‘) = limo m(u, E ’ ) . Note that m ( ~ ’ ) I: a*. We claim that (m(e’)I E’ < E ) isa-finite. The proof of this claim splits into cases exactly as did the proof of Lemma 2.2. (If a* is a regular a-cardinal then we can directly employ the combinatorial lemma mentioned in the Remark following the proof of 2.2.) In either case we see that there is a stage U ( E ) 2 u3 such that m(u, E ’ ) = m(E’) for all E’ < E ,

u 2 u(E) . From this and Lemma 4.3, Lemma 4.4 is immediate.

4.5 Lemma. C i s a-regular and a-hyperregular and C’ = a 0’.

Proof. Suppose C is not a-regular and a-hyperregular. Let K be the least ordinal such thatf”K is not a-finite.for somef<, C, K C: dom(f). As in the proof of Lemma 2.3 we can show that K is a regular a-cardinal. In particular K 5 a*. Note that {$ I K E 5 C} is a-r.e. since Cis. Hence there is E < a* such that for all y, 6


Thusy(y, e) = lim, y(u, y, e) exists for all y < K . Furthermore

The proof now splits into cases. If a* is a regular a-cardinal then K I m(e) obviously. If a* is not a regular a-cardinal, then K <a* so we may safely assume K 5 E hence again K 5 m(e). In either case Lemma 4.4 shows that {y (y , E)( 7 < K } is a-finite, a contradiction. So C is a-regular and a-hyperregular.

It follows that there are a-recursive functions h, k : a -+ a* such that foI dY,$

K7 C C' * h(y ) E C'

and

K , f l C'= 4 ++k(y) 4 C' .

By Lemma 4.4 there is an a-recursive function t : a* + a* such that for all €<a*

From the existence of h , k , and t it follows that C' I, 0'.

4.6 Lemma. For each e < a* there exists u such that f(u, E) = f(7, E) and g(u, e) = g(7, e) for all 7 L u.

Proof. Immediate from Lemmas 4.2 and 4.4.

Thus f ( e ) = lim, f(u, E) and g(e) = lim, g(u, E ) are a-finite. Define A = u {A"lu < a} and B = u ( B u ( u < a}. It is perhaps amusing to note that

188 S.G. SIMPSON

clauses (4) and (5) of the construction have played no role in the proofs of Lemmas 4.2 -4.6.

4.7 Lemma. For each E < a*,

and

Proof. First suppose ( ~ ( E ) , E ) E A . Let u be such that ( f ( ~ ) , f ) € A " - A'". It follows that f(< T,E) = ~ ( T , E ) = f(e) for all T 2 u. Put u = L(u). Then H(o,v) 5 E <H(o,v+ 1) and s o H ( ~ , p ) = H ( p ) for all p 5 v, T 2 u. Since A"+A<"wemus thaveg(u ,~ ' )= u + 1 foraIlE'2H(u,v)=H(u). Further- more there must be an 77 such that ( f ( ~ ) , e , q ) ERF and K , n B'" = @. Note that K, 6 77 < u. We claim that K , n B = @. Suppose not, say (g(u', E'), E') E

K, n (B" - B<"'), u' 2 u. Then g(u', E') u a contradiction.

stage such that f(< u, E) = f ( ~ ) and ( f ( e ) , e , q ) E RZ and H(u,L(u)) 5 e < H(u,L(u) + 1). Then ( ~ ( E ) , E ) E A " S A .

We have proved the f(e) part of the lemma. The g(e) part is similar. 4

Now for the converse suppose (~ (E) ,E ,? ) ERZ and K , n B = 0. Let u be a

4.8 Lemma. A 4, B and B $, A

Proof. In the first place ( 5 I K E C B } is a-r.e. since B is. Hence A <, B would imply the existence of an E < a* such that for all 7

But y = ( f ( ~ ) , e ) witnesses the contrary. 4

The proof of Theorem 4.1 is complete.


5. Further results

During 1970-1972 the theory of a-degrees has developed rapidly. Two of the prettiest theorems are to be found in the Ph.D. thesis of Richard Shore.

5.1 Theorem (Shore). Let A be an a-regular, a-r.e. set and let B be a non-a- recursive, a-regular, a-r.e. set. Then there are hyperregular a-r.e. sets A O , A , such thatA = A , U A , , A O n A , = 0, B $ , A , , B e , A , .

Corollary. Any non-zero a-r.e. degree is a nontrivial join of two hyperregular a-r.e. degrees.

5.2 Theorem (Shore). Let a and c be a-r.e. degrees such that a < c. Then there is an a-r.e. degree b such that a < b < c. If a is hyperregular then b can be made hyperregular.

Theorems 5.1 and 5.2 generalize two famous theorems of ordinary recursion theory due to Sacks. They are known respectively as the Splitting Theorem and the Density Theorem; for an exposition see Shoenfield’s degree mono- graph [ 141. The proofs of 5.1 and 5.2 are remarkable applications of Shore’s “blocking” device duscussed in the Remark on page 182 above.

We offer the following analysis of the proof of 5.1. A typical goal or requirement of the construction is cB * [e lAi where i < 2, f < a*. In the special case a = w , Sacks satisfies this requirement by preserving (with priority e) those computations which make cB and [elAi look equal on long initial segments of a. (Because of these preservations, Sacks is able to argue at the end that cB = [e lAi would imply that B is a-recursive.) In the case of an arbitrary admissible a, this will not work, because the preservations associated with an infinite, a-finite set of requirements can get out of hand if A 2 - cf(a) < a*. Instead Shore breaks up the set of a* requirements into A*..- cf(a) blocks each of size less than a*. A typical block Bv,i is {cB + [ E ]

H ( v + 1)) where i < 2 , v < A2 - cf (a). (See page 182.) Shore satisfies the requirements in block Bv,i by preserving with priority v those computations which make {y < a1 cB(y) and [eIAi(y) look equal for some E, H(v) 2 e < H(v + 1)) contain long initial segments of a. Thus the block of requirements Bv,j is handled as a single requirement, so the preservations associated with Bv,i do not get out of hand.

I H(v) 2 f <

190 S.G. SIMPSON

In another line of investigation, M. Lerman and C.T. Chong have attempted to generalize the theorems of Lachlan's Lower Bounds paper [3] to a-recursion theory. The results of this attempt are as follows:

5.3 Theorem (Lerman). If a and b are a-r.e. degrees with a n b = 0 then aub<O' .

5.4 Theorem (Chong). Zf d is any non-zero a-r.e. degree then there is a hyperregular a-r.e. degree c such that 0 < c < d and c is not a nonm'vial meet of two a-r.e. degrees.

Lerman and Sacks, with good reason, call an admissible ordinal a refractory if (i) there is a largest a-cardinal, call it N; (ii) A 2 - cf (a) < H; and (iii) there is no I;2 function f : a I s ' 0 where 0 < N. For example let a be the first constructibly uncountable admissible ordinal having constructible cofinality w ; then a is obviously refractory.

5.5 Theorem (Lerman-Sacks). Suppose a is not refractory. Then there are nonzero hyperregular cy-r.e. degrees a, b such that a n b = 0.

Of course one conjectures that the hypothesis of non-refractoriness can

Another area where the results to date are fragmentary is the question of be eliminated, but this does not seem easy to do ll.

minimal a-degrees. An a-degree m is said to be minimal if m > 0 and there is no a-degree a with m > a > 0. 5.6 Theorem (MacIntyre). Let a be an admissible ordinal such that I a I is regular and eveiy X E cy of cardinality less than I a I is alfinite. Then there exists a regular, hyperregular, minimal a-degree.

In particular, minimal a-degrees exist if a is countable. (For a = o this is an old theorem of Spector.) MacIntyre has conjectured that regular, hyperregular, minimal a-degrees exist for all admissible ordinals a.

The proof of 5.6 is not a priority argument. By exploiting priorities i la Shoenfield [ 14 : pp. 54-56] and the "a-finite injury method" of Sacks- Simpson [ 131, one enlarges the class of admissible ordinals a for which minimal a-degrees are known to exist. Namely,

lar, hyperregular a-degrees u, b such that u n b = 0. I ' Recently D. Posner has shown that for every admissible ordinal (Y there exist regu-


5.7 Theorem (Shore). Suppose A 2 - cf (a) = a Then there is a regular, hyperregular, minimal a-ciegree m < 0’.

One can probably adapt the proofs of 5.6 and 5.7 to get finite distributive lattices as initial segments of a-degrees. However, to eliminate the special hypotheses on a seems a difficult problem indeed.

This completes our survey of the current theory of a-degrees. Apart from what we have reported here, most questions concerning a-degrees are virgin territory. It is not always easy to appropriately generalize the statement of a theorem of ordinary degree theory to a-degree theory, much less the.proof. One obstacle is that the admissibility of a (i.e. of the structure (L,,€)) does not imply admissibility of the expanded structure (L,,€,C) where C C a, even if C is a-r.e. and a-regular. Therefore “relativization” to C (cf. Rogers [ 10 : p. 2571) may be difficult or impossible.

This suggests a typical question: for an arbitrary admissible ordinal a, are there X 2 setsA,BCa such thatA 3, B @ O’andBe, A @ Of? For a = w (indeed whenever A, - cf (a) =a) the answer is obviously yes by relativizing the Friedberg-Muchnik theorem to 0’. For aibitrary admissible ordinals a, the answer is probably still yes but the proof may require an “infinite injury” argument.

Carl Jockusch has proved the following theorem of ordinary degree theory: there exists a degree d such that for all b 2 d there is u < b such that there is no c with a < c < b. Jockusch’s proof is unique in degree theory in that it employs the powerset axiom of ZFC (via a result of Paris concerning Gale- Stewart games). Therefore, the following question is of exceptional interest: for which admissible ordinals a does Jockusch’s theorem generalize to a- degrees?

6. Appendix: the fine structure of L

The consmtctible hierarchy is defined by recursion on the ordinals as follows: Lo = {P}; L,,+l = { X C L,IX is first order definable over (L,,E) allowing parameters from L,,}; L, = U {L,I u < A} for limit ordinals A; L = U {L, I u an ordinal}. Ronald Jensen [ 11 has made a detailed study of the fine structure of L. (Jensen then uses his results to settle a number of open questions in set theory under the assumption V = L.) There is a close connection between some of Jensen’s ideas and some of the ideas in a-recursion

192 S.G. SIMPSON

theory. We now discuss this connection briefly.

(The equivalence proofs would be tedious but straightforward.) First, a glossary comparing some of Jensen’s terminology to some of ours.

6.1. Let a be a limit ordinal. Then L, is an admissible set iff a is an admissible ordinal.

6.2. Let a be an admissible ordinal and A C a. Then (i) A is Xn(L,) iffA is En in our terminology; (ii) A E L, iff A is a-finite;

(iii) The structure (L,,E,A) is amenab2e iffA is a-regular.

6.3. Suppose a is admissible andA I a is a-regular and letf be a partial function from a into a. Then

( i ) f i s CI(L,,A) i f f f l w , A ; (ii) the structure (L,,E,A) isadmissible iffA isa-hyperregular;

( i i i ) B C a i s A,(L,,A)iffc, &,A.

6.4. Suppose a is admissible andA Cr a is a-regular and a-hyperregular. Then B C a is A (La, A ) iff B <a A .

A decisive role in Jensen’s theory is played by the notion of En projection. For a an admissible ordinal and n 2 1, the En projecturn of a is defined to be the least /3 such that there is a C,(L,) function from a subset of /3 onto a. Thus the El projectum of a is just what we have been calling&*. Jensen’s main theorem reads as follows.

6.5.Theorem. For n 2 1, the C, projecturn of a is equal to the least 0 such that not every C,(L,) subset of /3 is a-finite.

The proof of 6.5 is essentially model theoretic rather than recursion theoretic in nature. One needs a delicate refinement of the Skolem hull- condensation method first used by Godel to prove the GCH assuming V = L. The reader is invited to study Jensen’s proof in [ 11.

For a admissible and n = 1, Theorem 6.5 is trivial and we have used it repeatedly in the present paper. For n = 2 Theorem 6.5 is non-trivial but has been used elsewhere in a-recursion theory, e.g., in [7] and in Chapter 3 of


[ 171. The interesting point is this: Theorem 6.5 for n = 2 can fail if one looks at admissible structures of the form (La ,E ,A) . Namely, one can construct an admissible ordinal a and an a-regular, a-hyperregular setA C a such that (i) not every A2(La,A) subset of w isa-finite;(ii) there is no Z2(La,A) function from a subset of w onto a. This means that some arguments of a-recursion theory, e.g., the proof of Theorem 3.2 of [7], do not generalize straightforwardly to recursion theory on admissible structures (L,,E,A). This is a sad but amusing state of affairs which deserves to be investigated further.

Bibliography

[ I ] R.B. Jensen, The fine structure of the constructible hierarchy, Annals of Math.

[ 21 G. Kreisel, Some reasons for generalizing recursion theory, in: R.O. Gandy and Logic 4 (1972) 229-308.

C.E.M. Yates, eds., Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 139-198.

[ 31 A.H. Lachlan, Lower bounds for pairs of recursively enumerable degrees, Proc.

[4 ] M. Lerman, Maximal a-r.e. sets, Trans. Amer. Math. SOC. (to appear). [5 J M. Lerman, Least upper bounds for minimal pairs of a-r.e. crdegrees (to appear). [6] M. Lerman and G.E. Sacks, Some minimal pairs of a-recursively enumerable

[7] M. Lerman and S.G. Simpson, Mximal sets in a-recursion theory, Israel J. Math.

[S] M. Machtey, Admissible ordinals and lattices of a-r.e. sets, Annals of Math. Logic 2

[9] J.M. MacIntyre, Minimal or-recursion theoretic degrees, J. Symb. Logic 38 (1973)

London Math. SOC. 16 (1966) 537-569.

degrees, Annals of Math. Logic 4 (1 972) 4 15 -442.

(to appear).

(1971) 379-417.

18-28.

(McGraw-Hill, 1967) 482 pp. [ l o ] H. Rogers, Jr., Theory of Recursive Functions and Effective Computability

[ 11 J G.E. Sacks, Post's problem, admissible ordinals, add regularity, Trans. Amer. Math.

[ 121 G.E. Sacks, Metarecursion theory, in: J.N. Crossley, ed., Sets, Models, and

[ 131 G.E. Sacks and S.G. Sinpson, The a-finite injury method, Annals of Math. Logic 4

[ 14) J .R. Shoenfield, Degrees of Unsolvability (North-Holland, Amsterdam, 197 1)

1151 R.A. Shore, Minimal a-degrees, Annals of Math. Logic 4 (1972) 393-414. [16] R.A. Shore, Priority Arguments in a-recursion Theory, Ph.D. Thesis, M.I.T. 1972. 1171 S.G. Simpson, Admissible Ordinals and Recursion Theory, Ph.D. Thesis, M1.T.

SOC. 124 (1966) 1-23.

Recursion Theory (North-Holland, Amsterdam, 1967) 243-263.

(1972) 343-367.

111 pp.

1971.

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., 19 74

MORE ON SET EXISTENCE

Frangoise VILLE University of Orleans

6 1. Standard sets of a theory: definition and general results

The structure of the standard sets is, by definition, the structure ( V , Ev) of the cumulative hierarchy; it is extensional and well founded. The transitive closure of an element a of V is denoted by [ a ] .

at least the binary relation symbol E. Realisations of k have thus the form vM,EM, ... >; i f m is an element ofM, [mIEM denotes the transitive closure of m with respect to €*.

To avoid confusion, membership in the metalanguage will be denoted E.

Let us suppose that (T) is a theory formulated in a language k, containing

Definition 1 . Let 311 be a realisation of L. An element a of V is called a standard set of% iff there is an ma in M such that:

ma is said to represent a in %.

If ( M , EM) is extensional, a standard set is represented by at most one element of M .

Definition 2. The element a of V is called a standard set of the theoxy (T) iff a is represented in every model of (T). S(T) denotes the collection of all standard sets of (T).

Note that (S(T),iEv I' S(T)) is a transitive substructure of CV, EV). When (T) C k,, (where L, = V, = H, = the collection of well founded

195

196 F. VILLE

hereditarily finite sets) and so, each formula of (T) is of finite length, it can be shown by a compactness argument that every standard set of (T) is hereditarily finite: thus S(T) C L,. I f (T) is a denumerable set of formulas (of denumerable length) C Lu1,, then one easily deduces from the Lowenheim- Skolem theorem that every element of S(T) is hereditarily countable.

A brutal generalisation of these facts would be: “LetA be standard and admissible, and let (T) be a theory in the language LA (associated withA, as in Kunen, Banvise etc.), then S(T) CA”.

But this cannot hold, as shown by the following example: TakeA = Lw,k, and the following theory (T):

- the binary relation symbols E and =, - for each integer n, a constant symbol c,, - two distinguished constant symbols c, and c,.

- extensionality - Vx x 4 co and for each integer n:

Language of (T):

Axioms of (T):

vx (x Ec,+l -(x = C n V X Ec,)) - vx (x Ec, +-+ Wn&,X = C,)

- vx (x E cw +x Ec,) - for each integer n:

c , ~ Ec, iff n & W

c, +cw iff n $ W ,

where W E Il - Xi, for example the set of Godel numbers of recursive well orderings.

Clearly (T) C

W q! Luck, so S(T) not I I ~ .

bounding the complexity of (T):

(*) Provided that (T) has an extensional model, if (T) is an L

and S(T) = o U { W} U {a}; but as is well known

LWck. Note that the theory (T) above is n: U 2: but 1 1

The principal aim of this paper is to show that S(T) C LUfk is obtained by

k-r.e.theoryof wF

‘LwFk> then s(T) Lwfk.

Below we shall use o1 to denote the least non recursive ordinal which we called oik above. So LW1 will be the set of all sets constructible with recursive

MORE ON SET EXISTENCE 197

order. We shall also make the following assumptions on (T): - there is at least one extensional model of (T) - (T) is a theory with equality In the following(T) will be an Lu1-r.e. theory of LL .

wi

92. Properties definable on S(T)

In this paragraph we assume that D is a subset of S(T) which satisfies the following condition: (D) Every element of D is defined in every fnodel of (T) by a formula of

So that for every a in D, for every model 311 of (T) there is an m, in 9?2 which represents a in 5% such that Vx (x E ma - @.,(x)) is true in %.

that we shall denote by @,(x).

Definition 3. a) A subset X of D is LU1-definable on D in a given model '% of (T) iff there is a formula GW(x , x l , ..., x,) of LLWl with n + 1 free variables, and parameters a l , ..., a, in nZ such that X = {a E V : (ma, a l , ..., a,) satisfies 4% in% 1 n D.

in every model of (T). b) A subset X of D is LUl-definable on D in (T) iff it is LU1-definable on D

Examples: model 371 of (T) by the formulax Exl with parameter ma.

graphs of + and X, and every arithmetic subset of w; here L,, or more precisely, its representative mL,, can be used as a parameter.

a) for each element a of S(T), a n D is defined on D in each

b) If L, E S(T) and L, C D, then o is definable on D in (T) as well as the

Below we shall need the following analysis of LU1-definability on D in terms of the consequence relation in (T):

Theorem 1. Under the condition (D), if X C D is LUl-definable on D in (T) then there are formulas $ (x) and $ 2(x) in LL, such that for evey a in D:

with only one free variable

198 F. VILLE

Theorem 1, which can be proved via the omitting types theorem for denumerable admissible languages, is fundamental for the proof of (*). The case for L, is treated in chap. 6 of [Kr.Kr].

Kp ( 5 3 ) or not (84). We shall distinguish two cases according to whether (T) is an extension of

$3. (T) is an extension of KP

In this case S(T) which obviously contains every element of L, will provide, in every model of (T) all the parameters we need.

Every model of (T) is extensional; then by the remark following Definition 1, every element of S(T) has at most one representative in every model of (T).

Every model of (T) is admissible; then if S(T) has no infinite element, S(T) = L,. If, on the contrary, there is an infinite element in S(T) then L, E: S(T) and we have LW1 C S(T) (cf. [8] p. 243). Note that in each case:

Lemma 1 . mere is an L, l-recursive mapping Q -+ @,(x)from LU1 into XL such that for every a in L, n S(T), @&) defines a in evety model of(T).

-

W 1

This is an easy consequence of the extensionality of every model of (T). Therefore LW1 satisfies condition (D) of $2.

Lemma 2. Let a subset X of L, be L, l-definable on L, in (T). Then X is L, recursive.

As in Theorem 1 we prove that there are $2 in LL such that for W 1

every a in LU1 :

For the definition of KP see [S] p. 232.


The applications: a 3x($,(x) A $l(x)) and: a + 3x(,(x) A J / ~ ( x ) ) are LU1-recursive by Lemma 1; therefore, consequence in (T) being LUl-r.e., X is an LU1-recursive subset of Lwl:

Theorem 2. w1 $ S(T).

Let us first prove one lemma about model of KP.

Lemma 3. Let % be a model of KP and R a linear ordering of w which is A l - dejinubb in%. For evep ordinal a of % if there is an isomorphism f of h,EM I’ a> onto an initial segment of the ordering R then f belongs t o m .

f is an isomorphism from (a, EM r a> into an initial segment of R (which we shall denote < R ) iff f satisfies the following equations, I ( f , a):

f(0) = a iff a is the least element in the ordering < R ,

f (h) = b iff b is the least element of R greater than any f ( 5 )

andforallX<a:

for ( < h in the ordering < R .

Now we want to prove that we have in% :

for every ordinal a, there is one and only one f such that Zcf, a).

Note that there is a Zl forrmlaIs(f,a) which describes the relation f is a fitnction of domain a ~ Z ( f , a ) in CYfC(we allow parameters) since R is A l in%.

By induction on ordinals we establish that 3 ! f I s ( f , a) holds in m for every a. The uniqueness is easily established, so w have to establish the existence; we proceed by induction:

- if a = $9 thenf= $9 and is the only fsatisfyingIs(f,a) in%.

- if a f (?J then by induction hypothesis there is a unique f , say f,, which satisfiesIssCfE,E) in %(for t < a). Thus,

- if a is the successor of 0, we put f , = f p U (0, b ) where b E w i s chosen as described in I. As f p belongs to W , f a obviously belongs to311 .

- if a is a limit ordinal, one can apply X l-replacernent, since Is E Z and, by induction hypothesis

200 F. VILLE

Thus there is an f in CM, such that: W/= (f = f, : ( <a). By Al-comprehension Uf belongs to CM. Puttingf, = Uf we have:

Proof of Theorem 2. Let Q be a recursive linear ordering of w with an initial segment IQ of length w1 (for example the ordering in Candy’s [GI). Since Q is a recursive subset of w 2 , Q can be defined in every model of (T) by a A o - . formula, using o as a parameter.

represented in) 371. By the choice of Q, there is an isomorphism f from (wl,EM r w,) into (ZQ, Q ) . Since the hypothesis of Lemma 3 are satisfied, fbelongstoW.ZQ can thenbedefinedby 3 $ € w 1 ( E , x ) E f w i t h x a s the only free variable and the two parameters w1 and f.

A:, since ni over w or over L, is equivalent to LW1-r.e.(cf. [BGM]). But th i s contradicts the fact that ZQ is a ll: subset of w which is not xi.

Suppose o1 E S(T); and let W be any model of (T): belongs to (is

So Ze is defined on w in (T) and, by Lemma 2, ZQ is L,, -rec; and hence

Thus we cannot have w , & S(T).

Theorem 3. S(T) C LU1.

Lemma 4. Let % be a model of KP and a be a subset of L which belongs to W. Then there is an ordinal a of W such that a C L,.

Since W is admissible, the function od which associates to each constructible x its order [ is expressible in 311 by a formula Od which is Z over CM and verifies %k V x ~ a 3 ! y Od(x ,y ) .

By C , replacement there is a b in% such that

%k b = {Od(x) : x EM a} .

Take a to be the least upper bound of b; it is an ordinal of 71’2 and a C L,.

Proof of Theorem 3. Assume a is an element of S(T) included in LW1 ; the example a) following definition 3 shows that a is LWI-definable on LU1 in (T)

MORE ON SET EXISTENCE 20 1

and Lemma 2 shows that a is LW1 -recursive. In every model 7 R of (T) there is, by Lemma 4, an (Y such that a C La, and

this a does obviously not depend on cPZ; therefore a € S(T) and by Theorem 2, a < wl. So a is an LUl-rec. subset of LU1 which is LU1 bounded, i.e. a E LW1.

An induction shows that every element a of S(T) belongs to L, 1 .

Theorem 4. S(T) = L, or S(T) = L, 1.

Either S(T) has only finite elements and S(T) = L, ; or there is an infinite element in S(T). In this last case we saw that LWt C S(T); by Theorem 3 S(T) C LW1, thereforeS(T) = Lwl.

Theorem 4 of Chapter 8 in [Kr.Kr] can now be generalised to JL,~.

Definition 4. A subset X of S(T) is uniformly LU1-defined in (T) iff there is a formula CP of JL with a single free variable, such that for every model 717 of (T)x= { a E ~ w h ~ @ ( a ) } .

Theorem 5. Every subset X of S(T) which is uniformly LU1 definable in (T) is L, -finite.

By Theorem 4 and Lemma 1 every element a of S(T) is defined in (T) by formula $&) such that the mappinga +$Jx) is LW1-recursive. If X C S(T) is uniformly LW1-definable in (T), it is a fortiori LUl-definable on L,, in (T), and by Lemma 2 it is L, l-recursive. Suppose qh is a formula of JL which defines X uniformly in (T). Let c be a constant symbol not occurring in (T). The set of axioms

w1

is Lwl-r.e. and is obviously inconsistent. By compactness there is a b E L, such that

is consistent. Therefore

202 F. VILLE

and, as c does not occur in (T)

This means that X is bounded by b, which implies X & Lwl.

94. (T) is not an extension of KP

It is convenient to introduce auxiliary theories (Tf) and ( E ) , formulated with an additional type of variables. (Ti) is needed to formalize the meta- mathematical notion of admissibility; let us call “metauniverse” the universe of models of (Tf). (E) relates standard sets of (T) to “real standard sets” i.e. standard sets of metauniverse.

Language of (Tf): We assume that (T) is consistent with extensionality.

- its variables, of a type different of the types in (T) are denoted X , Y , Z , ... - two binary relation symbols E‘ and =‘.

So the atomic formulas of this languages are of the form X E’ Y , X =’ Y ... . The axioms of (Tf) are those of KP formulated in this new language. Language of (E):

symbol of (T)) and of the type used in (Tf).

The new atomic formulas have the form x E X . The axioms of ( E ) are:

- variables of the type of the arguments of E (the distinguished relation

- a binary relation symbol E , as well as the symbols E and E’.

(1) 3!XVx(Vyy 4 x -+x EX) ( 2 ) V x ( V y E x 3 ! Y y E Y + 3 ! X x E X ) (3) VxVyVXVY(x E X A y E Y +. ( X E y * X E’ Y ) ) (4) VX(VY E’ x 3 ~ y E Y+. iix x EX).

Let (T’) be (T) U (Tf) U (E). Every realisation of the language of (T ’) has the form:


Definition 4. A standard set of (M U M’, ... ) is a standard set of the meta-universe, i.e. of ( M I , E&, , =;Mt).

Lemma 5. Let 311 i= (E) and (M, E M ) be extensional, Then the restriction of EM to the well founded part Mo of M is the graph of a function which maps representatives of standard sets in (M, E M ) onto representatives of standard sets in (M‘,E&~).

We prove that for all x in Mo 5% 3 ! X(x E X ) by induction along €* rMo with the help of axioms (1) and (2). Thus there is a function, say cp, defined every where on Mo whose graph is EM l- M o X M . By Axiom (3) this function embeds ( M o , €M PMo) isomorphically in (M’,€&t); by Axiom (4), writing € M for EM r M o

Therefore if ma represents the set a in ( M , E M , = M ? , @(ma) represents a in (MI, €ht,=L*).

Lemma 6. (TI) is consistent.

Let 311 = ( M , CEM, ... ) be an extensional model of (T) and Mo be its well founded part w.r.t. EM. Define by induction along EM, (i.e. € M 1 Mo) a function $ from Mo into V by:

$I is every where defined on Mo. Put: M i = $[Mo]

Mf = the least admissible containing M l El = e V I M f = = identity in Mf E l = the graph of $ 311’ = (MU Mf, E M , El, =1, El , ... ) .

311’ is a model of (T’) since: - (M, EM, z M ) is by hypothesis a model of (T) - (Mf , E 1, =1 ) is admissible and therefore a model of (Ti) - %’ satisfies (E) by definition of $ (cf the proof of Lemma 5).

204 F. VILLE

Lemma 7 . Every standard set of (T) is a standard set of (T’).

As (T) has an extensional model, (T) U {Extensionality} is consistent; it

By Lemma 5 every standard set of (T1) is a standard set of (T’): therefore is consistent with (Tf) U (E). Let (T1) be (T) U {Extensionality}.

as (T) C (T1), every standard set of (T) is a standard set of (T’).

Theorem 6. S(T) C LU1.

(T‘) is an Lwl-r.e. theory of .CL

is equal to either L, or LU1. By Lemma 7, S(T) C S(T’). Therefore such that (T‘) 3 Kp; by Theorem 4 S(T‘)

w1

S(T) c LW1.

This result evidently generalizes the results of [GKT] which establishes if a theory (T), formulated in the language of finite types, i n n : then subsets of w which appear in every a-model of (T) are hyperarithmetic and every collection of subsets which appears in every w model of (T) is Lwl-finite.

How to avoid this condition will be shown in a forthcoming paper. The only restriction we make on (T) is its consistency with extensionality.

References

[B] J. Barwise, Infmitary logic and admissible sets, JSL 34 (1969) 226-251. [BGM] J. Barwise, R.O. Gandy and Y. Moschovakis, The next admissible set, JSL 36

[GI R.O. Gandy, Proof of Mostowski’s conjecture, Bull. de 1’Ac. Pol. des Sc. VIII

[GKT] R.O. Gandy, G. Kreisel and W.W.Tait, Set existence, BuIl. de l’Ac.Po1. des Sc.

[Kr. Kr] G. Kreisel and J.L. Krivine, Elements of mathematical logic (North-Holland,

(1971) 108-120.

9 (1960).

VIII, 9 ( 1 960).

Amsterdam, 1967).

PART I11

INDUCTIVE DEFINABILITY

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., I974

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

S t a AANDERAA * IBM Thomas J. Watson Research Center, Yorktown Heights, New York

Abstract. We shall prove some theorems about inductive definitions and their closure I < I Z; I < 1 Al-monl ordinals. As corollaries we obtain the following results. I A; I <

= I A : l = I ~ ~ - m o n l < l n : l < IA:-monl.WealsohavethatInAI#IZ;:,I,InAI+ IZA-monl, IxAlZ l a - m o n l for all n. Moreover if n 2 2, then IAA-monI = IAAlS IZA-monI < IC;l< IALl-monl and aA-monI= I A A l I InA-monI < Inhl< I AA+l-mon I . Moreover, assuming the axiom of constructibility we have IAL I = I XA-mon I < Ix; I < Ink1 < IAA+l I, for all n 2.2. However, assuming that both Projective Deter- minacy (PD) and the axiom of Dependent Choices (DC) hold, then I & l = IXil-monl < IZ:~I < I I I ~ ~ I < I A Z ~ + ~ I = NIzi+l-mon~ < I I ’ I ~ ~ + ~ I < ~Z~i+~~.completeproofsaregiven for the most important results. Many less interesting results are stated without proofs, or with a sketch of a proof.

1 1 1

1. Introduction

Let w denote the set of nonnegative integers which coincide with the ordinals less than w.

Definition 1. By an inductive definition (abbreviated i.d.) we shall mean a mapping r : P(w) + P ( o ) , whereP(w) = {XlXE w } = the set of all subsets of w. (“Inductive definitions” is abbreviated i.ds.)

Definition 2. An i.d. r is monotone iff X C Y E w implies r ( X ) E r ( Y ) .

Definition 3. Let I‘ be an i.d. and let A be an ordinal number. Then we define I‘h to be a set of integers, defined by transfinite recursion as follows:

rO=O. * On leave from Institute of Mathematics, University of Oslo, Blindern, Oslo 3, Norway.

201

208 S. AANDERAA

For X > 0 we have

Definition 4. Let r be an i.d. The closure ordinal of such that I'h = r'w = rIr' is the set defined by r.

is the least ordinal X We shall let I rl denote the closure ordinal of r, and

Remark I . 1 rl exists for every i.e. r, and I is a countable ordinal.

Definition 5 . Let ww be the set of all functions of one number variable, i.e., the set of all total functions mapping w into w. Subsets of the product space

x = X , x ... xx, (X i = o or X i = P ( o ) or Xi = ww for all i = 1,2 , ..., k ) ,

will be called pointsets. Sometimes we think of these as relations and we write interchangeably x € A o A ( x ) .

Remark 2. It turns out that it is convenient for us to permit Xi = P(w). In this way we deviate from the definition of pointsets in Moschovakis 1970 and 1972.

(n ,m) = l /2(n2 + 2nm +m2 + 3m +m) as in Rogers 1967, p. 64.

according to the following definition.

For n , m E w, let ( n , rn) be a coding of pairs, say

We shall usefo,f1,f2, _.. to denote some functions throughout the paper

Definition 6. For each i E 0, fi is a mapping of w into w defined as follows: fi(x) = ( i , x ) , x E w, and& is also a mapping o f P ( o ) intoP(w) defined as follows: fi(S) = { f i ( x ) 1 x E S } . Moreover fcl is a mapping of P(w) into P(w) defined as follows:

Let a E w w . Write (a)i for the function h x a ( ( i , x ) ) , i.e. E ww and (a) j (x) = a( ( i , x ) ) .

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS 209

We do not need the full axiom of choice in this paper. At some places we need a weak axiom of choice:

Dependent Choices (DC). For each A C ww X ww we have

This follows from the axiom of choice. In some cases a still weaker axiom is sufficient:

Countable Choices (CC). For each A 5 w X ww we have:

The axiom of Dependent Choices implies the axiom of Countable Choices.

Note that the axiom of CC has to be used in order to prove that the follow- The axiom of Determinacy (AD) also implies CC.

ing prefuc transformations are permissible:

... vo31 ...+ ... 3lVO ...

... 3OV' ._. + ... v 3 1 0 ...

(See Rogers 1967, p. 375 and Exercise 16-2, p. 446.)

Definition 7. A point class C is a class of pointsets, not necessarily all in the same product space.

I t turns out to be convenient for us to represent each i.d. F by the pointset

instead of the pointset

{ (A , r (A) ) iA)cWwxWw.

We shall identify an i.d. with its representation. Hence r €C? means

2 1 0 S. AANDERAA

Given a pointset C? we usually in this paper pay attention only to the subclass(WwXw) n C o f e .

Definition 8. ZA is the class of all pointsets definable from recursive relations by EA prefix. Similarly for HA.

Definition 9. Let r be an id . , then fi = (wo X w ) - r (i.e. k(S) = w - r(S) for all S E w).

Definition 10. Let 6 be a class of pointsets. Then 8 = {x - A 1A E x and A E C}. (Here x is a Cartesian product as in Definition 5.)

Remark3. L e t r b e a n i . d . T h e n ? € l l f , i f f r E Z f , a n d f I f , = E f , , c f , = n f , .

Definition 1 1. Let C be a class of pointsets. Then the supremum ordinal for C is I C I = sup { I r I I r is an i.d. and l? is an i.d. and E C}.

Definition 12. LetA C w and l e t H be a pointset such that H E X , X X 2 X ... X X , where the Xi’s are as in Definition 5. Then

= d(x1t+x2, ..., x,-~) GI, x 2 , ..., x k b l , A ) € H I if Xk =P(w) and & = H otherwise. Let C be a class of pointsets. Then H A = {# I H € C}.

Remark 4. Note that if C =

Rogers 1967. (See pp. 409,374 and 304.) In the same way, if C= nA then then C A = ZAA, where is defined as in

eA =nAA.

Definition 13. Let C be a class of pointsets. The ordinal h is called a terminal ordinal for C iff there exists an A C w such that h = ICA 1. The set of terminal ordinals for C is denoted by ter(C).

Definition 14. Let C be a class of pointsets. The ordinal h is a transit ordinal for C iff there exists an A S o such that h < ICA I but I r I # h for a l l i.ds. rEe.

Definition 15. Let A and I’ be i.ds. Then A is one-one reducible to r up to h

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS 2 I 1

(notation: A ‘r) iff there exists a one-one recursive function f such that (Vt) (Vx) (t < h * (x E AE -f(x) E fl)). A is one-one reducible to r at everyordinal(notation: A < T I ’ ) i f f A I t r w h e r e h=(sup{lAl, Irl})+ 1.

Definition 16. Let A and I‘ be i.ds., and let h be an ordinal; and let g be a mapping of ordinals into the ordinals such that g ( t ) is defined if r; < A. Then A is one-one reducible to the g-l contraction iff there exists a one-one recursive functionf such that (Vt)(Vx)(t<h * (x E AE -f(x) E

at every ordinal (notation: A < 7 Sg I‘) iff A < i I r where h = (sup{lAl,lrl}) t l.(Weshall w r i t e A < f < r tomean t h a t ( A I t I g r ) holds for some g. A 5 7 5 I‘ is interpreted similarly.

up to h (notation A 5 $ I?)

A is one-one reducible to the g-’-contraction of r

Definition 17. Let A and r be i.ds., then A is one-one reducible to a reorganization of r up to h (notation A 5 = r) iff there exists a one-one recursive functionfsuch that the following two conditions are satisfied, for all ordinals 8, r;, where r; < A.

1. f-l(re) c AE * f - l ( r e + l ) c ~ 5 + 1

2 . f-l(re) - A’ # 0 * re+l - re + 6. A is one-one reducible to a reorganization of r (notation: A 5 T = r), iff A < t = r where A = IAI.

Remark 5. Let A and r be i.ds. and let h be a one-one recursive function. Then “A 5 T r via h” small mean that A 2 functionfmentioned in Definition 15 can be chosen to be equal to h , i.e., ~ ‘ ( f l ) = AT for all g <_ I A I . Similarly for the following expressions:

’< r via h”, “A 5” < r via h”, “A 5: N_ r via h” “A <1 hr via /I” “A II -g

and “A 5 ; s I’ via h”.

r where the recursive one-one

1 - z

Definition 18. Let inductively Ccomplete iff r E C and (VA) (A C o X o & A E C * A 5 7 r).

be a class of point sets, and let I’ be an i.d. Then r is

Definition 19. Let e be a class of pointsets. Then the spectrum of C (denoted: sp(e) ) is the following set of ordinals: {lrl IF€ e and F is an i.d.1.

Remark 6. We shall from mow XI assume that the axiom CC holds.

212 S. AANDERAA

2.

We shall prove the following results.

Theorem 1. Let C = ZA or C = Il;. Then there exist an i.d. r which is inductively e-complete and such that Ah S1 I’” for each i.d. A in e, and for each ordinal h Hence IF1 = I el E Sp(C).

Remark 7. It is proved in Aczel and Richter (1972) that I Xi I f IrIiI; and they have later obtained a lot of results including proofs of theorems 1-4 in this paper.

Theorem 3. Let e = II; or let e = E: for some n. Suppose I C! I L I c I. Let r and A be inductively C-complete and inductively C-complete, respectively. Let g(h) = h t 1 i f h is a successor ordinal, and let g(h) = h otherwise. Then A < T < g r a n d riiAl+l sg A.

Definition 20. Let K be a set of ordinals. An ordinal X is apoint of accumulation of K iff (Vt) ([ <A * ((31) ([ <r) < X & r ) EK)), and h is an isolated point of K iff h E K and h is not a point of accumulation of K . The set of points of accumulation of K is denoted by Acc(K) and the set of isolated points of K is denoted by I,(K).

Theorem 4. Let C be a class o f pointsets equal to J I A or EA for some n 2 1 Then

Acc(tra(C) U ter(C?)) = Acc(tra($)U ter(8))

and

tra(C) - Acc(tra(C?) U ter(C)) = t e r (6 ) - Acc(tra($) U ter(6)).

Remark 8. Tra( I l f ) = 6. Hence Tra( I l i ) # Ter(Ti) = Ter( Ei). We do not know whether Tra (Xi) = Ter (Ill), but we would like to state that as a con- juncture. We do not know if it is consistent with set theory to assume tra(nA) - ter(ZA) f 6 or tra(E;) - ter(II;) f 0 for some n. Theorem 4 will not be proved in this paper.

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS 213

To state the next theorem we need the notions C-norm and Prewell-ordering(C).

Definition 21. Anorm on a set p is a function 4 : p +. ordinals; we call a C-norm if there are relations s a n d <I i n .C and @, respectively, such that (VY (y € p =2. (VX) (x 5 Y - i 3 Y - [x E P c!?L 44x1 5 4(Y )I 1).

Definition 22. Prewellordering ( C ) every pointset p in C admits a C-norm.

Theorem 5. Let r and f. be i.ds., and let r E C and € C?, where C = C ,!, or c = n,!, for some n 2 1. Suppose r(s) - s + @ 3 D =+ F(s> - s c w. Then there exists an i.d. A E C such that I' 5; 5, A and I A I = I f I + 1, where g is as in Theorem 3.

Theorem 6. Let C = C,!, or C = Ili for some n > 1, and suppose Prewell- ordering (C). Then I C I < I 6 I .

Corollary 3. Suppose that every element in Godel, then lE i l< In,!,Iforall n 2 2.

w is constructible in the sense of

Corollary 4. Assume Projective Determinacy (PO) and the axiom of Dependent Choices(DC)hold. Then l ~ ~ i - l l < l C ~ i - l l a n d lCi i l<lrI i i l f o r a l l i 2 1.

These are the results which will be proved in this paper. At the end of the paper some recent results are stated which will be proved in a later paper.

2. Proof of Theorem 1.

Let C = or C = n,!, for some n 2 1. Then there exists a pointset E E C? such that E S w X w XP(w), which enumerates all pointsets in P(w XP(w)) n C in the sense that if A E C and A C w XP(w), then there exists number i such that

A ( x , S ) -E( i ,x ,S) for all x E w and S S w .

214 S. AANDERAA

We shall write x E Ai(S) for E(i,x,S). Then we have that for each i.d., A E C there exists an i E w such that Ai = A. Let r be an i.d. defined as follows: (1) r(s) = {XI ( 3 Y ) ( 3 z ) ( x = f z ( Y ) &Y EAz(f;l(o)}

= u ;=o fi (4 (fL1 (S))). (Recall thatf,(x) = (z,x), see Definition 6.) Then r E C , and by induction on h we can easily prove that (Vx) (x €A; - f z (x) E r,”). This proves that I‘ is C-complete since if A is an i.d., and A E C, then A = Ai for some i and Ai 57 r via fi.

To prove the last part of the theorem we shall prove a lemma.

Lemma 1. Let C = Ek or C = l l L for some n. Let r be an i.d. in e. Then there exists an i d . r0 in C such that I’ 57 ro and I” I I?; for each ordinal A.

Proof. Let ro be defined as follows:

We shall now complete the proof of Theorem 1. Let r be defined by (1) as before and choose ro E C such that r 5; ro

and rh < I ri for all ordinals h which is possible according to Lemma 1. Then ro = Ai for some j E o. Given i E o , we have A t I1 rh <1 ~$7 L1 I-”. Hence 45 <_ r” for all ordinals h. This completes the proof of Theorem 1.

3, Proof of the Theorems 2 and 3.

T o prove these theorems, we shall prove a lemma.

Lemma 2. Let C = E or C =n;, and let Aand I’ be i.d.5 such that A E and r E C. Then there exists an i.d. Z = ZA*r, such that Z E (? and

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS 2 15

Proof. We shall first outline the general idea of the proof, before giving the details of the construction. The main problem we have to solve is to construct E E 6 from A E 6 and I' E C? such that rh is coded in Z h in some way. First we note that E e. Suppose that we can recover rh from 2'. We cannot get r(rh) directly, but we can get f'(rh) = w -I"+'. If 4 < h then o - rh C w - fi. Hence we cannot save w - I" directly. But we can use the elements Ah as indices. Hence we can try to choose Z such that Z h = UElhAE X ( w - r E ) . We shall use a somewhat more complicated construction. Let I , = {x t 3 1 x E A,}. In order to generate the indices I , , code AX as fo(Ah), and I" is coded as U <, (IE X(w-I't)). In this way, we can go on simulating both A and r as long as A generates new indices. (In the proof of Theorem 5 we shall use another construction to obtain new indices.)

E-

In order to obtain IEl= IA l+ 1 if 1A1= Irl we addfi(0) = ( 1 , O ) when FA+' = rh. Moreover, in order to obtain I' Lir"' 2, Z, we add ( 2 , ~ ) to

when x E I".

We shall now give the details of the construction. LetH(S)= ( ~ ~ ( ~ U ) ( ( U + ~ , U ) ~ S & ( O , U > E S ) ) . We constructZfrom Aand r as follows.

v

Then Z is an i.d. and E E C. We can easily prove by induction on A.

i) ii) (vA)(A< IAI * rh=H(zh))

iv)

(VA) (VX) (x E A, - fo(x) E 2)

iii) (VA) (A < I A I * r A = fgl(zA+')) (Vh)((h< 1121 and X an limit ordinal) * rh =f;'(?')).

216 S. AANDERAA

Moreover, IZI = 1Al except when IA l= lrl. If lrl< IAl then ( 1,O) E Elr1*' - Elr'. Hence IZI = 1 I'l+ 1 = IAl + 1 if I A1 = I rl. Lemma 3 now follows easily and the proof of Lemma 3 is complete.

To prove Theorem 2, suppose I ZA I = In; I for some n. Choose A E Xi and r En: such that IAl= ICAl and l r l = IIIAI. Apply Lemma 2, and we get ZEZ,!, such that ! E l = IA(+1 = lZ,!,ltl, whichisacontradiction.Thisproves Theorem 2.

To prove Theorem 3, let Z = rl; or C = Xi and suppose I6 I 5 I C I. Then 161 < I C I by Theorem 2. Let r and A be inductively e-complete and inductively C-complete, respectively. Construct Z, =

Lemma 2 . Then El E 6! and Z2 E C and A 5; Z, 5; A and r <\A'+1 Sg Z,. Hence A <: 2, I'. This proves Theorem 3.

and Z2 = Zrl* as in

5;""' 5, A. Moreover, r <; E , 5; r and A 5; <g Z,. Hence

We shall not prove Theorem 4 in this paper.

4. Proof of Theorems 5 and 6 and their corollaries

Let C = XA or C =HA and let r and c satisfy the assumptions in Theorem 5. The proof is very similar to the proof of Lemma 2. The difference between the constructions in the proofs is the generation of indiceslA. We shallnowletZA= { ~ + 3 1 ( 3 . 9 ( ~ < h & x € f ' ( r ~ ) ) } . We shall as before now code I , asfo(ZA). Then ( 1 , O ) - Alrl. The details of the construction are as follows. Let as beforefi(y) = ( i , y ) (see Definition 6) and letH(S) =

{ u I ( 3 u ) ( ( v + 3 , ~ ) ~ S & ( O , ~ ) E S ) } , a s i n theproofofLemma2. LetA(S) be defined as follows:

- I , # fl if h < I I'l. As before, we let

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS 2 1 7

Then A is an i.d. and A E 6. Moreover, we can easily prove by induction A. i. rh = H ( A ~ )

ii. F(H(A~)) c rh+l

iii. iv. rh = ~ Y ' ( A ~ + ' ) v.

Hence / A / = /PI+ 1, since lrl I IAl and( l ,O)EAlr '+ ' -Air'. Moreover, by iv. and v. above we have r 5; 5, A. This completes the proof of Theorem 5.

rX+l - r h # @ * F(H(A~)) - H ( A ~ > # 0

rh =fF1(Ah) if h is alimit ordinal.

We shall now prove Theorem 6. We need one more definition.

Definition 23. An i.d. r is single-valued iff x E r(S) andy E r(S) implies x =y. The set of single-valued XA i.d.'s is denoted by '"X; and '"IT: denote the set of single-valued IT: i.d.'s.

Theorem 6 is going to be an immediate consequence of Theorem 5 and the following lemma.

Lemma 3. Let E = X,!, or C = n,!, for some n 2 1, and suppose Prewellordering (C). Let r be an i.d. in C . Then there exist i.d.3 i; in e and in (?, such that f' is single-valued and i f r(S) - S # $ then $9 # p(S) = f(S) C r(S) - S. More- over, i f F(S) - S = @, then l?@) = w and f'(S) = $.

Proof. Let r'(S) = r(S) - S . Then I" E C. Consider the set A = { (x , S ) I x E r ' ( S ) } . By Prewellordering ( C ) we have that there exists C-norm 6 of A . Then there are relations I in C and 2 in 6 such that if T c w andy E w, and i f y E r"(T), then ior all x E w and all S C_ w we have

Then we define f as follows.

Then fi E C . Given S, and suppose r'(S) # 6. Let hs be the least ordinal h such that h = @((x,S)) and x E r'(S) for somex. Let xs be the least x E w

218 S. AANDERAA

such that A, = @((x,S)). Then r(S) = {x,}. If I?'@) = 0 then r(S) = 6. Hence r is single-valued. We now define r as follows

x € i.(S) - (Vy) ( y E F(S) * y = x)

Then r(S) = F(S) if F(S) # 0 and f(S) = o if f'(S) = 0. Lemma 3 now follows immediately.

Theorem 6 follows from Theorem 5 , Theorem 1 and Lemma 3 , by choosing r € C such that lrl= IC I . Then we obtain a A € C such that IAl= Irl+l = I el t 1 by Lemma 3 and Theorem 5. This proves I CI < ICI, and the proof of Theorem 6 is complete.

Remark 9. We may obtain a slightly stronger result than Theorem 6 by using a weaker assumption than Prewellordering (C). It is enough to assume that there exists a norm on A and a relation R S o X w X P(o) such that if ( y , S ) € A (i.e.,y E r ' ( A ) ) then we have for all x E o

In this case we can define i. as follows

Then r(S) - S # fI * fI # e(S) - S C r(S).

We shall now show that Corollaries 1-4 follow from Theorem 6. We have that Prewellordering (Hi ) and Prewellordering ( E i ) (see

Moschovakis 1970, p. 3 3 ) . Hence In: I < I Xi I and I I $ l < Inil by Theorem 6. This proves Corollaries 1 and 2.

According to Moschovakis 1970, p. 33 , the arguments of Addison 1959a, suffice to show that if' every real number is constructible in the sense of Godel, then for each k > 2, Prewellordering (EL). By using this fact we obtain Corollary 3 from Theorem 5 immediately.

Finally, we have that PD and DC imply Prewellordering(Hii-l) and Prewellordering ( X i i ) . (See Moschovakis 1970, p. 3 3 , or see Martin 1968.) Hence Corollary 4 follows.

INDUCTIVE DEFlNITIONS AND THEIR CLOSURE ORDINALS 219

5. Further results

We shall now state some results without proofs. A full treatment will be published elsewhere.

Remark 10. Let w , be the first non-recursive ordinal. Spector showed that l n l -monI= 1 I I I~ -monI=o l<IA~l .Accord ing toT .Gr i l l i o t , IC:-monl= IE!l.

S. AANDERAA

Acknowledgement

the problem of deciding the order relation between I I l i I and I E; I .

on the preliminary draft of this paper.

I wish to thank Jens Erik Fenstad for having encouraged me to work on

I am also indebted to Peter Aczel and Wayne Richter for helpful comments

I am also grateful to Leo Harrington for helpful discussions.

References

[ 11 Aczel, P. and W. Richter, Inductive definitions and analogues of large cardinals, in: Conference in Mathematical Logic, London, 1970. Lecture Notes in Mathematics Nr. 255 (Springer, Berlin, 1971).1-9.

[2] Addison, T.W., Some consequences of the axiom of constructibility, Fund. Math. 46 (1959) 123-135.

[ 31 Addison, T.W. and Y .N. Moschovakis, Some consequences of the axiom of definable determinateness, Proc. Nat. Acad. Sci., U.S.A., 59 (1968) 708-712.

[4] Martin, D.A., The axiom of determinateness and reduction principles in the analytical hierarchy, Bull. Amer. Soc. 74 (1968) 687-689.

151 Moschovakis, Y.N., Determinacy and Prewellordering of the continuum, in: Math. Logic and Foundations of Set Theory, Y. Bar Hillel (ed.) (North-Holland, Amsterdam, 1970) 24-62.

[6] Moschovakis, Y.N., Uniformization in a playful universe, Bull. Amer. Math. SOC. 77

[7] Rogers, Hartley, Jr., Theory of Recursive Functions and Effective Calculability

(81 Spector, C . , inductively defined sets of natural numbers, in: Infinistic Methods

(1971) 731-736.

(McCraw-Hill, New York, 1967).

(Pergamon Press, Oxford and PWN, Warsaw, 1961) 97-102.

J.E.Fenstad, P. G. Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., I974

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

Douglas CENZER University of Michigan and

University of Florida

1. Introduction

An inductive operator r over a set X is a map r fromP(X) to P ( X ) such that for all A S X , A C r (A) . r determines a transfinite sequence {P: a € ORD (ordinals)}, where F a = U{Yp : f l < a } for a = 0 or (Y a limit and Fa+' = r(r7. r is monotone if, for all A , B inP(X), A E B implies r(A) C r ( B ) .

clearly I r I always has cardinality less than or equal to F. The closure r of r is rIr', the set inductively defined by r.

Inductive definitions are basic to the development of recursion theory. Following the methods of Kleene [ 1 1,121, we will define ordinal recursion and recursion in a partial functional by means of inductive definitions.

Given a class C of inductive operators, one would like to characterize the closure ordinal ICI = sup{lrl : r € C} and the closure algebra c= {A : A is 1-1 reducible to T for some r E C}. We write C-mon for the class of monotone operators in C.

The following is a brief summary of results on inductive definitions over the natural numbers.

The first significant results on inductive definitions were obtained by Spector [24], who showed that (IIl-monI = wl, the first non-recursive ordinal, and that n y - m o E l l i -mon = nt. Candy (unpublished) later showed that In: I = w 1 and l l y = n!. We make use of slightly generalized versions of

I t is easily seen that In: 1 > w1 and lli # II!. Richter [19] demonstrated that even In!l is a rather large admissible ordinal. Anderaa [ 11 recently proved that In{ 1 < 1 1. On the other hand it follows from the work of Aczel [ 21 on

The closure ordinal I I of r is the least ordinal a such that Fa+' = Fa ;

0 ~ _ _ _

Spector's results in section 3. -

221

222 D. CENZER

Ei operators that I Ei 1 = I Ei-monI. Aczel and Richter [3,4] have characterized I II; /, 1 Ei /, and, for all n , lII:1 in terms of reflection principles in the constructible hierarchy.

Pu tnam [ 181 showed that I A; 1 = Si, the first non- - A; ordinal; it is also known that A; = A;. More generally, for all n > 1, A: = A:, IIA-mon = II,, and CA-mon = EA. (However, E i -mon # E .) Using the techniques of Lemma 9.1 2, we can now show that I A: I = 6; for all n > 2. (See Cenzer [8] .)

1

2. Summary of results

I n this paper we explore further the relation between non-monotone inductive definitions and ordinal recursion. As pointed out above, ordinal recursion can be defined by an inductive operator, and in return the theory of inductive definitions can be developed within the framework of ordinal recursion.

greater than a and let a* be the least stable ordinal greater than a. Let S be the class of stable ordinals. (See section 3 for a brief development of ordinal recursion theory.)

For any ordinal a , let a+ be the least recursively regular (admissible) ordinal

Theorem A. (a). III; I is the least ordinal a which is not @+-recursive; (b). in: I is the least ordinal a which is *+-stable;

(c). 1 1 is the least ordinal a such that L , is a E1-elementaiy submodel of

L a , '

Part (b) was proven independently by Aczel and Richter [3 ,4] .

Theorem B. (a). I Eil is the least ordinal a which is not a*-recursive in S ; (b). I Zi I is the least ordinal a which is a*-stable in S ; (c). I E;l is the least ordinal a such that L , is a E2-elementary submodel of La*.

-

Theorem C. (a). II; is the class ofsets of numbers which are 1 II: I-semirecursive (the - domain of a 1 KIi !-partial recursive function); (b). Zi is the class of sets of numbers which are I EiI-semirecursive in S.

We derive results similar to Theorem C regarding E; and IIi inductive operators.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS 223

We extend the above results to combinations of operators in which the components need be inductive. For example, if denotes the class of inductive operators which are the composition of two ll: operators, then l ( l l~j21 is the least ordinal a which is not a*-recursive. Similar extensions of Theorems A, B, and C obtain for the classes (ll: j" and (C!)" for all n.

We generalize this result to the ordinal recursive arithmetic hierarchy and thus obtain results concerning constructibly analytic inductive operators. (A relation is constructibly analytic if definablt by means of quantifiers restricted to the constructible reals.)

inductive operators, we define a functional GT and prove the following theorem.

Theorem - D. (a). Ill: I = w?', the least ordinal not recursive in GT; (b). is the class of sets of numbers which are semirecursive in G t .

I t is well known that for A E o and a admissible, A EL, iff A is a-recursive.

Analogously to the results of Aczel [2] on Ef and

3. Ordinal recursion

This section is intended to provide the necessary background of ordinal

Let ( ) be a natural sequencing function from U,,,ORDn to ORD. recursion theory. Proofs omitted here can be found in Cenzer [7].

( ( a , ,..., a , ) ) j=a j ; l n ( ( aO ,..., a,))=n+ l ; ( a o ,..., a,)*(f10 ,..., on)= (a,, ..., an,fl , , -.,&).

We give the inductive operator (actually a class of operators) which defines ordinal recursion in full detail here since we will later want to discuss two related operators with reference to this definition.

Definition 3.1. For any ordinal y, any I < w , and any f = (fo, ..., fi- 1), n,[f] is the monotone operator such that for all k and n < w , all i < k and j < I , all a = (a,, ..., ak-1) with each aj < y, all fl, u, 7, ,$, and { < y, and all A E ORD:

(1) ( ( l , k , I , i ) , a , a j ) € n,[f](Aj; (2) ( ( 2 , k , l , i ) , a , a j+ 1)E Q,[f](A); ( 3 ) g < { implies((3,k+4,I),g,{,u,7,a,o)EQ,[f](A);

E > { imp l i e s ( (3 ,k+4 , i ) ,~ ,{ ,~ ,7 , a ,7 ) E n , [ f ) ( A ) ; (4) for all m, b,cO, ..., and cmPl < a, and all T ~ , ..., 7m-.1 < y, if for all i < m,

(0) (( 0, k 3 1, n )> a > n ) E Q, [ f I ( A ) ;

224 D. CENZER

(c i , a, 7) € A , and ( b , f , p ) E A , then ((4, k , f, b, c), a,B) € R,[f] ( A ) ; (5) i f V ~ < o 3 5 ) < ~ [ ( ( 5 , k + l , I , b ) , 4 . , a , 5 ) E € A ] , a n d VC <73,$< o 3 p < ~ [ p > { A ( (5 , k+l ,Z ,b ) ,4 . , a ,p )EA] , and (b ,T ,u , a ,p )EA, then((5,k+l,Z,b),o,a,P)€R,[f](A); (6) i f V o O [ ( b , o , a , 7 ) E A ] , a n d ( b , p , a , O ) E A , then

(7) i f ( b , a , P ) E A , then ( (7 ,k+ l , I ) , b , a ,p )€a , [ f ] (A) ; (8) iffi(Lyj)=p, then ( (8 , k , f , i , j ) , a ,p )ERyLf ] (A) ; (9) p E R,[f] ( A ) iff p is put in by one of clauses (0) to (8).

( (6 , k , t , b),a,P) E Q,lfl ( A ) ;

We write f 3 for (a, and 5, [f] for a,[ f ] ; [f] is u{a;[f] : E ORD).

Definition 3.2. (a). {a},(a, f ) -0 iff (a, a,p) E c,[f); (b). F is y-recursive iff 3a < w.F = {a},; (c). F is weakly y-recursive iff 3a < w 3a < y . F = ha, f . fa},(a, a, f). (F may be partial in the above definition.)

We point out that OUT definition of a-recursion yields the usual w-recursive functionals and that every o-recursive functional is y-recursive for all y > w. Notice that for any a and any (Y < 0, {a}, C {a}p. The following lemma is easily verified.

Definition 3.4. y is recursively regular (RR(y)) iff for all y-recursive functions F, all a and 0 < y, if Vo < 0. F ( o , a ) 4, then sup,<pF(o,a) < y.

As in recursion on the natural numbers we have the “recursion theorem”

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS 22s

Proposition 3.5. For any a < w there is an Esuch that for all y, a < y, and

f: {el,(a,f> = {aI,(Ca,f) .

In order to prove that various functionals are y-recursive, we want to show that the set of y-recursive functionals is closed under the following two schema: (1) Strong Composition (G, Fo, ..., Fm-l, Go, ..., Gn-l) = H iff for all a and all f: H ( a , f ) =FF(FO(a,f), ...,Fm- l ( a , f ) , hB .Go(P,a, f ) , ..., W -Gn-1@,a , f ) , f ) ; ( 2 ) Strong Primitive Recursion (C) = F iff for all /3, a , and f: F(P, a , f) = G(P, a, ( h a . F(o, a , f N r B,n,

do), i f u P . where in general g 1; (u) =

The following two propositions, which demonstrate the desired closure, can be proven by means of the recursion theorem. See Cenzer [7], pp. 14-18, for details.

Proposition 3.6. For all m, n < w, there is a primitive recursive function Cmpmrn such that for all a, b,, ... , b,-,, co, ..., c ~ - ~ , all y, all a< y, and all

..., {bm-II,(a>f)j f: {Cmpmrn(a, (b ) , (c))},(a,f> = {aI,({bO}Ja,f), ...

.{CO>,(P,a,f), ..., X P {c,_l},(P,a,f),f).

Proposition 3.7. There is a primitive recursive function Spr such that for all a < w , all 7, all a,P < y, and all f :

{Spr(a)Iy(P, a , f > 1: { ~ I ~ ( P , a , (hu{Spr(a)Iy(u, a , f ) ) I 8 , f ) .

Definition 3.8. Sup ( a , f ) 1: ifffis total on a and = sup { f ( u ) : u < a}.

It is easily seen that the functional Sup is y-recursive for any y. For y > w, w = l e a s t a < y [ a # O ~ S u p ( a , h ~ . ~ + 1 ) = a ] , sow isy-recursive. Let {c,} = Sup and {c,} = w . We now distinguish an important class of functionals over ORD.

Definition 3.9. (a). POR is the smallest set of numbers containing cs, c,, (0, k, l ,n) , (1, k, l , i ) , ( 2 , k, l , i ) , (3,k+4,1), and (8 ,k , l , i , j ) for all k,l ,n, i<k, and j > I , and closed under CmpmJn for all m , n <w and Spr.

226 D. CENZER

(b). F is primitive ordinal recursive (p.0.r.) iff 3a E POR. E = {a}_

The usefulness of p.0.r. functionals lies in the following proposition, which can be proven by induction on POR.

Proposition 3.10. For all a E POR, if F = { a } _ , then (a) F is total on total functions; (b) for any recursively regular y > w , any a< y, and any y-recursive f: F ( a , f ) = {a>,(a,f).

We say that a relation or predicate is p.0.r. or a-recilr+~e iff its characteristic function is. We list some properties of the p.0.r. relations and functions.

Proposition 3.1 1. (a). The followingfinctions and relations are p.0.r.: (1) for all i, ( )i, ( ), In, and * ; ( 2 ) <, I,>, >, and = ; ( 3 ) lim (a) i f f a is a limit ordinal; (4) the operations +, * , and exp of ordinal arithmetic; ( 5 ) T, defined by a TP = u iffP+u = aor ( a 2 0 A u = 0); ( 6 ) all arithmetic relations over wk X (“a)’, for all k , 1 < w ;

(b). the p.o.r. functions andlor relations are closed under the following: ( 1) union, intersection, and complementation; ( 2 ) bounded quantification; (3 ) definition by cases; (4) bounded search operator (least a,<,, f ( a ) = 0 v a=y) . -

Our next goal is to define a p.0.r. T-predicate for ordinal recursion. First, we need to study p.0.r. inductive definitions (such as

we want the function F , defined by F(<,a) =

[f]). If r is p.o.r., 1, i f a E r c , to be p.0.r. 0, i f a e r c ,

Lemma3.12.IfGisp.o.r.andFisdefi:nedbyF(<,a,p,~,f)-

SUP ( E , hoe F(o, a , p , Y, f)) for lim (8 or < = 0, and F(E+ 1, a , p , 7 , f) = G(ff,p,Y,(XP.F(<,P,p,Y,f)) r;;,f),thenFisaZsop.o.r.

Proof. F can be defined by recursion on the lexicographic ordering of ORD X p .

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS 2 27

Definition 3.13. For f = (fo, ..., J - - ~ ) , Tz(g,y, ~ , f ) iff T E nt[f] Proposition 3.14. T‘ is p.0.r. for all 1.

Proof. 0 [f] is an inductive definition over p = s ~ p ~ < ~ ( y , . . . , y ) , aY[f] is seen to be p.0.r. by inspection of Definition 3.1. It follows from Lemma 3.12 that T is p.0.r.

n - Y

Proposition 3.1 5. (a) For all recursively regular y, all a, p < 7, all Q < a, and all y-recursivef, {a),(a,f) -0 iffYt,o<y.~’(t ,u,(a, a ,p ) , f ) iff T‘(y, 7, (a , cx,P),f). (b) for any ordinal y and any f, y is recursively regular in f iff 52r1 [f] = Q;[f]

Proof. We can show by inspection of Definition 3.1 that u closed under 52, I f ] , and therefore equals E[f] ; (a) anzla?f;?(b) follows from this. If y is not regular inf then we have a functional {a}, and a , p with s~p,<~(a},(u, a , f ) =y; it is then not difficult to construct a functional (c} , w i t h ( ~ , a , @ , O ) E ~ ~ [ f ] - 52;.

u Q i I f ] is

Corollary 3.16. RR is p . 0.r. n

Proof. Let h(y) = supn (y, ..., y). h is p.0.r. and RR(y) iff “7 <h(’Y)[T’(’Y+l,’Y,T) + Tz(Y,Y,7)].

r2h-r

4. Recursive analogues of large cardinals

The class of recursively regular ordinals is intended as a recursive analogue of the class of regular cardinals; w1 obviously corresponds to N,.

Definition 4.1. (a) w, is the a’th ordinal which is recursively regular or a limit of recursively regular ordinals; (b) a+= leastp [B>a A RR(p)], the “next” recursively regular.

Notice that wo = w , (a,)+ = and for limit ordinals a supp<,wp= w,.

Definition 4.2. (a) a is recursively inaccessible (RI (a)) iff RR (a ) and a is a limit of recursively regulars;

228 D. CENZER

(b) a is recursively Mahlo (RM (a)) iff RR(a) and every normal weakly a- recursive function f from a to a has a recursively regular futed point ( f is normal iff strictly increasing and continuous at limit ordinals).

The following is easily verified (part (c) using Propositions 3.15 and 3.16.)

Proposition 4.3. (a) RI (a) i f f RR (a) A a, = a; (b) Rh4 (a) + RI (a): (c) RI and RM are p.0.r.

We could defined properties like recursively hyper-Mahlo and hyper inaccessible, but our interest here is in stronger notions of recursive largeness.

Definition 4.4. (a) a is absolutely projectible t o p iff there is a functionfa- recursive in parameters 6 < p mapping a I- 1 into 0; (b) a is projectible to /3 iff there is a weakly a-recursive functionf mapping a 1-1 intoo; (c) a is non-absolutely projectible (NAP(&)) iff RR (a) and there is no /3 < a such that a is absolutely projectible to up;

(d) a is non-projectible (NP(a)) iff RR(a) and there is no 0 < a such that a

is projectible to up.

The following concept turns out to be very useful in the study of recursively large ordinals and particularly relevant to the two notions of projectibility.

Definition 4.5. For all ordinals a and all Q and < a: (a> p is a-recursive in u iff there is a a-recursive function F such that /3 = F(a); (b) p is a-recursive iff there is an index a such that {a}, = p.

We derive the following equivalences for projectibility .

Proposition 4.6. For all recursively regular ordinals a and all /3 < a,P closed under >; (a) a is absolutely projectible to Vr <a 3 a < /3 [ r is a-recursive in u] ; (b) a is projectible to /3 iff 37 <a Vr<a 3a < p [ i is a-recursive in y, a].

iff


Proof. We give the proof for (a); (b) is similar. (-+) Suppose we have t< and F such that A T , F ( T , ~ ) is a projection of a top. Given ~ < a , let u = F ( r , k ) ; t h e n r = l e a s t f < a . F ( { , t ) = u , andT isa-recursivein(u,t) (+) Suppose Vr<a 3u<P[7 is a-recursive in u]. Then p in particular is a- recursive in parameters less than 0. Let f o ( T ) = least ( a , (3, < a [(a, u ) < p A To(t , F,(a, u, T))] , and let f (T) = ((fo(7))o,(fo(7))1). This f is the canonical projection of a top.

Corollary 4.7. (a) NP (a) iff NAP (a) A RI (a); (b) NF' and NAP are p.0.r.

In order to prove that ordinals exist which are recursively Mahlo or non- projectible, we first discuss the concept of stability.

Definition 4.8. (a) a is y-stable in f iff a is closed under all functions y-

recursive in f. (b) a is stable in f iff a is -stable i n k (c) a is stable (S(a)) iff a is stable in $9.

We need the following lemma and proposition to prove that stable ordinals exist. See Cenzer [7], pp. 47-49, for a proof of Lemma 4.9.

Lemma 4.9. For any ordinal y, any D E y, and any f from D to D such that D is closed under all functions y-recursive in f, i f IT is the collapsing function mapping D 1- 1 onto an ordinal in an order-preserving fashion, then for any a,p E D and any y-recursive functional F ,

F ( a , f ) --p implies F(n(a) , f ) = @).

Proposition 4.10. For any ordinal y, any a < y, and any f from a to a, (F(5, f ) : s<a, F y-recursive} = D i s an initial segment of y.

Proof. Notice that a E D ; it follows from Lemma 4.9 that for t< a, F ( ~ , f ) - - - / 3 i m p l i e s F ( ~ , f ) = n ( ~ ) = f l . N o w g i v e n a n y p € D , p = F ( t , f ) f o r some t< a, and therefore IT@) = 0.

Corollary 4.1 1. (a) For any ordinal - - a and any function f from (Y to a, there is an ordinal 6 stable in f with 8 = (Y.

2308 D. CENZER

(b) For any f from w to w, there is a countable 6 stable inJ

We can use a different technique to find ordinals stable in functions from ORD to ORD.

Proposition 4.12. For any f : ORD -+ ORD and any a, there is an ordinal 0 > (Y such that 0 is stable in f:

Proof. Let Do = (F(5 , f ) : 5< a and F ism-recursive) and uo = SupD,. For all n < w, let D,, = (F(6,f) : [ < u, and F ism-recursive and a,+1 = SUPD,,~. Then 0 = Sup,<,u, is stable in f .

Definition 4.13. (a) 6 2 = the a’th ordinal stable in x A ; (b) 6 * = least 0 [p > 6 A S@)] ; ( c ) 6 , = 6,. 0

We point out that F is the least non-00-recursive ordinal and for all n , is the least ordinal which is not w-recursive in 6,.

Proposition 4.14. (a) For any 0 and any y > 0, if0 is y-stable, then 0 is recursively Mahlo ; (b) 0 is non-projectible iff 0 is a limit of ordinals which are 0-stable; (c) the relation RS, defined by RS(a,P) i f f a is 0-stable, isp.0.r.; (d) S is not m-recursive.

Proof. (a).Given a normal weakly 0-recursive function f , let for lim (a);

g(a)= i f ( a ) , otherwise. Theng is y-recursive, g(a) - f (a ) for all 0 <0, andg(0) =p. But then a. = least a < y [RR (a) A g(a) = a] < 0 by stability, so that f has a recursively regular fixed point. (b) I f 0 is non-projectible, then for any a <0, a, = Sup {F(k ) , 5 <_ (Y A F 0-recursive} is 0-stable and a < ool; if 0 is projectible to a < 0, then by Proposition 4.6, 37 < 0 {F(LJ, y) : a < a} = p . It is clear that there can be no 0-stable ordinal greater than max(a,y). (c) This follows directly from Proposition 3.15. (d) IfS were m-recursive, then 6 , = least 6 .5(6) would be w-recursive, contradicting its stability.

Sup (a,Xu . f ( u ) ) ,


Corollary 4.15. (a) NP(a) implies Rh4 (a); (b) there are non-projectibles less than 6,.

Given any p.0.r. relation R on ORD, p = least a. R(a) is -recursive, but further, p = least a < p+ . R(a), and so p+-recursive. For example, w1 = least a < w2 [RR(a) A a # 0 1 . We have the foilowing.

Proposition 4.16. If a is any of the following: wl, the least recursively inaccessible, the least recursively Mahlo, the least non-absolutely projectible, the least non-projectible, then a is a+-recursive.

Definition 4.17. (a) a(*) = a ; (b) y, = least a. a not a(n)-recursive.

= (a("))+;

Since for any stable ordinal 6 , 6 is not dn)-recursive, it is clear that the y, exist and are all less than 6 , . Proposition 4.18. For all n > 0, (a) RR(yn) and RI (7,); (b) VT < 7, . r is y,-recursive; (c) VU, yn i u < y r ) , u is not yF)-recursive.

Proof. We give the proof for n = 1. (a) If 7RR(y1) , then y1 = sup,&'(u, 61, for Some 0, 6 < 71 and 71- recursive F ; but then 0, t a r e y recursive ( 0 is p+-recursive and P' < y t ) and so is F. This implies that y1 is y,-recursive, a contradiction. If -7RI (yl), then y1 = 0+, for some 0 < yl ; but then y1 = least y < y; [RR(y) A P < 71. (b) T < y1 implies r r+-recursive, but since RI (yl), r+ < yl. (c) If u is 7;-recursive and y1 5 u < yt , then y1 = least y < y t . Vr <u[RR(T) -+ r IT].

li

The following is easily verified.

Proposition 4.19. (a) For all n > 0, y, = least y. 7 is $")-stable; (b) yi is the least non-absolutely projectible ordinal.

Corollary 4.20. (a) For all n > 0, RM (7,); (b) the least recursively Mahlo ordinal i s less than yl .

232 D. CENZER

The following characterization of y, will be useful.

Proposition 4.21. For all n > 0, yn is the least ordinal y such that Va < [{a}+)(y) J-, 3a < Y . {aIa(,>(4 41.

Proof. (2) Suppose {a}yt(yl)J.. By 4.18c, P = least pP<,,; [T ’ ( (P )~ , ( ~ ) 2 , (a , (p) l , (010)) (012 < (PQ is less than yl. (Notice that u < T+ iff V $ 5 u [RR($)+ $ 5 T ] is p.0.r.) Let (y = ( P ) ~ ;

a n d a < y l . (I) Given y < y l , we know that y = {a} + for some a < w. Let {b)o+(U) =least tEe0+. {a}o+ =

The proof for n > 1 is similar.

7 A $ = u ] . Clearly, {b},,+(y)4 and for any

a < 7, {b}a+(a)T.

Corollary 4.22. For any n > 0 and any i < w:

(a) i f { a } , p ( ~ ~ ) N i, then 3a < Y, I {a),(,)(a) = i; (b) if . {aIa(n)(a) N i and if {a},p’(yn) 4, then {a),lt”)(yn) N i.

In section 6 we make use of the foregoing material to prove that In: I = yl . First it is necessary to study the class of IT: relations with reference to ordinal recursion.

5. H! relations

In this section we prove the following two generalizations of the Kreisel- Sacks [I41 result that forA 2 o , A En: iffA is wl-semirecursive.

Proposition 5.1, If Q C w X P(w) is IT;, then there is a p.0.r. functional F with rg(F) E (0, I} such that for all m and A : Q ( m , A ) iff 30 .F(U,m, XA) 1 iff 30 < w;’. F(u,m,x~) 1.

Proposition 5.2. The relation K , defined by K ( ( a , m , n ) , A ) iff {a} A(m) 11 n, is ni. a 1

We want to code ordinals into the natural numbers. Let {a}A be the a’th function partial recursive in A .


Definition 5.3. (a) W(@) iff @ is the characteristic function of a non-strict well-ordering of a subset of w ; (b) Field (@) = {s : @(s, s) = 1); if W(@), 14 I is the ordinal isomorphic to the well-ordering defined by @ and, for s E Field(@), Is I, is the image of s under the map from Field(4) to 141 ; s 5, t iff Isl, 5 Itl@ iff @(s , t ) = 1 .

Lemma 5.4. (a) Wis nk; (b) there are Z relations M and M’ such that for all Cp and all $ such that W($) ,M(@,$) i f f W(@) A I@ILI$I ,andM’(@,$)i f f W(@) A I@I<I$I; (c) there are l l i relations L and L’ such that for all @ and $: L ( @ , $) iff W ( @ ) A W($) A l@l<_IJ/ I ,a~dL’(@,$) i f f~(@) A W($) A l 4 l< l$ l .

hoof. (a) W ( @ ) iff Cp is a linear ordering A V8 3 p -+ (O(p+l) <, q p ) ) . (b)M(@, $) iff 38 V m Vn. @ ( ( m , n ) ) = $(@(m),O(n)));M’ similar. (c)L(@,$) iffW(4) A W($) A-M’($,@);L‘ similar.

The following lemma is an easy relativization of the standard result; see Shoenfield 1221, p. 184, Cenzer 171, pp. 29-3 I .

Lemma 5.5. (a) For all A E w, {a: W({a}A)} is IIi -A complete; (b) (Boundedness Principle) For any A 5 w and any there is a u such that for all u in V, L( ( u } ~ , { u } ~ ) .

V C {a: W ( { a } A ) } ,

We generalize Spector’s results [24] on’n: monotone inductive definitions; see Cenzer 171, pp. 32-34 for proofs.

hopositjon 5.6. I f K is a n: relation such that for all A , FA, defined by F,(B) = {m: K(m,B ,A)} , isa monotone inductive operator, then: (a) the relation P, defined by P(rn,u,A) i f f m E F$u}A1, is ni; (b) the relation S, defined by S ( m , A ) i f f m E Fw$, is ni. ( c ) fo reveryA ~ w , i r A i s w f ; (d) the relation Q, defined by Q ( m , A ) i f f m E G , i s ni.

The following lemma completes the preparation for the proof of Proposition 5.1.

Lemma 5.7. If Q is a IIi relation on w X P(o), then there is a IIy relation K

234 D. CENZER

such that for all A , r,, defined by r,(E) = {m: K(m, B, A)}, is a monotone inducfive operutor such that for all m, Q(m, A) iff (m, 1) Ec.

Proof. Suppose Q(m,A) iff V@3p - R ( m , ( p ) , A ) , where for all s,t.R(rn,s,A) A s C t +R(rn,t ,A). Let K((m,s),B,A) iff ( r n , s ) € B vR(rn,s ,A) v V p . (m, s* (p ) )EB,and le t FA k d e f i n e d f r o m K as above. I t is easily seen that for all m , s, and A , (m,s) E FA iff VI) ( sLI )+ 3p.R(m,$(p),A),sothatQ(m,A)iff(m, 1 ) E & ( 1 beingthe empty sequence).

Proposition 5.1. I f Q 5 w X P(w) is H i , then there is a p.0.r. functional F with r g ( F ) c (0, 1 ) wch that for all m and A : Q(m,A) $f 30 . ~ ( u , m , x , ) = 1 i f f 3 u < o f . F ( o , m , X A ) = 1.

Proof. Let K and r, be defined from Q as in Lemma 5.7. Then Q(m,A) iff 3a . m E rs iff 30 < wf . in E I';, where the latter equivalence follows fiom Proposition 4 . 6 ~ . Since K i s n y , i t is p.0.r. by Proposition 3.1 1 . I f we se tF(u .m,XA)= 1 i f f ( r n , l ) E r ; , thenFisp.0.r. byLemma3.12.

To prove Proposition 5.2, we code up the ordinal recursive functions on wf in a nf - A fashion.

Definition 5.8.N(u,m,A) iff W ( { U } ~ ) A I{u}'l = m .

The relation N is easily seen to be arithmetic.

Proposition 5.10. For all 1 > 0, the relation R', defined by R'(m,$,A) iff nz E K; 141, is n i . Proof. Kfi [$I is defined by a ni - $ , A monotone inductive operator f l A [@] which parallels SZwA [+I. From a slight generalization of Proposition 5.6b it follows that Kfi [$I*, which equals R, [$Iw:' is rI; - $,A uniformly in @ , A . We give some cases of the definition of a, [$] : (0) GO.k,i.n), ~ 4 ~ ~ . ..., uk.. u ) E QA [+](B) i f f ~ ( { u ( ~ } ~ ) A ... A ~ ( { z i ~ . , > A > A N(u,n ,A);


Proof. (a) This follows from Proposition 5.10. (b) Let uf = a ; by Proposition 5.6c, laA [xA J I 5 a. Since i l a [xA] parallels a A [ ~ A l , w e h a v e ~ E [ x A I = a , t x A j , so {~>,(IC~>~I,XA) = I { U > ~ I iff TYa,a,(a, I~~I~I , I{uPI),x~). (c) This follows from (b) above and Proposition 3.14b.

Proposition 5.2 is a corollary to 5.11

Proposition 5.12. (a) For any functions f, any a recursively regular in f, and any function qf~ weakly a-recursive in A i f W ( @ ) , then I@ I < a; (b) the relation W , , defined by Wo(a,@) i f f W ( @ ) A I@ I = a, is p.0.r.

Proof. (a) Define H by recursion on u so that H ( u , @) = the unique m . Iml, = (5

Let G(m, @) 2 least u,<, . H ( u , @) y m , if @(m, rn) = 1

if @(m,m) = 0. Then H and G a;e a-recursive and I@ I = suprn <w G ( m , @) < a by regularity. (b) The graphs of G and H above are p.0.r. in a, 4.

236 D. CENZER

Corollary 5.13. (a) For all A C w , w.;' is the least ordinal # w recursively regular in A ; (b) for any recursively regular a > w and any A C w, if x A is weakly a- recursive, then m.;' < a.

Proof. Suppose a is recursively regular in A and w < a < 0.;'. Then there is a well-ordering @ w-recursive in A of type a. But 9 must be weakly a-recursive in A , since a > w, and then by 5.12~1 I@ I < a. (b) Any well-ordering @ recursive in A must be weakly a-recursive, but then by 5.12a. I@I<cu.

6. Hi inductive definitions

In this section we prove

Theorem 6 . l . (a ) lnil~y~; (b) for all A 5 w , A E n; i f f A is y -semirecursive.

Definition 6.2. Given an f operator r, we say that F and I , rg C (0, l}, are p.0.r. functionals associated with r if for all rn,A,O,r, (a) m E r ( A ) iff 30. ~ ( o , rn, x A ) - 1 iff 3a < w.;' . F ( U , rn, x A ) = 1 (as in 5.2) (b) 1(7, m ,p) = Sup (7, A 5 . I (5 , rn, p)), if lim ( T ) or T = 0; 1(r+l,m,p)~Sup(p,ha.F(o,m,hn .Z (~ ,n ,p ) ) ) .

We see by inspection that if rp is defined by m E r p ( A ) iff 3a < p AF(o,m,XA) = 1, then m E Fi iffI(.r,rn,p) = 1. For sufficiently large p (e.g., p = H I ) , rP = I'. We get a better bound on p by looking at the rr individually.

Lemma 6.3. For any ni inductive operator r with associated I , and for all ordinals r: (a), For all rn arid all /3 2 w T , +(m)

(b), 0;' I w , + ~ ; I (m, T , @ :

Proof. We break the proof into four parts: (1) (a)o and ( l ~ ) ~ are trivial.


(2) (a), A (b), + (a),+, by Definition 6.2. (3) (a), + (b), by Corollary 5.13b. (4) [lim (7) A Vu < 7 . (a),] + (a), by Definition 6.2.

Proposition 6.4. For any IIi inductive operator r, I r I 5 yl.

Proof. For any y, N(y) implies by 6.3 that for all m, Xry(m) =I (y ,m , y) and m E 1'7+1 iff 30 < y', F(o, m, Xp , I ( y , p , y) = 1. Given m E PI+', choosea so that {a},(a)=least u < a [ F ( o , r n , X p ~ ~ ( a , p , a ) ) ~ I ARI(a)] Then {a}y; (y l )$ , so by Proposition 4.21, there is an a < y1 such that {a},+(a)$, so that m E E

We next define a n : inductive operator A, such that 1 A, 1 = y l . Recall from Proposition 5.2 that the relation K ( ( a , rn, n),A) iff {a}&(m) = n is n,. 1

Proof. The proof is broken up into four parts: (1) and (b)o are trivial. (2) (a), A (b) , -+ (a),+, by the definition of A, and the regularity of w,+~. (3) (a), -+ (b), for all T 5 y l , as follows: for any T 5 y1 and any $ < o,, there exists some a < w and u < o, such that = {a}o. Choose b so that {b},(rn,n)-Oiff {m}, 5 {n},. Then { (rn ,n) : (b,m,n,O)EA,} is apre- well-ordering of length a,. Refine this to a well-ordering by choosing a least member of each equivalence class to obtain a well-ordering recursive in A, of

A type w,; it follows that o1 2 w,+~ . On the other hand, it follows from Proposition 3.15 that A, is p.0.r. in w, and therefore weakly w,+,-recursive; then by Corollary 5.13(b), 4 5 (4) (lim (7) A Vo < T. (a),) -+ (a), is trivial.

Corollary 6.7. /A , 1 = y,.

Proof. ]A, I 2 y1 by Proposition 6.4. By Proposition 4.18(b), for any 7 < y l ,

238 D. CENZER

there is an m such that {mjYl = wT+l, so that {mj,?. Let b be as in (3) of the proof of Lemma 6.6 above: then (b ,m,m, 0 ) E A:] - Ai , so that I A I I ~ Y ~ .

This completes the proof of Theorem 6.1(a). We prove 6.l(b) (that y1 -semirecursive over P(w)) as follows: (c) For any inductive operator r, by Lemma 6.3(a), r = {m: 30 < y1 [ I (o ,m,o) = - 1 A RI(o)]} and is therefore yl-semirecursive. Any A _C w such that A r is also y1 -semirecursive. (2) Suppose that m € A iff {a}Yl(m).l. Choose b so that {b},l(m) = 0 - {a}yl(m). Then by Lemma 6.6(a), m E A iff ( b , m , 0 ) E A,.

=

-

7. A functional GT with = y1

Recall that for any partial functional F , wf is the least ordinal not equal to I @ 1 for any well-ordering @ recursive in F. I t is easily seen that for any F, there is an a such that 0;” = w,, although w, need not be recursively regular, as indicated by the remark following Lemma 7.8.

Hinman has defined the functional E f from ww to {0,1).

The following results are proved by Aczel [ 2 ] :

# Theorem 7 . 2 . (a) wyl = 12;L-monI; (b) fi)r all A C w , A E X2;f-mon ijyA is Ef-semirecursive.

Grilliot first pointed out that, using Ef, one can prove that I Ei-monI = I C l I.

Definition 7.3. G r ( f ) = n iff u(O)} f + ( f ( 1 ) ) = n. ( W J

I n this section we prove the following theorem.

# Theorem 7.4. (a ) wyl =A ni I = y, ; (b) tor all A C w , A E n f iff A is G y-semirecursive

i f l A is y , -semirecursive.


We define recursion in Gf as in the definition of ordinal recursion (3.1); it is possible to consolidate the first seven clauses into five, since we are doing w-recursion. Clause (8) is replaced by the following to define Q [ G a (8)# for any f€ ow, if for all s and t , f(s) = t implies ( b , s,m, t ) E A and Gf(f) e n , then ( 6 , k , b ) , m , n ) E R{Gf](A).

Lemma 7.5. (a) Gf is consistent, that is, for any f and g in ow and any n < w ,

(b) for any a and m, there is at most one n such that ( a , m, n ) E R [Gf] . ( f C g A ~ f ( f ) = n) + Gf<g) 3~ n .

Proof.(a) GivenfCgand v(O)} g( 1) =f( I), and 04 2 a{. (b) This follows directly from (a).

Defmition7.6. (a) {a}'] (m)-n i f f ( a , rn ,n )E a [ G f ] ; (b) fe ww is Gf-recursive iff there is an a < w such that f = {a}G1 .

J- '(f(l))=n.we haveg(0)-f(O), b,)+

#

#

We need the following definition and lemma to prove that wy? 2 y1

Definition 7.7. F,(a, rn) = f l iff {a},(m) = f l A

Lemma 7.8. For any recursively regular a, with w < a < yl, wFa = a.

< w.

Proof. (5) Since Fa is a-recursive, it follows as in Corollary 5.13(b) that

(2) As in (3) of the proof of Lemma 6.6, we have for any w, < (Y a well- ordering @ of type c3, which is then for some a, @ = A t . Fa(a, t ) and is therefore recursive in F,.

w p 5 a .

-recursive and therefore a-recursive. But

We remark that for non-regular limits a of recursively regulars, if one sets F,(m) N n iff 3 u < a. {a},(rn) = n , then w p = QI as in Lemma 7.8.

Proposition 7.9. wff >_ y l .

Proof. Suppose w$+ = 7 < yl. Let e ( a , m, t ) = n iff G# ( t = 0 A n = a ) v ( t = 1 A n = m) v 3c, p( t = (c, p ) A {c} 1 ( p ) = n). Then 0 is

Gf-recursive and for all a,m, w ~ t ' e ( a ~ m , r ) = 7. Hence for all a,rn,

240 D. CENZER

C f ( h t . 0(a,m,t)) -F7+(a,m). But then F,+ is Gf-recursive and by Lemma 7.8, wyp 2 I-', a contradiction.

Our next goal is to show that IR[Gf] I 5 yl. We need to be able to compute 4 from the graph off in order to construct associated functionals F and Z analoguous to those of 56. The following lemma is a rather technical application of Proposition 5.12(b); we refer the reader to Cenzer [7] for the proof.

Lemma 7.10. The relation Ro, defined by RO(a, @) i f f there is an f E ow

such that @ = Xgraph(f) and of= a, isp.0.r.

We now define a p.0.r. functional F associated with R[Gf] .

Lemma 7.1 I . There is a p.0.r. functional F with rg(F) 5 (0, 1) such that for all ordinals 0 and all A C w, if(wf)+ <_ p, then for all s, s E R [Gf] ( A ) iff F(P,s,x'4)= 1.

Proof. F is defined by cases and the only interesting one is, of course, case (S)', the application of Gf . Let

1, if 3 s , t ( p = ( s , t ) ~ O ( ( b , s , m , t ) ) = 1; 0, otherwise.

J(b,p,(m),O) =

Then for anyb ,m, f , a n d A , i f f ( s ) = t i f f ( b , s , m , t ) E A , then h p - J (b , p , (m) , x A ) = Xgraph(f). We now define F(P, ((8, k , b ) , m, n) , 0) iff O(((S,k, b), m,n), 0 ) = 1 V g t , u <p3a , t[Ro(u, h p . J ( b , p , ( m ) , e ) ) A 4 < u+ A 0 ( ( b , O,m,a)) =

0((b, I , m , t ) ) = 1 A T o ( ( , t , ( a , t , n ) ) J . I fR,(u,hp . J (b ,p , (m) ,xA) ) , then u 5 of, so taking p 2 (wf)+ is sufficient.

1

We next define a p.0.r. functional 1 associated with R[Gf]

Definition 7.12.1(7+ 1 , m , b) = F(0 , m, hn . Z(T, n,p)); Z(T, m , p) = Sup (7 , hu. Z(u, m,p)) , if lim ( I - ) or I- = 0.

Proposition 7.1 3. For all m , n < w, all limit ordinals a, all r = a + n , and all

B2w,+Zn: (a), m E R ' [ G ~ ] i f fI(I- ,m,P) = 1;


Proof. As usual, we break the proof into four parts: (1) (a)o and (b)O are trivial. (2) (a), A (b), -+ (a),+l by Lemma 7.1 1. (3) (a), -+ (b), follows from Corollary 5.13(b). (4) (lim (7) A Vo < r .(a),) j . (a), is trivial.

Corollary 7.14. (a) For all recursively inaccessible ordinals a, all b, m < q and all f E uw such that f ( s ) = u iff (b,s, m, u ) E f2*[G?], f is a-recursive; (b) for all recursively inaccessible a and all m < u, rn E P + l [Gfl i f f I(a+I,m,a+)= 1.

Proof. (a)f(s) =(least p < a!. Z((P)~, ( b , s , n ~ , ( p ) ~ ) , ( p ) ~ ) = 1 A

(b) NOW whenever f comes from Lemma 7.1 I , taking 0 2 a+ suffices.

( P I 2 2 U(p),+w)l 0 . [Gf] as above, uf 2 a, so that as in

Proposition 7.15.(a) lCl[Gf]I < y l ; (b) for all f E ow, i f f is Gf-recursive, then f is Yl-recursive; (c) uf? i 71.

Proof. (a) Suppose 9 E Q 7 1 + l [Cf] - WI [ G t ] ; the difficult case is 9 = ( ( 8 , k , b) ,m ,n ) . By Corollary 7.14(b), for all recursively inaccessible a,

q E Cl0l"[Gf] iff 3g,u,a,t <a+[Rg(o,hp,f(b,p,(m),hs .I(a,s,a))) A

6 < u' A I(&, (b , 0, m,a), a) - - I (a , (b , 1, m, t ) ,a ) = 1 A To(g,.$, ( a , t , n ) ) ] . We can write 6 < u+ as Vr 5 E(RR(r) -+ 7 5 u); thus the inside of the brackets is a p.0.r. relation, say,P([, u,a, t ,a). Choose c so that for all a,

(c}*+(a) *least ( [ , u , a , t ) < a + [ ~ ~ ( o l ) AP(~ ,CJ ,~ ,~ ,CY) ] . Since q E Cl71+l [Gf], {C}~;(T~)$, and therefore by Proposition 4.21 there is an a < y1 such that { C } ~ + ( ~ ) J . . Then q E Llay"[G#] a71[G#], contradicting the hypothesis and completing the proof of (a). (b) By (a), alca] = function is y1 -recursive. (c) Any Gf-recursive well-ordering r$ is now y1 -recursive so that 1 $ 1 < y1 by Proposition 5.12(a).

[@I, so by Corollary 7.14(a), every GT-recursive

242 D. CENZER

Combining ( c ) with Pioposition 7.9, we have

Corollary 7.16. = y l .

Finally, we obtain the other directions of (a) and (b) of Proposition 7.15.

Proposition 7.17. (a) I!2[Cf] I = y1 ; (b) f o r all f E w w , f i s Gy-recursive iff f is yl-recursive; ( c ) fiir all A 2 w . A is GT-semirecursive ijj'A is y I -semirecursive.

Proof. (a) Suppose 1Q[Gf] 15 OL < y l ; without loss of generality we assume that OL is recursively inaccessible. Then it follows as in Proposition 7.1 5(c) that w I 1 5 1y < y l , contradicting Proposition 7.9. (b) By Corollary 7.16, we have wyp = y l ; as in the proof of Proposition 7.9, it follows that the function F recursive function f~ w w , for some a, f ( m ) = {a},,(m) = {a}r:(m) =Fr:(a,rn) for all m. (c) This follows directly from (b).

c#

is Gf-recursive. Now for any partial y l - r:

This completes the proof of Theorem 7.4.

8 . Extensions of the class of II; operators

Riclitcr [ 191 defincs a method of combining operators as follows.

rO(A), if r o ( A ) # A ; Definition 8.1. (a) [To, r, ] ( A ) =

The fullowing results, due to Richter, are of interest

Theorem 8.2. (a) I [KIf , II!] 1 is the least recursively inaccessible ordinal, 1 [ rI:),@, n:] I is the least recursively hyper-inaccessible ordinal, arid similarly .tor I [nl.n0. 0 0 ..., n:] I ; (b) 1 [n1,nl] 0 0 1 is the least recursivelj'Mahlo ordinal, I [ n ~ , H ~ , I I ~ ] I is the

least recursivelis hj>per-Mahlo ordinal, arid similarly f o r I [I$, ..., ny] 1.


Using the techniques of 6, it is not difficult to prove n

Proposition 8.3. For all n, I [m] I < y2.

We obtain a result analoguous to Theorem 8.2; the proof is omitted since i t is similar to that in Richter [ 191.

Proposition 8.4. (a) 1 [lli,ll:] I is the least recursively regular ordinal y such that y is not y+-recursive and y is a limit of ordinals a which are recursively regular and not a+-recursive; (b) 1 [II:,@] 1 is the least recursively regular ordinal 7 such that y is not y+- recursive and such that every normal weakly y-recursive funcfion has a fixed point QI which is recursively regular and not a'-recursice.

While the ordinals of Theorem 8.2 present a nice hierarchy of recursively large ordinals, those of Proposition 8.4 and its obvious extensions seem of less interest. The natural ordinals to consider above y1 would appear to be y2, y3, and so forth. It turns out that there are natural classes of inductive operators with these as closure ordinals.

For r = Po * ... * r,, we do not require each operator Ti to be inclusive, but only the composition r. For example, C since for any operator r, if ro(A) = w - r(A) and F1(A) = w - A , then r = ro * r l .

We now state the primary result of this section.

Theorem 8.6. For any n > 0,

(b) for all A E w , A f (ll!)" iff A is y,-semirecursive. (a) I ( m " I = 7,; ___

Theorem 8.6 is proven in a manner similar to the proof of Theorem 6.1.

Proposition 8.7. For all n > 0, I(I'Ii)" 1 2 y,.

244 D. CENZER

Proof. We sketch the proof for (lli)2. Given r = ro- rl E (IIi)2, we have by Proposition 5.l(a), p.0.r. functionals Fo and F , with rg(Fi) S {0,1} such tha t fora l lm andA: m E F i ( A ) i f f 3u.Fi(5,m,xA)= 1 iff 3 u < w l .Fi(u,m,xA)= 1. We definep.0.r. functionsHandI by: I ( r , m , p ) - - S u p ( ~ , X ~ . I ( ~ , m , P ) ) , iflim(7) or r = 0; H(T+I ,m,b) = Sup(P,Xa.Fo(a,m,hP.J(r,P,P))); I (T+l ,m,b) N_ SUP (b , XQ.Fl(@,m, X P . l ( T , P , P ) ) ) .

A

The following lemma is proven in the same manner as Lemma 6.3.

Now for recursively inaccessible ordinals a , m E r" iff I (a ,m, a ) = 1, and m E r a + ' iff 3u<a++Fl(a,m,hp .H(a+l ,p ,a+) )= 1. Asin Proposition 6.4, it follows that if m E P z + ' , then for some a < y2, m E

thus I l - lSyz. so m E rrz;

Next we construct operators A, E (llf)" such that lA,I = 7,.

Definition 8.9. A(A) = {(x,y) : L ( { x } ~ , b}A)}; A, = (A),-'. A , .

The following lemma is easily verified.

Lemma 8.10. For any A C w, let wf = a; then (a ) A(A) is a pre-well-ordering of length a; (b) $ A is p.0.r. in a, then w t H ( A ) = a(,); (c) i f A isp.0.r. in a, then A,+'(A) = { ( a , r n , t ) : {a},(,,(rn) N t } .

Applying this lemma and proceeding as in the proof of Lemma 6.6, we have


Proposition 8.12. For all n > 0,

(a) (b) IAnI = 7".

= { ( a , m , t ) : {aITn(m) = t ) ;

It follows from Proposition 8.7 that I\n = A?, which equals { ( a , m , t ) : {a},(m)-tby Lemma8.11. (b) T h i s follows from (a).

This completes the proof of Theorem 8.6(a); 8.6(b) ((n;)" = yn -semirecursive) is easily obtained. (c) This follows from Lemma 8.8(a) and Proposition 8.7. (2) This follows directly from Proposition 8.12(a).

Finally, we can extend the results of 97 to (n;)" and 7,.

Following the development of $7, we can prove

Proposition 8.14. For all n > 0 and all k < w ,

(b) for all f : wk -+ w, f is G,#-recursive i f f f is 7,-recursive; ( c ) for all A C w , A is G,#-semirecursive iff A is y,-semirecursive.

# (a> = yn ;

9. Z! inductive definitions

The main result of this section is a characterization of 1 E! 1 which is the dual of the characterization of I lIi I = y1 given by Proposition 4.2 1 (in combination with Theorem 6.1).

Theorem 9.1. 1 Zi I is the least ordinal u such that for all a < w,

(VT < 0 {a}T+(T>J) --f {a}o+((J)i.

We also prove an analogue to Theoren 6.1 (b). -

Theorem 9.2. For all A C w , A E Zi i f f A is 1 Z; 1-semirecursive.

Anderaa [ I ] has recently obtained a significant result regarding dual

246 D. CENZER

classes of inductive operators which implies that I IIf I < I Z! I and ICi1< In:\. Combining the former inequality with the techniques involved in Theorems 9.1 and 9.2, we are able to characterize the spectra of the two classes.

Definition 9.3. For any class C of inductive operators, Spectrum(C)= ((1'1: r E C ) .

Theorem 9.4.(a) Spectnim(;F.i)= {a:+a< y1); (b) Spectrum (Xi) = {a 5 I Zi 1 : (Y is IY -recursive}.

We begin by applying Proposition 5.1 to Cf inductive operators.

Lemma 9.5. For any Z; operator I?, there is a p.0.r. functional F with rg(F) 2 (0 , l} such that for all m and A , rn E r ( A ) ijj ' tlu. F(a,m, x A ) = Oijy Va < w.;' . F(o,m, x A ) = 0.

Lemma 9.6. For any r a n d Fas in Lemma 9.5, i f / is defined by f ( r , m, p ) = Sup (7, h l .I ( t , m, p ) ) , i f lim (7) or 7 = 0, and f ( r + l , m , p ) = 1 - Sup(P,ho.F(o,m,hn.f(7,n,P))), then

(a) for all rn, all r and all >_ w,, x,,(m) = I(T, m, p ) ; (bjfor all 7, wyr 5 o,+, .

Lenima 9.6 is proven after the pattern of Lemma 6.3.

Proposition 9.7. For any Xi inductive operator r, i f y = 1 r I is recursively inaccessible, then y is y+-recursive.

Proof. For all inaccessible y, m E P iff /(y, m, y) = 1 and m E rY+' iff Vo < y'. F(a , rn, A n . I ( y , n,y)) = 0. Hence rY+l= l?Y iff V m < o [ I ( y , m , y) = 0 + 3a < y+ - F(o,m, hn .I(y,n, y)) = 11 iff 37 <y+Vm < w[/ (y ,m ,y ) = O + 3 0 < 7 . F(o,m, An .I(y,n, y)) = 11, the last equivdence by the regularity of 7'. It is clear that if y is the least such that rY+' = rY, then y is 7'-recursive.

This gives half of Theorem 9.4(b). Since, as we pointed out in 88, Xi 2 (nil', we also have the following.

ORDlNAL RECURSION AND INDUCTIVE DEFINITIONS 24 I

Three rather technical lemmas are needed for the proof of Theorem 9.1. We refer the reader to Cenzer [7] for proofs of these lemmas.

Lemma 9.9. For any non-recursively-inaccessible ordinal CY < y, (a) ifa is a limit of inaccessibles, then a is a'-recursive; (b) if there is a largest recursively inaccessible p < a, then @'-recursive in p.

Lemma 9.10. There is a l l i relation K * such that for all a < w, for all a, and all A 5 w such that A is a prewellordering of length a, K *(a, A ) i f f {a}oi+(")&.

Definition 9.1 1 . For any operator r, let r be defined by r ( A ) = { (m, n ) : m , n E r( { p : ( p , p ) E A } ) A ( n , n ) $ A } U A .

Lemma 9.12. For any inductive operator r, any ordinal r , (a) F7 = {(rn, n ) : m , n E r7 A (least u . rn E ru) I (least o . n E r u)} ;

(b) Irl= 11'1. (c) i fr is X i ( l l i ) , then so is r.

We are now ready to prove Theorem 9.1, that 1 (9 For any u < I Xi 1, we find an index iisuch that V r < u . { i T } 7 + ( ~ ) . 1 , but {T}u+(u)T. The proof splits into two parts. (a) Let a be recursively inaccessible. We have a X{ operator I? with associated F and1 and an rn E POrt1 - roi. It follows from Lemma 9.6 that there is an index Tisuch that for all r , {if}7+(r).1 iff -7 M ( r ) v rn 4 r7+l.

(b) It follows from Lemma 9.9 that there is an index a such that for any a! < 1 E! 1 , at least one of the following holds: (i) {a}&+(&) - 0 and RI(a);

(ii) {a}&+(.) - c and { c } ~ + = a; (iii) {a}a+(a) = ( P , b ) , /3 is the largest recursively inaccessible ordinal less than a!, and {b}&+(P) =a. It is important to note that the function ha. {a}&+(.) is 1-1 on the set of non-recursively-inaccessibles less than I Xi 1. If (i) holds, we apply (a) above. If (ii) holds, choose ?i so {T}r7+(r) = least [ < r+[ {a}7+(r) # c ] . If (iii) holds, apply (a) t o p to find 6 such that V r < p a {6}7+(r)& but {z }p+(p)T ; then let

1 = least u - va < w [ ( V r < u . {al7+(7)&) -+ ta3,+(0)-11-

0, if { d j 7 + ( r ) < w v ( { U } ~ + ( T ) ) ~ f b ; {6}7+(( {a},+(r))o), otherwise. GI,+ (7)

24 8 D. CENZER

(2) Let u = I X; I and suppose that for some a, {a}7+(r).1 for all r < u but {a}u+(u)T. Let r be a Xi operator such that lrl= u, and define r, as follows, usingLemmas9.10and 9.12: m E r l ( A ) i f f m E r ( A ) v ( m = O A-K*(a,A)). rl is Xi and 0 E r'"' - r'y, so that lrl I > u = I X I , a'contradiction. (Notice that since ?' contains only pairs, 0 $ ?.)

This completes the proof of Theorem 9.1; we have as a corollary to the proof the following.

Proposition 9.13. I Ei I is recursively inaccessible.

Proof. If not, then part (b) (ii-iii) of (I) will provide a contradiction to the conclusion in (2).

We now complete the proof of Theorem 9.4.

Proposition 9.14. (a) For any u < 1 Ei 1, i f u is u+-recursive, then u ~ ~ p e c t r u m ( ~ i ) ; (b ) forany u < Il l :I ,uESpectrum(lli) .

Proof. (a) Given u 1 {c>,+, choose b so that { b } 7 + ( ~ ) =least {<r+. { C } ~ + . = T .

Then V r < u . {b }7+ (~ )T but {b}u+(u).l. Let r' be a universal then by Lemmas 9.10 and 9.12, V r < u . - K*(b, p) but K*(b, Yo). We define ro with lrol = u by r 0 ( A ) = {m: m E r ( A ) A - K * ( ~ , A ) } U A .

(b) Since In: I < I Xi 1, there is an index a such that Vr < u . {a}7+(r).1, but {a}o+(u)T. Let r be a universal I l i operator and let r ' l (A)={m:mEi ' (A)AK*(a ,A)}; then l r l I = u .

operator;

For the proof of Theorem 9.2 we need the following lemma.

Lemma 9.15. There is a A f w such that A is a prewellordering of length a , K(t , A ) iff To(&, a, t).

relation K such that for all a < w, all a, and all

Proof. Since T is p.o.r., this is an easy consequence of Lemma 9.10. -

We now prove Theorem 9.2, that Xi = I I-semirecursive. Let u = I X f 1. (g) Given a Xi operator r, we have by Lemma 9.6 and Theorem 9.1,


m E F i f f 3 a < u ( R I ( c u ) A I(cu,m,a)== 1). (2) Let r be a universal operator and let Fl be defined by rl(A) = A u ((0,s) : s E r ( { p : (0, p ) a})}

u {( 1, t ) : K(t , { p : (0, p ) € A } ) } . We have 1 I'l 1 = u and for all a , m, and n < w , {a},(m) N n iff ( 1, (a , m, n)) E q.

10. Za relations, stable ordinals, and H l-recursion

In 94 we defined, for A 2 ORD, the ordinal 6f and showed in Corollary 4.1 1 that forA C W , ~ ? < K, and Hl is stable in A . It follows that, for any A and B in P(w), B is m-semirecursive in A iff B is H l-semirecursive in A iff B is 6f-semirecursive in A . Recall that 6:-A is the least ordinal not isomorphic to a well-ordering A: in A . In this section we present the following results, parallel to those in 95 regarding ni and uf.

Proposition 10.1. If Q 5 w X P(w) is Xk, then there is a p.0.r. functional F with rg (F) C { 0,1} such that for all m and A : Q(m, A ) i f f 3a .F(CY, m,XA) N 1 i f f 3a! < Hi. F(a ,m, X') N 1 i f f 3a<61 A .F(cu,m,xA)% 1.

Proposition 10.2. The relation K,, defined by K2((a,m,n), A ) i f f

id,p(m,XA)'n, is xi.

Proposition 10.3. Forall A C w , Sf = 6; -A .

The proof of Proposition 10.1 is basically an adaptation of Shoenfield's [21] Absoluteness Theorem to ordinal recursion theory with a set parameter; the proof follows:

Proof of Proposition 10.1 : The second and third equivalences follow by stability; we prove the first. Suppose we have Q(m, A ) iff 34 V $ 3 p .R(m,$(p), $ ( p ) , A ) , where R is recursive and p < t A R ( . . ~ . . ) + R ( . . ~ . . ) . L ~ ~ s , , @ ~ = {s: seq(s)~-R(rn,~(ln(s)),s,A)}; let 3 be the Kleene-Brouwer sequence ordering. Then, as in the proofs cited for Lemma 5.5, Q(m,A) iff 34 [Sm,@, is well-ordered by a] iff 3 4 3 7 3 f ( f : w+7)vsvt[s,tESm,*,A A s&jt + f ( s ) <f ( t ) l iff

250 D. CENZER

30 3g(g: o + u ) Vp . C(m,&), xA) = 1, where G is p.0.r. with rg (G) 2 { 0, I}. The direction (-+) of the last equivalence involves lettingg code up a y e n @

and f and taking u = (0 ,~) for a given T. Let h(u) = Sup (a, An . (m)) so that for any g: w -+ u and any p , &) < h(u). As in Lemma 5.7 there is a p.0.r. relation K such that for all ordinals u and all A C w , ro,A, defined by 7 E ro,A(q iffK(.r, u , A , ~ , satisfies (m, I ) E rA iff Vg(g: w-+u)3p. G(m,g7p),xA)-0.ThenQ(m,A)iff 3 u . ( m , I ) @ rA iff 3 u 3 T [ r;,L1 = r;,A A (m , I ) ri4 1 .

Let F(a, m, xA) =

F i s easily seen to be p.0.r. and Q(rn,A) iff 3a. F(a,m,xA) = I.

10.2.

ProofofProposition 10.2: Since Sf is stable in A , K2((a, m, n ) , A ) iff {a}H,(m,XA)=n iff 38 .{a},(m,xA)=n iff 3B3u,u(N(u,m,B) A N ( u , n , b ) A (a ,u ,u)EK;[xA]) , whereNandKB are taken from Definitions 5.8 and 5.9. It is clear from Proposition 5.10 that this is a L; relation.

The proof of Proposition 10.3 depends heavily on the Novikoff, Kondo [ 131, Addison Uniformization Theorem: (See Shoenfield [21], p. 188.)

(a)i+l= r ( 4 A (m, 1) r-t;Ap ; if r(ol),/l (cu),P

0, otherwise.

This completes the proof of Proposition 10.1; next we prove Proposition

Theorem 10.4. For any nf relation P there is a Il relation Q such that for all m, 4, $, and A :

(a) Q(m,$, $ ,A) -+P(m,6, $ , A ) ; (b)3$.P(m,4,$,A)tt3!6.Q(m,4,$,A).

Corollary 10.5. (a) I f Q is II;, then for all m, $, and A : 3$.2(m,$, $ , A ) i f f 3 $ E A ; - A .Q(m,$, $;A); (b) If Q is Xi, then for all m, $,and A : 3$.Q(m,O,$,A)i f f34€ A i - A . Q ( m , ( a , $ , A ) .

Proof of Proposition 10.3, that Sf = 6; - A for all A E w :

(I> We show that S!, - A is stable in A . Suppose {a}-(a, xA)$ for same a and s o m e a < 6 i - A ; c h o o s e Q , E A i - A such that 1 ~ 1 = a . T h e n 3 B 3 u , u ( l ~ l ~ = I { u > ~ I A (a ,u ,u) E K h [ x A ] ) ; applying Corollary 10.5 there is a B which is


there is a B which is A; in A . I t follows that {a),(a,XA) < S i - A . (2) For any u < 6; - A , we show that u is not stable in A . Given u < 6: - A , we have a A: - A well-ordering @ of type u. I t follows from Proposition 10.1 that @ is 00-recursive in A , so if u is stable in A , then @ is u-recursive in A . But then by Proposition 5.12, I @ / < u, a contradiction.

We remark that Propositions 10.1 and 10.2 could be combined and re- stated as follows:

Proposition 10.6. A relation over natural numbers and sets of natural numbers is iff it is -semirecursive iff it is H -semirecursive.

11. Stability and the ordinal arithmetic hierarchy

A crucial point in the study of ni inductive definitions in 9 6 is the fact that the relations RR and RI are p.0.r. A comparison of $5 with 5 10 makes it clear that in the study of Z i inductive definitions the relation “stable” will have a large part.

Definition 11 . I . (a) Sl(01) iff 01 is stable; (b) for all n > 0, Sn+’(a) iff 01 is stable in Sn.

We say that 01 is n-stable if Sn(a). By Proposition 4.12, n-stables exist for all n > 0. It is interesting to note, however, that for n > 1 there need not be any countable n-stables and uncountable cardinals need not always be n-stable. Let So be all of the ordinals for the sake of simplicity. Recursion in the S n is closely related to the ordinal arithmetic hierarchy, defined similarly to the usual arithmetic hierarchy.

Definition 11.2. R & ORDk X (OmoRD) is y - Xn iff there is a p.0.r. relation P and an alternating sequence 30, < y ... QnPn < y of ordinal quantifiers such that for all a and f: R(a, f ) iff 3p1 < y ... Q, 6, < y .P(fl, a, f); y -n, and 7 - En are defined analogously.

These y-arithmetic classes are comparable to the usual arithmetic hierarchy for sufficiently regular y.

Proposition 1 1.3. (a) For any f, any y recursively regular in f, and any partial function F, F is y-recursive in f iff graph (fl is 7 - C in f ;

252 D. CENZER

(b) For any regular cardinal K (or K = -) and any partial functional F, F is K -

recursive i f f graph ( F ) is K - X 1 .

We need a notion of relative n-stability.

Definition 11.4. a is n ---stable (RS"(a,P)) iff RR(@ and for all

relationsR and all t < a , 3 y < @ . R ( t , y ) + 3y<a .R( t ,y ) . - E n

Lemma 11.5. For all n > 0, RS" is p.0.r.

Proof. This is an easy application of Propositions 3.14 and 3.15.

In contrast to Lemma 11.5 is the following result.

Lemma 11.6. For all n, Sn+l is not -recursive in S".

Proof. If S"" were 00-recursive in S", then least a! .Sn+l(a) would also be 03-recursive in Sn, contradicting its n +I-stability.

We can now prove an ordinal arithmetic "Hierarchy Theorem".

Theorem 1 1.7. For all n > 0, (a) for all a , S"(a) i f f for any 00 - X,, relation R and any t < a, 30. R ( t , P) ++ 30 < a tR(t ,P); (b) for any n-stable ordinal P and any a! < (3, Sn(a) i f f RSn(a,p); (c )Sn i s m - n , bu tnotm-X n' . (d) for any R E ORDk, R is 00 - Zn+l i f f R is 00 - in S n .

Proof. Let n = 1; for n > 1 the proof is similar but more involved. (a) This follows from Proposition 11.3(a). (b) This is immediate from (a). ( c ) sl(a) iff V ~ V U V r< atla < w [ ~ ' ( u , o,(a, r,y)) 3 y < a] ; ifS1 were also m - El , it would be.-recursive, contradicting Lemma 11.6. (d)(+) 3PVy.P(P,y,a)iff 3P30[S1(a) A ( a , P ) < o A Vy<u ,P(P ,y , a ) ] . (+) 3 ~ . {a)_(@, a,S1) N_ 1 iff 303to[~l(o) A (a , P ) < u A

Tl(o,o,(a,P,a,l),X~. RS1(7,u))], whichism- E2 since by(c)S' i s m - l l , (We identify S" and RS" with their characteristic functions for the sake of simplicity.)


We can relativize Theorem 1 1.7 (d) to large ordinals.

Proposition 1 1.8. For all n > 0, all ordinals a such that a is a limit of n-stables, and all R 5 ORDk, R is a - &+, i f f R is a - XI in Sn.

In particular, R is H, - Z2 iff R is H, - El in S.

Although the S n are not w-nn complete (for example, since for any 00-

recursive f and any m, f (m) < 6 and is therefore not stable, the only A s w reducible to S by an m-recursive function is the empty set), they play the role in Theorem 11.7 of the n’th “jump” of 4. We can think of stability as a jump operator in the following sense (proven as I1.7(c)).

Proposition 11.9.ForanyA CORD, { a : cuisstableinA}ism-llI i n A but not 00- ZI in A.

For the remainder of the section we discuss S’ or S for short.

Definition 11.10. (a) E ( a , f ) iff a is recursively regular in f ; (b) a is inaccessibly stable (IS (a)) iff E ( a , S ) and a is a limit of stable ordinals.

Proposition I I . I I . (a) IS (a) iff= (a, S ) A a = 6,; (b) is p.0.r. and I S is p.0.r. in S; (c) for all a, 6,1 is recursively regular in S; (d) a stable in S implies a is recursively regular in S.

Proof. (a), (b), and (d) are similar to results on RR, RI, and stability. To prove (c), notice that by Theorem 1 1.7 (b), S(p) - p = 6, v RS’( p, 6,) for

regularity of a,+,. < so that S 1 6,, is weakly 6,+1 -recursive; (c) now follows by the

It is clear that Sf must be inaccessible stable, but as 6 s need not be countable (see 6 13), we need something else.to construct a countable inaccessible stable ordinal.

Lemma 11-12. For all ordinals 0, S(p) i f f for all a< and all -recursive F, F(a) # 13.

254 D. CENZER

Proof. By Proposition 4.10,Dp = {F(a) : a< able initial segment of ORD such that Sup (Dp) is the least stable ordinal greater than or equal to 0. The lemma follows directly from this fact.

A F is w-recursive} is a count-

Proposition 11.13. There are countable inaccessibly stables.

Proof. The least ordinal which is not w-recursive in S is clearly regular in S and will be stable by Lemma 11.12.

We are interested in ordinals much larger than the first inaccessibly stable because of the following result, which is parallel to Proposition 4.16.

Proposition 11.14. If a! is any of the following: 6 the least inaccessibly stable ordinal, the least hyper-inaccessibly stable ordinal, then a is a* -recursive in S.

Proof. For example, 6, = least a < 6,. S(a).

n For any a, let a*" = cr*...*_

Definition 11.15.0, = least 0.0 is not p*n-recursive in S.

It is clear that the 0, exist and are less than the least ordinal not w-

recursive in S, and therefore countable. The 0, are large with respect to stability as the yn of 94 are with respect to regularity.

Proposition 11.16. For all n > O,p, is inaccessibly stable.

Proof. For any a< P n , each ai is a;'-recursive in S and therefore 0;"- recursive in S ; any w-recursive function F is equivalent on pn to a p i " - recursive function Fo by the stability ofpi". By the definition of &, F(a) = F,,(a) + 0,. Hence by Lemma 11,12,0, is stable. The proof that 0, is regular in S and is a limit of stables is parallel to the proof of Proposition 4.18 (a).

We can characterize the 0, with a proposition similar to Proposition 4.21. We state our result for n = 1.


Proposition 11.17, pl is the least ordinal such that for all a < w ,

{a}p*(P,S)J. -+ 3a<O .id,*(a,s>.l.

12. Zi inductive definitions

In this section we present the following theorems.

Theorem 12.1. For all n > 0, (a) I( Z12>" I = fl , ; (b) for all A C_ o, A E (Ei)" iffA is &,-semirecursive.

Theorem 12.2. I Ili I is the least ordinal 0 such that for all a < w, if Va < p . {a),*(a,s)J, then {aIp,(P,S)J..

Theorem 12.3. For all A C a, A E "i iff'A is I II; I-semirecursive in S. -

Theorem 1 2 . 4 . ( a ) ~ p e c t r u m ( ~ i ) = { a : a < ~ ~ } ; (b) Spectrum (ni) = {a < IIIi 1 : a is a*-recursive in S } .

We begin by showing that I' E implies I I' I I ol. For any E i operator I', we have by Proposition 10.1 a p.0.r. F so that m E F(A) iff 3a .F(a,m, xA) = 1 iff 3a < Sf . F(a, m, xA) N 1. Let I be defined from F as in Definition 6.2. Parallel to Lemma 6.3, we have

Lemma 12.5.(a)Forall manda,xr , (m)Yf(a,m,SLY); (b) for all a, I

Now for any inaccessibly stable ordinal f l and any m: m E 3a<P*. F(a,m,Xn.Z(fl,n,P))- 1. I t followsfrom Proposition 11.17 that foranym,P1 cannot be the leas tpwi thmEI 'P+l , so that II'15P1.

for aUa<pl ,P = U,<6,{(a,m,n): {a},(m,S>-n}=B,.

iff

To show that 1 Z i l = /3,, we define a Xi inductive operator T such that

We need the following rather technical lemma.

256 D. CENZER

Lemma 12.6. There is an index s such that for all ordinals 0, ifall a < SP are Sp-recursive in S (for example, any 0 5 pl) or 0 = 0, then for all a < cs)s&, X B P ) = S(4-

B Proof. As in the proof of Lemma 6.6, there is a well-ordering {d } P (d for short) of length 6P such that for b in the field of d , {b}, ( S ) N Ib I d . Choose c so that for all a and 7, {c},(a,S) = 0 iff {a},(S) is stable. Recall from Proposition 5.12 the relation W , such that Wo(u,@) iff W(@) A I@ 1 = u. Choose

P

1, if wo(a, d) or 3a [(c , a , 0 ) E Bp A

0, otherwise. s s o that { ~ } s p + ~ ( " ' x ~ p ) ~ w,((Y,d r o { b : d (b ,a ) = 1 A a.itb)l;

Proposition 12.7. There isa Xi relation K* such that K*((a ,m,n) , B P ) iff {a),,+,(m,S)-n, for alla,m,n, ando.

Proof. Begin with K 2 from Proposition 10.2 and apply Lemma 12.6 and the recursion theorem (Proposition 3,5) .

Definition 12.8. T(A) = A U {s: K*(s,A)}.

The following proposition is easily verified and completes the proof of Theorem 12.l(a) for n = l .

Proposition 12.9, (a) For all a < P I , (b) for all (Y 5 Pl , Ta = B, :

= S a + l ;

(c) lTl= 01. ~

It is now easy to see that =pl-semirecursive: (c) We have r= { ( a , m , n ) : {a},,(m,S)-n} by Proposition 12.9. (2) For any Xi operator r with associated1, by Lemma 12.5, m E r i f f 3 (~<01[IS(a) A 1(&,112,(Y)"- 11.

The proof of Theorem 12.1 for n > 1 is straightforward except for the construction of an operator T, with IT, I = 0,. We need the following lemma, a fairly difficult corollary to the Uniformization Theorem ( 1 0.4). (See Cenzer 171 for a proof.)

Lemma 12.1 0. There is a relation L such that for all A 5 w ,


{(u,u) : L2(u,u,A)} isa well-ordenngof type 6: Let ri+l(A) = {<(i , (u ,uN: L2(u,u,A)} U A .

We can generalize Lemma 12.6.

Lemma 12.1 1. For all n > 0, there is an index s, such that for all p and all limit ordinals r <On, $A = (r, .... I'n-l)(Br+np), then forall a < 6r+n(p+l),

{'n '&r+n(p+ 1) (a, xB> N_ s(a).

As in Proposition 12.7 each A: is Xi, so T,

Parallel to Propositions 8.3 and 8.4, we have:

I t is not difficult to check that for all n > 1, lTnl = p n .

Proposition 12.13. (a) I [Xi, ...,Xi] I < p 2 ;

(b) I [Z i ,ll t ] I is the least ordinal p which is not p*-recursive in Sand which is a limit of ordinals with the same property.

The results regarding inductive definitions are parallel to those in 59 on E inductive definitions. Since all proofs are basically adaptations of the

techniques of 3 9, we omit them.

The fact t ha tn i -mon =ni, whereas lli is muchlarger suggests that In;-monI<lll:I.Ontheotherhand, IXi-monl= IXiI and,althoughwe have no functional for ll; corresponding to Ef, many other results on l l i and Xi operators carry over to Xi and Ili operators, so perhaps the two ordinals are equal.

An interesting open problem in this area is whether or not Ill;-mon I = In; 1.

13. Ordinal recursion and the constructible hierarchy

Our definition of wecursive is equivalent to the Kripke 151 - Platek [ 171 definition of Z1-definable (without parameters) over La. In this section we explore the relationship between ordinal recursion and constructibility.

258 D. CENZER

Let 4, and \Ir denote formulas of the language of ZF (Zermelo-Fraenkel set theory - see Shoenfield [21] for details). We follow Levy in classifying formulas as A o , C , and so forth.

Proposition 13.1. For all recursively regular ordinals a > w (or a!=w), all k and n > 0, and all R E ak, R is a -C, iff R is C, definable over L , (without parameters).

Proof. (+) By induction on the class of p.0.r. functionals we see that all p.0.r. relations are A -definable over L,; the result follows by adjoining quantifiers on each side. (+) We first prove a lemma; let {F(o) : a€ORD} be Godel’s [9] enumeration of the constructible sets.

Lemma 13.2. The relations E, defined by E(o, T) i f f F(a) E F(T), and C, defined by C(CJ,T) i f f r = F(o) , are p.0.r.

Proof. E and Care defined by an induction similar to that which defines F. The result follows from this lemma by induction over formulas.

For structures SQ a n d q f o r the language of ZF, we write d <cM iff A i s an elementary submodel o f q a n d PQ <,,%I iff 4 c?i3 and for all X,

formulas 4, and all a E I 4 1, PQ + 4, [a] implies d: /= @[a] .

Proposition 13.3. For all a!, 0, and n , (a) a! i s n-0-stable iff La< Lo; (b) S n ( 4 i f f La< ,, L.

(See Levy [ 161 for a definition of satisfaction for C, formulas in proper classes like L . )

(a) This follows from Proposition 13.1. (b) This follows from Theorem 1 1.7 (a) and Proposition 13.1.

Combining Proposition 13.3 with Theorems 6.1 and 12.1, we obtain new characterizations for In; I and /Xi[.


Proposition 13.4. (a) In: I is the least a such thet La<,, La+; (b) IEiI is the least a such that La<== La,.

There is no comparable characterization of I Ci I or init, since for any a, La+ iff L, <*, L i + , which is true for any regular ordinal a and La* iff La<,, La*, which is true for any stable ordinal a.

There i s a characterization for each of the four closure.ordinals in terms of certain reflection principles.

Definition 13.5. (a) (a!,a+) is X l ( l l , ) reflecting iff for any Xl(nl) formula @,

(b) (a,a*) is Ez(nz) reflecting iff for any Ez(nz) formula a, La+ i=@[a] -t3P<a.Lp+ + @ [ P I ;

La* l=a[a] + 3 P < a . L p * +@[PI.

Combining Proposition 13.1 with Propositions 4.21 and 11.17 and Theorems 9.1 and 12.2 we obtain

Proposition 13.6. (a) In! I is the least ordinal a such that (a, a') is C, reflecting; (b) 1 Ci I is the least ordinal a such that (a, a') is Ill reflecting; (c) 1 Cil is the least ordinal a such that (a , a*) is Z2 reflecting; (d) l l l ~ l is the least ordinal a such that (a,a*) is 112 rejlecting;

There is an obvious extension of this result to classes like (n;)" which we leave to the reader. In 5 14 we will obtain similar characterizations of ICA - L 1 and Ill: - L 1 for n > 2.

show that in fact it is a rather large constructible cardinal. The following result is very helpful.

We indicated in 3 11 that 6 s need not be countable; our next goal is to

Proposition 13.7. There is a p.0.r. function FL such that for all ordinals a, L {Xfl .FL(p,a,a) : oEORD}

= (hp .FL(P,U,(Y) : ZSZ].

Proof. Let FN (a, a) iff F(o) E aa iff 'do< a .3y < a! . 3 ~ < a[C(7, (0,~)) A

E(T,u)] . Define FL so thatFL@,a,a)=(F(a))(P) i f F ( o ) E a a :

FL(p' a'a!) N- 0, otherwise. least y < (Y . 3 ~ < U[C(T, @,y)) A E(T, a)], if FN(a, a)] ;

260 D. CENZER

The second equality follows from a collapsing argument using Lemma 4.9.

Definition 13.8. (a) LC(a) iff L + a is a cardinal; (b) is the a’th infinite constructible cardinal.

Proposition 13.9. (a) LC is a - r l l ; (b) LC is m-recursive in S: (c> ha. N; is m-recursive in S.

Proof.(a)LC(a)iffVo--XP.FL(/3,0,a)mapsa 1-1 into some ~ < a . (b) and (c) follow from Theorem 11.7 (d).

Corollary 13.10. For any ordinal a , (a) S2(a) + LC(a); (b) S2(a) + a = Ni

Now if Y = L , then the least 2-stable is a rather large uncountable cardinal. Andreas Blass has pointed out in conversation that it can be proven by a simple forcing argument that CON(ZFC) -+ CON(ZFC+ 3 countable 2-stable).

Since Sn n Nf = @ for n > 1, rc4-recursion in Sn is the same as N, - recursion; in 3 14 we obtain results connecting the En+l - L relations over w with n - $-stability.

It is interesting to note that although by Proposition 4.12 Sf” exists for each n , the two-place relation Sn(a) is not definable in Z F (being in fact “equivalent” to a satisfaction relation for L ) , so that the existence of an ordinal which is n-stable for all n is independent of ZF.

L

14. The constructible analytical hierarchy

It follows from Proposition 13.1 that for any a and any n , all a - En relations are constructible. On the other hand, it is consistent with ZFC that there be a A$ non-constructible set of natural numbers (see Jensen- SoJovay [ 101). Therefore it need not be the case that every E$ relation over w be ~0 -En for any n. However, we are able to generalize Proposition 10.6 if we restrict the discussion to constructibly IlA relations.

Definition 14.1. A relation R is constructibly EA (R E EA - L ) iff R can be defined by a En formula with function quantifiers restricted t o L n “ w . Il; - L and other constructible definability classes are similarly defined.


Theorem 14.2. For all n, k and all R 5 wk, R is Zh+l - L iff R is H l L - En.

Proof. (+) Begin with Proposition 10.1 and replace additional function quantifiers by ordinal quantifiers using Proposition 13.7. (+) This is proven from Proposition 5.10 as was Proposition 10.2.

We need some terminology for n - Hfi -stability.

Definition 14.3. (a) Sy(a) iff RSn(a, Nf) iff a is n - Hi stable; (b) 6, - A is the least ordinal n - (c) is the a'th n - t$ stable ordinal; (d) scn(a) is the least n - Nf-stable greater than a.

stable in A ;

We generalize Theorem 14.2(-+) by adding a set parameter.

Proposition 14.4. For any CA+2 - L relation Q on o X P(w), there is an Nf - X , relation P such that for all rn and A, Q(m, A ) i f f 3a< Nf,P(a,rn,A)iff 3a<6n+1 -A.P(a,rn,A).

We obtain further results parallel to those in 5 10 from the following corollary to Addison's [5] uniformization theorem for IIi - L.

Proposition 14.5. For any n > 2 , any Xi - L relation Q, any m and any con- structibleA C w , 3 @ € L .Q(m,+,A) i f f 3$(6is Ah-Lin A ) .Q(m,4,A).

Parallel to Proposition 10.3, we have

Proposition 14.6. For aN n 2 1 and all constructible A E o, &,-A = (I~,,+,-A)~

Theorem 11.7 can be relativized directly to n - Hf -stability and Hfi - L, relations. We leave the details to the reader.

Definition 14.7. For any n 2 1 , un is the least ordinal u which is not %"(a)- recursive in Sr.

Notice that u1 is the ordinal p1 defined in 5 1 1. The following results are proven as in 8 1 1.

262 D. CENZER

Proposition 14.8. For all n 2 1, (a) S:(o,) and un is afixed point of the n - tsf stables; (b) un is the least ordinal u such that u is ntl-scn(u)-stable; (c) u,, is the least ordinal u such that for all a < w ,

{a)sc n (Ll) (u , s? ) J + 37.< 1s - {a)scn(T)(7,s;)J.

We can now prove the main result of the section.

Theorem 14.9.Foralln>_l,1X~+11=u,

Sketch of proof: Let n = 2 for simplicity. (9 For any Xi - L operator F, we have by Proposition 14.4 a p.0.r. functional Fwi th rg(F) G {0,1} such that for all m,A: m E r ( A ) iff 3 ~ < 6 , - A .Vp<6,-A .F(m,a,P,xA)- 1.DefiningIby bounding the quantifiers we have as in Lemma 12.5:

Lemma 14.10. (a) Forall manda , X,,(m)----((y,m,62,a,62,(u); (b) foralla,62 -r" L62,a+l; (c) for all a = 6,,,, m E r " + l iff3o<sc2(a) . v T < ( ( u , ~ ) ) * . F ( ~ , u , T , x ~ . I ( o I , ~ , ( Y , ~ ) ) " - 1

It is now easy to check that for any m, m E I'az+l + m E FUz. (2) As ir. the proof of Theorem 12.1, we can define a such that for all CY <_ u2, 6, -Ta = 6,

- L operator 'Y and

T" = U o < 6 2 , a { ( a , ~ , n ) : {a},(m,S,) 2 %n}.

Combining Propositions 13.3(a) and 14.8(b), we have

Proposition 14.1 1. For all n 2 1, 12: A + l - L 1 is the least ordinal a such that L

L"-%z+l scfl(a) '

We can prove directly an extension of Theorem 12.2.

Theorem 14.12. For all n 2 1, all a , if V r < a . {a}scn (T{~ ,Sr )4 , then {a}

Definition 14.13. For any n >_ 1, (a, scn(a)) is X n ( l l n ) reflecting iff for any

- L I is the least ordinal a such that for (a,Sy)J. sen(")

q n , ) f0rmula a, Lscn(,) t= @[a1 +. 30 < . LSC"(@) t= @[PI.


Parallel to Proposition 13.6 we have

Proposition 14.14. For all n 2 1, (a) I En+l - L 1 is the least ordinal a such that (a, scn(a)) is Xn+l-reflecting;

(b) - L I is the least ordinal a such that (a, scn(a)) is IIn+l-reflecting.

1

The results of this section can be extended in an obvious fashion to obtain characterizations for l ( X A -L)kl and -L)kl

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J.E.Fenstad, P.G. Hinman (eds.). Generalized Recursion Theory @ North-Holland Pu bl. Comp., I 9 74

INDUCTIVE DEFINITIONS

Robin 0. CANDY Mathematical Institute, Ox ford University

$0. Introduction

Mathematical Logic is certainly permeated with inductive definitions. Here are some examples of concepts which are usually or readily defined in this way. In syntax; the notions of expression, well-formed formula, proox theorem. In semantics and model theory; the satisfaction relation, validity, Morley’s notion of rank. In set theory; well-founded set, ordinal, constructible set, the forcing relation, Bore1 set. In recursion theory the question is rather: are there any fundamental notions which are not inductively defined? All this suggests that a study of inductive definitions in general should produce interesting and applicable results. Of course it could be that it is always particular features of the definitions which are significant, so that a general study will only yield trivial results. But in fact this is not the case. One example is Barwise’s completeness and compactness theorems: the theorems are consequences of the form of the inductive definition of derivation, not of its particular details. (This is discussed in 52 below.) Another example is Moscho- vakis’,notion of hyper-projective; the original definition was by rather elaborate schemata. Under the new title hyper-elementary Moschovakis (in 141) has reworked the material as a study of first-order positive inductive definitions; the proofs are more general, shorter, and more transparent. A final example is the study of extended systems of notations for ordinals which flourished in the 1950’s. The authors (no names, no pack-drill!) were often at pains to verify, case by weary case, that their systems of notation had certain simple properties (e.g. of belonging to A;). But this verification was quite unnecessary; all that was needed was to observe that the definitions were built up using arithmetical (not necessarily monotonic) clauses, and then to apply a trivial theorem about such definitions.

265

266 R.O. GANDY

General studies, then, are worth pursuing. And once this has been accepted, it would be unreasonably Draconian to de-ny them autonomy. The view taken here is that inductive definitions are interesting in their own right. Of course we are also interested in applications; but we do not have to back up each line of enquiry with a promise of applicability.

My original intention was to give in this paper a fairly systematic account of first-order inductive definitions on an admissible set. But the recent work of Moschovakis [4], Aczel [ 11 and Barwise (this volume) made my account obsolete. So what is presented here consists, in effect, of remarks and reflec- tions.

In $ 1, I discuss the various methods which have been used to investigate certain particular classes of inductive definitions. Section $2 represents the residue of the first draft; it describes the method of semantic tableau which may have further uses. The results of 2.3. are certainly, the result of 2.4. possibly, more expeditiously proved by other methods. The results of 2.5. about I l l positive inductive definitions are new; but it is not clear if that class is significant. In $3 an effort is made to present Kleene's theory of recursion in a type 2 object as a branch of inductive theory. In so far as the effort is successful (when 'E is recursive in F') it gives a clear indication (already discussed by Aczel in [ 11) of how to set up the theory for structures other than the natural numbers. There is some inconclusive discussion of the contrary case. In $4 I draw attention to some of the problems which were ignored in $ 1-3, and make propaganda for the investigation of the forms of inductive definition which occur in proof theory. (This propaganda is directed as much at recursion-theorists as it is at proof-theorists.)

5 1. Preliminaries and a discussion of methods

1.1. Let 'u be an arbitrary first-order structure with domain A . An n-place inductive operator CP is a map P(nA) + PC.4) (P for power-set, nA for A X ... XA). For simplicity we shall always suppose that @ isprogressive, i.e., R C CPR; if not, replace @ by a', where CP'R = R U CPR. The a-th iterate, CPQ(Ro), of @ applied to R , is defined by

INDUCTIVE DEFINITIONS 267

If R, = @ we write simply CPa. The closure ordinal 1 @(Ro) I of @ from R, is defined by

The closure, @-(I?,) of CP fromR, is simply CP"(Ro) where a = ICP(Ro)(. If a E aa(R0), the stage I a I a(Ro) of a is defined by

It is often convenient to set ( a on we supposeR, = 6.

= I@(R,) I for a 4 (Pm(Ro). From now

1.2. We shall be interested in these things as CP ranges over some given collection C. So we define

The class Cm of C-fixed points is defined by Cm = {ap" : CP E C}. Of greater interst is the class C(") of C-inductive relations:

(where R is m-place and CP is n + m-place). A relation is C-co-inductive ('E C(-w)') iff its complement is C-inductive. It is C-bi-inductive ('E @('=)')

iff it is both C-inductive and eco-inductive.

1.3. One is normally interested in inductive operators which can be defined in some language L. Naturally L must contain variables ranging over A , and a relation symbol R. We do not automatically assume that L contains '=', nor that identity is a relation of%. Then CP is defined by a formula cp(x,d) of L (which shall not contain any free variables other than x (= xl, ..., x,)) iff

(1) CPR = {a E "A : (%, ..., R) k cp(ii,k)} ;

here ._. indicates whatever enlargement of % is necessary to give a realisation of L. We adopt the convention that cp, @; $, 9; ..., are always related as in ( 1 ) .

268 R.O. GANDY

Note that this useful convention may sometimes conceal the true nature of @. For example, if

where R is one-place and $ is quantifier-free and T is a term then the corresponding cp is not quantifier-free. In what follows, however, we shall mostly be concerned with rather broad classes C for which this difficulty does not arise, and we shall use syntactic classes C of formulae to characterise the corresponding classes of inductive operators. If 9 is a class of formulae we shall be particularly concerned with the class 9+ of inductive operators defined by formulae of 7 in which R occurs only positively, and the class T m of inductive operators which are defined by formulae of 9 and which are also monotonic; i.e. which satisfy R C S + @R C W.

There are two classic cases; much of the recent work on inductive definitions stems from trying to understand them and to generalise them. In both, the underlying structure is % = IN, O,S,=).

(R) (Post-Smullyan). If (? lies in the range from Rud + to Zym then (?("I = Xy and (?('m) = Recursive. (Here Rud is Smullyan's 'rudimentary' (= constructive arithmetic)).

(H) (Kleene-Spector). If e(-) = Iii and

lies in the range from ny+ to ll:m then = Hyperarithmetic.

1.4. The standard problems of inductive theory for a given a, C are to determine I C? ), to characterise (?("), to determine the closure properties of (?(-) and (?(""), and to uncover any additional structure which these sets may have. Further problems arise from relativisation - that is by considering @ which depend on parameters. The methods which have been most used may be summarised as follows.

1.4.1. Direct methods. One proceeds by constructing particular inductive definitions. A good example is the definition and use of 0 in hyperarithmetic theory. This way of proceeding appeals naturally to the purist. For many recursion theorists it also has a psychological attraction: there is a pleasure in working out the details of an intricate recursion which is akin to the pleasure of constructing a tangible object. And for investigating the fine structure of


C(”) i t seems to be the only method available. In classifying the r.e. sets, for example, one has actually to construct simple sets, maximal sets, and so on, in order to prove their existence and discover their properties.

1.4.2. Use of higher lype recursion. In case (H) it is possible to consider as the class of relations which are recursive in the jump operator. This

was shown by Kleene in [ I ] . The first study of the generalisation of (H) (by Moschovakis in [ 1-31) was based on this method. But it has turned out that the results are more easily obtained by other methods (in particular 1.4.1. and 1.4.4.).

1.4.3. 7’he use of normal forms. In case (H), for example, many of the closure properties of C(”) and d’”) and a certain amount of the structure of these classes can be most easily derived from the fact that C(”) = ll:. Recently (in [4]) Moschovakis has obtained a normal form for the generalisation of case (H) to arbitrary structures by using a ‘game quantifier’ (the idea underlying this is discussed in $2.3 below). But I think it would be a mistake to place too much reliance on this method. For I believe that the future development of the subject will be concerned with finer classes C. These will not have the sort of broad and simple syntactic characterisation of the classes so far considered; i t is not to be expected then that C(-) will have a simple syntactic form.

1.4.4. The method of embedding. We can enlarge a first order structure 91 to a structure (91 ,S,E), where S is a subset of the cumulative hierarchy of types VA formed with the elements ofA as individuals (urelements). Le., VA = u( V; : a E On} where V; = u { P( Vp U A ) : 1.3 <a}. In the language of (%,S,E) we can use Levy’s classification of set-theoretic formulae. If 9 is a class of formulae in this language, then we denote the class of relations on A whch can be defined in (a, S, E) by formulae of 7, by ?/(a, S , E).

We gjve three examples in which, given e, S can be chosen so that

Example (A): e = Z 7 t (i.e., first-order existential positive operators), S = the set of hereditarily finite members of VA .

21 0 R.O. GANDY

Example (B): C = Ah+ (i.e., first-order positive operators), S = the smallest subset of VA such that (a ,S, E) is a model for the axioms

KPU' of Kripke-PIatek set theory over a set of urelements.

(For the description of KpU+ see Barwise's paper in this volume; also 2.4.1. below.) In both these examples some conditions must be placed on % in order to ensure that RHS ( 1) C LHS (1). A sufficient condition is that \u should have only finitely many functions and relations, and be such that C('") contains a pairing function for %. Barwise gives beautiful, short proofs of both these results, and example (B) is his theorem. (These proofs will appear in some lecture notes which Barwise is preparing, see the introduction to Barwise [ 3 ] .) Barwise-Candy-Moschovakis had previously proved it when % itself is the structure of an admissible set. I think example (A) should be credited to Grilliot [ I ] ; but the class had been investigated from various points of view in Moschovakis [ 11, Montague [ 11 and Gordon [ 11. What is so fine aboutsBar- wise's proof is that one does not need to prove first any intricate results (e.g. the stage comparison and pre-well-ordering theorems) about ccm); but these theorems are easily proved once (1) has been established. In Moschovakis [4] the reader will find statements and proofs of these theorems, valid even when (1) fails, and some discussion of this case. So far as I know the following problems are open for both examples.

Problem I . Find minimal conditions on % for (1) to hold.

Problem 2. Are there any examples where (1) fails, but ecm) is an interesting class in its own right. (In the simplest failure of (1) for example (B), e(") is merely the class of %-definable relations.)

Problem 3. If the answer to problem 2 is 'yes', is it possible to find a restriction of the RHS which makes (1) true?

Example (C). Here % is '37 and we identify N with w E V A . For C we take one of the classes nf+l, E:+~, II:, E: of non-monotonic operators considered by Aczel and Richter in [ 11. Then S = L, e ,. Of course this does not solve the problem (as they do) of characterising IC 1. It would be good to have a Banvise type proof of their results. As an interim measure we mention

INDUCTIVE DEFINITIONS 27 1

that the structure of L, (>, can be coded in C(”“); this does simplify the Aczel- Richter proofs.

To sum up: the wearisome feature of so much earlier work on generalised (and even ordinary) recursion theory has been the excessive use of notations and codings. The detads of these were frequently irrelevant to the results; bu without keeping track of the details one could not prove the results. This criticism also holds, I think, for those more elegant versions (e.g. Richter 111 where, rather than some particular system of coding, one deals with a type o system. The method described in this section avoids these Zongeurs. One passes as rapidly as possible to an equation like (1); thereafter one can use the actual objects (ordinals, cumulative sets) instead of their codes.

1.4.5. Invariant definability. For the most studied cases, C(”) has received elegant characterisations in terms of invariant definability. The classic references are Grzegorczyk, Mostowski and Ryll-Nardzewski [ 11, Mostowski [ 11 and Kreisel [ I ] . More recent work is in Kunen [ 11, Moschovakis [ 31, Grilliot [2], Barwise, Gandy and Moschovakis [ 11, and the papers by Barwise and Ville in this volume. Just as the direct methods appeal to those who like to think in terms of constructions (Pascal’s ‘espirit geomtrique’) so do these to those people who prefer to think in terms of structures for a language or theory (‘espirit analytique’). But, so far as I know, there is no general approach to the problem of characterising a given e(”) in terms of invariant definability. Two particular problems might serve as first steps.

Problem 4. Give an invariant definability characterisation of (Il:+)(-) for any arbitrary structure, or for an admissible set. (Grilliot in [ I ] has shown that for acceptable % , this is the class of relations which are semi-prime-computable in E.)

Problem 5. Characterise by means of inductive definitions the classes defined by Grilliot in his paper in this volume.

I t is also relevant to seek invariant definability characterisations of the classes C of inductive operators. The only work known to me along this line is Feferman [ I ] . He showed that I: :+ is the class of operators which are monotonic both with respect to the relation argument and with respect to

21 2 R.O. GANDY

end-extensions of the structure %. This suggests:

Problem 6. Develop a theory of the connection between invariant definability characterisations of C and those of d").

52. The method of semantic tableaux

The results we prove here are not essentially new. The method is an obvious one, but I have not seen it used elsewhere; it is described in the hope that it may prove useful in other contexts. And we take this opportunity of showing how easy it is to work in the system of set theory with a structural collection of urelements (as introduced by Banvise in his paper in this volume). Given a structure (% ,S, E) of the kind described in 1.4.4., we shall use a, b, c to range over A , r , s, t to range over S , and x,y, z , u , u , to range over A US.

2.1. Definition. The class €-Prim (a) (or 3 for short) of €-primitive recursive functions over ,ZI is defined to be the last class 3 of functions over A U VA satisfying the following conditions.

(i) If F is a function of gI, and

F ' x = F ? if ; € " A ,

= 0 otherwise,

then F' E 3. (ii) If R is a relation of \u and

R*x=O if x E n A and Rx

= {O} otherwise,

then R* E 3. (iii) The following functions E 3.

( x ; y ) =Df {z : z E x v z =Y}.

ux C X ~ U V = D ~ x if u € u ,

=Df ( y : ( 3 u E x ) ( y E u ) } .

=Df y otherwise.


(iv) P i s closed under explicit definition. (v) P i s closed under €-recursion; i.e. if G, H E 9 and

Fxy = Gx? if x € A ,

= H(F r x,y)xy otherwise

then F E 3, (where F r x , p {<u,y, z ) : u E x A z = Fzfy}).

Mutatis mutandis all the results proved in Jensen & Karp [ 11 hold also for 3. In particular the characteristic function of any A. relation (without parameters) lies in 3, and the graph of any function in 3 is A1. Also, U E 3 and (v) can be dropped in favour of (v’) which is obtained from (v) by substituting the equations:

Fx? = Gx? if x € A ,

= H(U{FuU : u Ex})xy otherwise.

Note. With the definition of c given above, identity on A becomes a primitive recursive relation. I do not know if a satisfactory theory of 3 can be developed which avoids this.

2.2.Notationsand hypotheses. Let 3 =(%,S,€) be such that D ( = A U S ) is closed under 3. Since D is closed under pairing we need only consider inductively defined subclasses of D (we reserve ‘set’ for members of S ) ; X ranges over P(D), and we introduce a 1-place predicate symbol ‘EX’. Let L = Ls,u(x) be the infinitary language for (3 , X ) . We use a, b, ..., x, y, _.. ..., r, s, t, to denote variables of L of the appropriate sorts. We suppose that 3, V,M, W, are all primitives of L, so that we may suppose without loss of generality that negation is applied only to atomic formulae. A basic formula is an atomic formula or the negation of one. We suppose that L contains constants for all elements of D. We suppose that a 1 : 1 coding function g : L+D has been defined, and we identify formulae with their codes. We further suppose that g has been chosen so that all the functions of elementary syntax, and the truth definition for atomic sentences not containingx, belong to 9. (This imposes some limitation on ‘u; it is certainly possible if ‘u has only finitely many functions and relations.) The only realisations of L

214 R.O. GANDY

we consider are (3 , X ) , so the truth of a sentence varies only with X . We denote the class of predicates definable by formulae in which V

(resp. 3) does not occur by C, (resp. ll,). We use A o , E l , ..., to refer to predicates definable using conjunctions and disjunctions over finite sets. We introduce V, for the closure of El under (finitary) boolean operations. Finally, as in 5 1 , a ‘+’ indicates that ‘EX’ only occurs unnegated.

2.3.1. Definition. We define the semantic tableau for a sentence cp by induction on the level n. (1) At level 0 there is a single point P , the vertex and the sentence at P is cp.

(2) L e t P be a point of level n, and let the sentence at P be $. (i) If $ is basic, P is a tip and there are no points below it.

(ii) If $ is Ws, or (3x)O(x) then P is disjunctive. (iii) If $ is M s , or (Vx)O(x) then P is conjunctive. (iv) If $ is Ws or M s , then for each O E s there is a point Q, of level

(v) If Ic/ is (3x)O(x) or (Vx)O(x), then for each y E D there is a point

(vi) ‘below’ is the transitive closure of ‘immediately below’.

n + 1 immediately below P at which the sentence is 0 .

of level n + 1 immediately below P at which the sentence is O(y).

Evidently a semantic tableau is well-founded and every ‘branch’ through it comes to a tip.

2.3.2. Definition. Let an assignment X to 2 be given; we define the grounded points of a tableau T for cp and their ordinals as follows.

(i) If P is a tip, it is grounded iff the sentence at P is true, and in t h s case

(ii) If P is a disjunctive point it is grounded iff some point immediately IPI = 0.

below it is grounded, and in this case

IPI = Min { lQ 1 : Q immediately below P} + 1.

(iii) If P is a conjunctive point it is grounded iff all points immediately below it are grounded and in this case

I P I = Supf { I Q I : Q immediately below P)

(where Sup+Y = Sup {a + 1 : a E Y } ) .


(iv) The tableau T is grounded iff its vertex V is, and then 1 TI = I Vl .

From this there follows straightforwardly:

2.3.3. Lemma. Under the assignment X for X , cp is true i f f the tableau T for cp

is grounded.

2.3.4. Now let CP E L+ be a positive inductive operator. The complete tableau T ( x , @ ) for x E @("I is defined by modifying the definition of the tableau for @ ( x , X ) . Clause (2) (i) of 2.3.1. is altered to:-

(i) (a) If $ is basic and does not contain k, then P i s a tip (i) (b) If $ i sy EX, then P is (conventionally) disjunctive; there is just one

point immediately below P , and the sentence there is cp(y,k).

The definition of grounded for points of the complete tableau for x E CP" is again 2.3.2. Notice that 2 does not occur in the formula at a tip, so that t h s definition does not depend on an assignment for 2.

2.3.5. Lemma. x E a" i f f its complete tableau is grounded.

This is readily proved, by lemma 2.3.3., using transfinite induction on lylQ and PI.

2.3.6. Remarks. (1) . I believe the notion of trees with both conjunctive and disjunctive points was first introduced by Beth in [ 11.

(2) As Moschovakis first pointed out, and has greatly exploited, the notion of 'grounded' can be given a very intuitive explication in terms of game theory. Players R and v choose in succession a sequence of points on a branch through the complete tableau, starting at the vertex. Suppose P I , ..., P,, have been played and P,, is not a tip; ifP,, is conjunctive, A must play next, otherwise V. In either case the appropriate player must choose for Pn+l a point immediately belowP,,. IfP, is a tip then the game is finished; it is a win for V if the sentence there is true, for A if it is false. Then player V has a winning strategy just in case the tableau is grounded. From this it is plain that x E CP" can be expressed using an o-sequence of quantifiers. If we code consecutive turns by the same player into a single turn, then the o- sequence becomes an alternating sequence of V's and 3's.

21 6 R.O. GANDY

(3) It is almost obvious that 'grounded' has been given a V 1 + inductive definition. This is verified in the proof of the following:

2.3.7. Theorem. 7'here is a J E ( V, +)("I such that for any 4, E L+, any x E D ,

Thus J is universal for @+)("I. The theorem generalises to the infinitary language theorem 6 of Moschovakis [2], and its proof derives from his.

Proof. We code a point P on a complete tableau by a finite sequence u(P) = (uo , . . . , u ~ - ~ ) ; the u i are just the sentences ( O , O ( y ) or cp(y)) of 2.3.1. (iv), (v) or 2.3.4. (ib) chosen to lead one from the vertex (coded by ( )) to the point P.

of the form Ws or f h and z E s, or (ii) u is a sentence of the form y €2 and z is cp(y), or (iii) u is a sentence of the form (3y) O(y) and z is of the form O(y). Then R E 3 by the stipulation of 2.2. To see that this is true in case (iii), observe that either O(y) does not contain the variable y free, or the constant y belongs to the transitive closure of O(y). Using R it is easy to construct a function G E 3 such that G p x u = 1, 2 or 3 if u codes a tip, a conjunctive point or a disjunctive point on T ( x , @ ) , and = 0 otherwise (this shall include the case that p is not a formula of L+ with at most one free variable). Let F be defined by (1) F p x ( ) = p(x) if Gpx( ) # 0,

Let R p u z be the relation which holds just in case either (i) u is a sentence

F p x u = ( u ) ~ ~ - ~ if Gpxu # O and lhu>O, = @ € fl otherwise.

Then if u codes a point on T(x , p), F p x u is the sentence at that point.

(2) Now we give an inductive definition of a class K.

-+ (cp,x,u) E K ,

-+ (p, x , u ) E K ,

+ ( p , x , u ) E K ,

Gpxu = 1 A F p x u is a true basic sentence not containingx

Cpxu = 2 A (Vz)(Gpx(u * Z ) = 0 V (p,X, u * z ) E K )

Gpxu = 3 A(2z)(Gpx(u*z) # 0 A ( p , x , u * z ) E K )

where u * z codes the sequence got by adjoining z at the end of the sequence coded by u .


Note that if G q x u = 0, then (qxu) @ K . I t is a straightforward matter to verify that (3) Evidently wo = ($3 4 @,9, ( ) ) E K

Then there are H , , H , E '? which satisfy (4)

( p , x , u ) E K - u is a grounded point of T ( x , p).

a n d w l = ( @ E @ , @ , ( ) ) $ K .

H l ( p , x , u ) z = ( p , x , u * z ) if G ~ p x u = 2 and G p x ( u * z ) f 0,

H 2 ( q , x , u ) z = ( p , x , u * z ) if Gpxu = 3 and G p x ( u * z ) f 0, = wo otherwise,

= w1 otherwise. So we can rewrite (2) in the form: (5) Q(w) v ( V z ) ( H l w z E K ) v ( 3 z ) ( H 2 w z E K ) + w E K where, since the truth function for basic sentences E 3, Q E 3'. But 3'5 A1 ; thus we see that K E (V';>".

Finally we set

By (3) and 2.5.3., J satisfies the conditions of the theorem. QED

2.3.8. Remarks to compare ordinals using (v l+)(m) relations. More precisely the relations

(a) Besides providing a universal set for (I.,+)("), K allows us

are both (V;)(-). For, from 2.3.3. and (5) we have

Q(w) + l W l ~ = 0 , . sup+ IH, w z I

'1 = (;in IH2wzI+1 . 3 G(W)O(W)l(W)2 =

And from these it is a straightforward matter to write down a V1+ simultaneous inductive definition for < K , GK. (For details see Moschovakis [2] .)

27 8 R.O. GANDY

Further, if one of wl, w2 E K , then either w1 < w2 or w2 < wl. Hence there is a partial function (or selection operator) with graph in (V,)@) which is defined if at least one of its two arguments is in K , and which wdl then select one of its arguments which is in K . For a general discussion of the principles involved see Grilliot [ 3 ] . For more refined theorems of the same kind see Moschovakis [4].

an initial member, or if all or some of its 'predecessors' Hiwz belong to K . Recently Aczel [ 11 has shown that the introduction of a special inductive class with this property is not essential. If (2 is sufficiently closed, then <@, <@ are (?-inductive for any @ E C . And, of course, for the case we are actually discussing, Barwise's method (cf. 1.4.4.) can be applied.

Of course 0 E (II:+)("). The realisation that in general one needs (V:+)@) to get a universal set for (Ah+)@) is due to Moschovakis. We discuss the problem of when V y + can be replaced by IIy+ in 52.5 below.

(b). The arguments in remark (a) depended on the fact that w E K if it is

(c). The inductive set K has a similar role to 0 in hyperarithmetic theory.

2.4. For our next application of tableaux we need:

2.4.1. Definition. (Barwise). 9 = (2i ,S, E) isadmissible iff i t satisfies the axioms of (KPU). I t is an admissible beyond (KPV). We give an alternative characterisation:

iff it satisfies the axioms of

(i) 2 is admissible iff (a) it is closed under 3, and (b) it satisfies the axiom of A,-collection:-

(VX E s) ( 3 Y 1 cp -+ (3 t) (VX E ( 3 Y E t) cp

where cp is any A, formula. (ii) 3 is admissible beyond %if it is admissible andA ES.

It is easily shown that an admissible 3 satisfies the axioms of A , - separation and Z,-collection. We write o ( % ) f o r S n On.

2.4.2. Theorem. If 2 is admissible then

IE,I=o(a) and ( E D + p = E D .

This theorem makes plain why admissible sets carry a recursion theory


similar to ordinary recursion theory. It is inherent in the original development of admissibility theory by Kripke and Platek, but I believe it was first stated (for C in lectures which I gave in Manchester and UCLA in 1968.

Proof. By a subtableau with vertex V of a complete tableap T = T(x , p) we mean a subclass Y of the class of points of T such that:

(i) V E Y and all other points of Y lie below V ; (ii) if Q E Y is conjunctive, then all the points of T immediately below Q

(iii) if Q E Y is disjunctive then at least one of the points of T immedialy belong to Y ;

below Q belongs to Y . A subtableau Y is well-founded if:

(iv) the sentence at every tip of Y is true (v) the relation 'below' on Y is well-founded.

2.4.3. Lemma. A subtableau is well-founded i f f all its points are grounded points of T.

For 'if use induction on IPI; for only if use induction on 'below'.

Corollary 1. 7he union of a collection of well-founded subtableaux with vertex V is itself a well-founded tableau with vertex K

For the union will be a class of grounded points of T which satisfies (i)-(iii).

Corollary 2. The sentence a t the vertex of a well-founded subtableau is true under the assignment of CP" to X .

2.4.4. Lemma. For vertex uo of the complete tableau for x E am ' is Z in Y , uo, x , cp.

EL+ the relation 'Y i s a well-founded subtableau with

We refer, of course, to the coding of points introduced in the proof of theorem 2.3.7. Observe that the relation 'u is below u' is simply u C u ++Df lh(u) < lh(u) A (Vi < Ih(u)) ( ( u ) ~ = ( u ) ~ ) . Using the functions G and F it is easy to construct a primitive recursive predicate L ( Y , uo,x, p)

which expresses conditions (i) -(iv), for the quantifiers in those conditions

280 R.O. GANDY

are all restricted to Y . And (v) can be expressed by:

M( Y ) -Df ( 3 f f ) (ff is a function with domain Y and

range c On A ( v u , u E Y ) (u C u + H u < Hu)).

This is obviously I;, , and so

gives the required relation. Now let Q, E E D + . The proof of the theorem rests on the crucial.

2.4.5. Lemma. I f uo is a grounded point of the complete tableau for x E Q,=

then there exists a well-founded subtableau Y with vertex u0 such that Y is a set (i.e., Y E D or Y is 'D-finite').

The proof is by induction on ( U ~ I , , ~ . If u0 is a tip of T the required subtableau is {uO}. If u0 is a disjunctive point, then by IH we have a well-founded subtableauy for some point immediately below u0 and ( j ; u O ) is the required subtableau.

If uo is conjunctive, then the sentence at uo is M s , say, and, by IH,

Hence, since D satisfies El -collection, there is a t E D for which

(ve E ~ ) ( g Z E t )N(p ,x , u0 * e , z ) .

Y = u,,, U { ( z ; u O ) : z E t AN(p ,x ,uo*e ,Z)} .

Now set

For any 0 E s

Y , = U ( z E t A N ( ~ , x , u g * 8 , z ) }

is a well-founded subtableau with vertex uo* 0 , by Corollary 1 to 2.4.3. But then Y is evidently a well-founded subtableau with vertex uO. Finally since D satisfies the A 1 -separation axiom and is closed under 3, Y E D as required. '

' See footnote on page 299.


This completes the proof of the lemma.

Finally, to prove the theorem, we claim:

For if x E am, ( ) is grounded on T(x , @) and the RHS follows by 2.4.4. Con- versely if the RHS holds, then q(x, @)") by corollary 2 of 2.4.3.

QED

2.4.6.Corollary. WithD,@asin2.4.2. , i fweput

then \Im = am.

For, since the subtableau Y of 2.4.4. belongs to D, so does

2.4.7. Remarks. (1) A corollary of this theorem is lemma 2.5. of Barwise [ 1 J , which may be stated thus: the class of derivable sequents of LD,w, where D is admissible is C1/D. A comparison of Banvise's proof with ours shows, I think, the advantages both of working on forms of inductive definition, and of using the complete tableau. We have only 4 cases to consider(tips, 3,W,m) against Barwise's 10. And because the complete tableau contains all the different possible justifications of x E am we have avoided having to deal with derivations which contain, hereditarily, sets of derivations as subderivations.

( 2 ) P.W. Grant has given (unpublished) a rather neat proof of the theorem for the case (C /D)(") based on the second recursion theorem for D-recursive functions.

(3) Suppose we consider a relativised inductive operator q(B) where B is a given subset o f D and atomsz E B occur only positively in &I). In general lemma 2.4.4. fails: we can only assert that Y belongs to some extension D' of D in which the A&) axiom of collection holds. [This is in contrast to the particular case when D is the hereditarily finite structure overA.1 However, if B E ZD, then 2.4.4. still holds, and so, if x E (@(B))" b = { y : y E B occurs

+

282 R.O. GANDY

on the subtableau for x E am} belongs to D. 1.e. if x E (a($)" then x E (@(b))" for some 'D-finite' b ED. The Barwise compactness theorem is just a specid case of this fact. so, in this case, i f 1 E (+(h))('"), then x is weakly D-metarecursive in B. This suggests

Problem 7. Are there easily characterised subclasses C(B) of EA +(B) such that

(e = ( X : X is weakly (strongly) D-metarecursive in B}?

2.5. In this section we consider (no+)-. First we note that by using the proof of 2.3.7. it is easy to show:

2.5.1. Theorem. I j D satisfies the stipulations of 2.2., then

Now we prove our main result, which concerns those D all of whose members are countable.

2.5.1. Theorem. Let Y be a function such that for all s ED, {vsn: O<n <a) = s. Then there is a relation R ', primitive recursive in v, such that, for any @ E no+,

x E @" - ( ( y , z ) : R ' q x y z } is well-founded.

Proof. We use the coding for points on the complete tableau T for x E am used in 2.3. Since x, remain futed throughout the argument, we omit all further mention of them. We use p , q to range over the set Seq of codes for finite sequences from w. For definiteness we suppose 1 is the code for the empty sequence, and we suppose p C q impliesp < 4 . Let p : Seq -+ D; we establish a mapping up from Seq into T as follows:

( 1 4 uo( 1) = ( ) (the vertex of T ) ;

(lb) if u , ( p ) is a conjunctive point of T

UP(P*O) = q P ) * P(P*O) ;


( Ic ) if u,(p) is a disjunctive point of T , at which the sentence has the form Ws, and if n > 0

up(p*n) = u,(p) * vsn ;

(Id) if u,(p) is a point of T at whichy E X stands

U p ( P * 1) = U , ( P ) * P(Y) ;

( l e ) in all other cases

We consider a tree I'whose infinite branches correspond to different choices of p . A point p ( p ) of this tree is determined by the values of p up to and including the value at p :

We say the branch p is secured at p if u,(p) is a tip of T. Let

I" = { p ( p ) : (Vq <p) G(q) is not immediately secured)} ,

be the tree of non-past securedpoints of I'. I" is well-founded if every branch is secured at some point. And this is so iff the relation

restricted to I", is well-founded.

2.5.2. Lemma. I f T is grounded, I" is well-founded.

Let T be grounded. For a given p consider the sequence

284 R.O. GANnY

Pj+l = Pi * 0 if u p ( p j ) is conjunctive,

= pi * (pm > 0) ( u p ( p j * m ) is grounded)

if u,(pi) is disjunctive,

if u,(pi) is a tip. - - Pi

Since T is grounded, p i is always defined, and u,(po), u p ( p l ) , ... is a sequence of grounded points of T , each one immediately below the one before. But this sequence must terminate at a tip, u p ( p k ) say, and then p is secured at p k .

2.5.3. Lemma. I f r’ is well-founded, then T is grounded.

Suppose r’ is well-founded; we prove the following by induction up I”:- (3) If p ( p ) E r’, there is a 4 < p such that up(4) is a grounded point of T.

If p ( p ) is immediately secured, then u p ( p ) is a tip and so is grounded. If not, let the points of I” immediately below p ( p ) be p ( p ) * p ( p ’ * m), where p’ < p , and p(p‘ * m ) ranges over D. By IH, for each of these points there is a q < p ’ * m , with u,(4) grounded. If for any of them this 4 is < p , then (3) holds for p ( p ) . If not, u p ( p ’ * m) is grounded, where p takes the values assigned by p ( ~ ) at arguments < p , and can take any value at argument p‘ * m. By (lb), ( lc) or (Id) this means that all the points of T immediately below u p ( p ) are grounded points of T. Hence u p ( p ) is a grounded point of T ; so (3) is proved.

is grounded. But then, since p( 1) E T , up( 1) (= ( )) is a grounded point of T , and so T

To prove the theorem we have to show that R (as given by (2)) restricted to r” is primitive recursive in v. Since obviously, R 9 3, we have to show that ‘E r” is primitive recursive in v.

But, given p ( p ) , equations (1) determine up(4) for q < p by an w-recursion which uses v and the G and F of 2.3. This concludes the proof of 2.5.1.

Remark. An alternative proof could be constructed using semantic tableau rules similar to (but simpler than) those used by Lopez-Escobar in [ 11.

To see the interest of this theorem, let


wy = Sup+ {R : R is a $-recursive well-ordering whose field E D } .

We consider only admissible D of the form L, (so we are takingA = @). Let q be the least admissible ordinal such that for some x EL, , x is not w-

enumerated by any L,-recursive function. Certainly q is quite a large ordinal. For example 1) is much greater than Po the closure ordinal for ramified analysis.

2.5.4. Theorem. If a is an admissible ordinal less than q then

Proof. LHS >, RHS follows from considering the inductive definition for the initial well-ordered segment of an L,-recursive ordering. For FWS 2 LHS we apply 2.5.1. Since an La-recursive w-enumeration of any s EL, is L, bounded and so belongs to L, , we can find it by search; thus there is an La-recursive v as in 2.5,1. And since there is an La-recursive well-ordering of L,, we can pass from the well-founded relation R’ to a well-ordering relation Q with 1Q12 IR’I.

ordinals concerned. (1) If p ( p ) and q are related as in (3), then

Further, it is straightforward to verify the following by induction on the

Thus l Q l 2 Ix RHS 2 LHS in 2.5.4.

(recall that R’ , Q depend on x, p). This suffices to prove

It is known that there are admissible ordinals < o0 for which

286 R.O. GANDY

(see Gostanian [ I ] ; he calls them bad ordinals). Thus there are certainly ordinals a for which I V + /L,I > III, + /La/. I have found it hard to get much insight into @ID)(-!. Here are two problems which may well be easy, but which I could not solve.

Problem 8. Are there admissible D (preferably with A = @) for which

Problem 9. Is it true that

53. Inductive definitions with type 2 parameters

3.1. The natural way of relativising a class C? of inductive operators overA to a given total or partial function { : A + A is well-known and well understood: one enlarges C by allowing atoms fxy to occur positively in the defining fomiulae for 2 , where 3- is the graph of {. If C satisfies some simple closure conditions all goes very smoothly, and the theorems about C(") are easily relativised to theorems about C?({). We begin an investigation here of relativisation to a type 2 functional F : AA + A ; in particular we wish to connect this relativisation with the recursion theories of Kleene and Platek. We consider only the case where C? is A:+ (first-order, positive) or A!+ (quantifier free, positive), or Xy+. Further we assume thatA contains a pairing function ( ) with projection functions ( ),,,( and a copy of the natural numbers (so that it is an expansion of an 'acceptable' structure).

3.1 . l . Notations. Capital italics stand for subsets of " A . To avoid a plethora of qualifying marks we shall not always distinguish between a relation over A and its coding as a subset o fA . ThusR, {(x,y) : xRy}, {(x,y) : xRy} may all be denoted by R. The variables {, 77, range over (A +p A ) , the set ofpartial functions; f , 4 denote their graphs in the above ambiguous sense. For Y C A , (x, Y ) denotes { ( x , y ) : y E Y } , and conversely, Y,, the x-th component of Y is { y : (x ,y) 6 Y } . Letters F,G stand for partial functionals from (A to A . In order to code their graphs we define, for a binary relation R ,


[R : y ] =(O,R)U { ( I , y ) } ,

F= {[f : FC] : { E D o m F } .

and then set

F is consistent iff

3.1.2. Let z E P(P(A)), and let 2 be a formal variable for it. Let the class of formulas C be given by initial and closure conditions; we enlarge i t to C(2) as follows:

(i) c p ~ c -+ c p ~ qi); (ii) if cp E C (Z), {x : cp} is an abstract of C(Z);

(iii) if T is an abstract of C (Z), c E 2 is a formula of C (2); (iv) other closure conditions as for C . Now let be included in a language for positive inductive definitions, and

consider a progressive operator @ given by @X = {u : cp(x,X, 2)) where cp(x,X, 2) EC (Z), and has no free variables other than x. Aczel (in [ 11) has pointed out that if z is monotonic (i.e. X 5 Y A X E 2. -+ Y E z) then @ is a monotonic operator. Further, for C = AA+ much of the theory of lifted up to (e (Z))(-); in particular Aczel's proof of the stage comparison theorems goes through. Aczel makes use of the fact that Z can be treated as a quantifier: (zx)cp holds just in case {x : cp} E 2. We add the remark that provided z is monotonic we may introduce 2-rules into the calculus of sequen ts: 1 : from { r k cp(x), A : x E Y } for some Y E

infer r k (Zx) cp(x),A 2 I : from {I-, cp(x) k A : x E Y } for some Y which meets every X E Z

infer r, (Zx) ~ ( x ) I- A . These rules permit cut-elimination. This should make it possible to extend Barwise's method (cf. 1.4.4) to this case.

can be

3.1.3. Provided F is consistent, it makes sense to introduce the monotonic set of sets F" defined by:

F" = { Y : (3XEF)(XC Y ) } .

zag R.O. GANDY

And then

(1) F{ = y +-+ [f : y ] E F O

From now on we always assume that F is consistent. We note that a consequence of the definition of F” is:

(2) y E R g e F + + [A XA : y ] E F o .

3.2. We use Kleene’s schemata SIbS8 to define.the class %(F) ofA-theoretic functions partial-recursive in F. We take 9( = %, and for simplicity ignore relativisation to a function - that is we drop S.7. We do not insist that F be defined only for total functions, so that S.8. reads:

where U = u ..., u,, z = (8, (m, 0, 1 ), 2, z ); both sides are defined iff

A t - {zl) (L, t , F) E Dom (F) .

For the case considered this is, in effect, Platek’s definition of ‘recursive in F ’ and we shall refer to it as ‘Kleene-Platek’, or ‘K-P’ recursion. If the domain of F consists exactly of the total functions we say that F is clean, and write F€%:

this purpose, but a thumbs up sign seems more appropriate). We set We write ‘{z} (a)? ’ for ‘{z} (a) is defined’; (Moschovakis introduced ‘4’ for

D = D(F) = {(z,U) : { z } (L, F)?} ,

V = V ( F ) = {((z,U),y): {z}(Z,F)=y}

The class 92+(F) of sets semi-recursive in F is defined by

%+(F)= { Y : ( 3 z ) ( Y = D , ) } .

Similarly

%*(F) = { Y : Y , N - Y €9 ‘(F)}


(This would be pronounced as 'bi-semi-recursive in F'.) I t is known that not every non-empty set of 'wz+(F) is the range of a total function recursive in F; that is why, following a warning from Kreisel in [ 2 ] , we avoid using 'r.e. in F' for W(F). Also, in general, 72'(F) # %?(F).

3.3. We now seek connections between the definitions in 3.1. and those in 3.2.

3.3.1. Theorem. For any consistent F over %

(i) 72+(F) C(XY+(Fo))(m), (ii) (E:+(F"))(") = (A;+(F"))(") .

Remark. We separate out the two parts because (i) can be generalised to other structures, but (ii) depends essentially on the fact that 9 is finitely generated.

The proof of (i) consists of writing out formally an inductive definition CP which follows Kleene's definition of D(F) (for (ah+)). An existential quantifier is essential in the clause corresponding to substitution (S.4.). Further details will be found in Grilliot [ 11, which also introduced the 'counting down' technique necessary for the proof of (ii). We illustrate this by a simple example. Suppose an inductive definition for X has just one existential clause:

We replace X by the component X , in the other clauses, and replace the above clause by:

3.3.2. Theorem. If F satisfies the condition (*) below, in particular if F €% , then

290 R.O. GANDY

(*) There is a total function 77 E (Z:(F"))(-) in the domain of F, such thai none of the partial functions 7) { i : i < n} (n = 0, 1,2 , ...) are in the domain of F.

This theorem is implicit in GriUiot [4]. We illustrate the method of proof by a particular example. Let @En!+ ( F " ) be associated with the formula (Vt )p l ( t , x ,k ,2 ) , a n d l e t q = ern withOEZy+(Fo);(ingeneral oneshould consider q = 0,"). Consider the inductive operator * E Ly+ (F") given by:

(whereX3$ denotes {z : (3,(x,z))EX}). We claim that q: =@.".To see that this is so it is sufficient to observe the following facts.

(a) For some0 < Is/,@ = q . (b) I fxE@", then for some?< l * l , ~ ~ , x = q . (c) If n = (p t ) 7 pl(t,x, Grn, F"), then

For the general case, one applies the above trick repeatedly, starting from the inside, so as to get rid of all the V's from p. The 3's can be got rid of by 3.3.2.( ii).

QED

3.4. Now we would like to be able to establish a converse to 3.3.1. Under any reasonable definition of 'function recursive in F' however, we would not expect Rge F to be recursive in F. But by 3.1.3.(2) RgeF is definable from F"! Because we required F" to be monotonic, we have allowed [K : y ] E F" even when the relation R is not functional. But in the proof of 3.3.1. and 3.3.2. we never needed this extension from 6 to Fa. In fact those


theorems remain true if 6 be substituted for F" throughout. The disadvantage is, of course, that the operators in F are not monotonic, and we cannot apply Aczel's results to them. SO we make the following definition:

3.4.1. Definition. The operator CP(F") determined by substituting F" for 2 in the formula q (x ,x , 2) is said to be functional (at F) iff

(NF"))" = (@(F))" .

As the example of V(F) and D(F) shows, it may be obvious from the form of cp that @(F") is functional at all F. We write CP E Fn-e (Fo) to mean that CP E C(F") and is functional at F.

We recall that E E% and satisfies E{ = 0 if (3x) ({x = 0), E{ = 1 otherwise.

3.4.2. Theorem. If F E 'X , then

(i) (Fn - Ai+(F")>(") = %+(F, E) = (Fn - A;+(F"))(")

(ii) (Fn - A;+(F"))('") = %(F, E) = (Fn- A;+(F"))('") .

The equality of the extreme members is given by 3.3.2. For (i), %+(F, E) C RHS is proved by a simple extension of the proof of 3.3.1. To prove LHS (i) Cc>;! +(F, E) we apply the first recursion theorem. We first show that for cp E A;+ (2) there is a functional Jv, with indexj,+,, defined by Sl-S9 such that if Rget C ( 0 ) then (1) (V$(cp(&Dom{,F) +J,+,GC,F,E) = 01, and (2) (V$(JvG {, F , E) = 0 + cpG Dom t, F")). We proceed by induction on the construction of p; we only exhibit relevant occurrences of variables and constants. Case (a). cp(i;) is basic, not containingk nor 2. Then Case (b). cpo is T($ EX. Then J , O ={(TO). Case (c). qo is {z : ~(i;,z)} E Z. Then JJG) - J ~ ( u , ( I , F ( h t . 0 1 u ) ( J ~ ( u , ( O , ( t , u ) ) ) = 0)))). Suppose {z : $(u,z)} E F. Then

Jv($ = 0 if cp($ and is undefined-otherwise.

(Vt) ( 3 ! u) J/ (z, (0, ( t , u)) ) .

29 2 R.O. GANDY

So, assuming (1) holds with $ for cp, the h term in the definition o f J9 is a total function, 77 say, and $@,( 1, Fq)) is true. But thenJ9(Z) = 0, so (1) holds.

Conversely, suppose J,J$ = 0. Then the h term must define a total function 7, and so by (2) with $ for cp, to,<> E (z : $6,~)). Similarly (1,Fq)E {z : $(U,z)).Hence {z : $(U,z)}EFo;i.e.(2)holds. Case(d).rpis$l A $2. Then J,(ii) =Ji l ( i i ) *J$&ii). Case (e). cp is v $ 2 .

Then J90- {h}O'$L,ji2jU,t,F,E) where h is an index such that the RHS is defined with value 0 just in case

Such an index h exists by Gandy's selection operator theorem. (For a proof see Moschovqkis [ 51 .)

to obtain a partial function { E 32 (F, E) which is the minimal solution of Now we can Fpply the first recursion theorem (for a proof see Platek [ 11)

But then, by ( l ) , (@(F))- C Domt , and by (2) Domt L(@(F0))-. So, if @ is functional at F, (@(F"))" = Dom {, which suffices to prove (i).

To prove (ii) we note that we can strengthen the result about h (which shows that *(F, E) is closed under the 'strong or') to get a k (corresponding to 'strong definition by cases') satisfying

This completes the proof of the theorem.

3.5. Discussion. I believe I was the first to realize that %(F, E) (or 32(F) if E E 9?(F)) had a 'nice' recursion theory. Since then Sacks and his colleagues at MIT have carried it to stratospheric levels of soplustication (see, e.g. Sacks [ 13 and MacQueen [ 11). Theorem 3.4.2. represents an attempt to explain why this was possible. I t also suggests a different way of developing the theory. Namely one starts from (Fn - A A + (I?")(") and then follows the lines sug-


gested in 3. l., checking functionality where necessary. The theorem is not easily weakened or generalised. If we drop the require-

ment F € cK* then we cannot any longer use E to compare ordinals of computations. However, the use of Hinman's E# (E'S = 0 if (3x) ({x = 0), = 1 if (Vx)(fx > 0), undefined otherwise) will yield an h as in (c) above, so that (i) holds with E# replacing E. But, as Platek pointed out, a functional K which satisfies Kcgx = 0 if {(x) = 0, = 1 if g ( x ) = 0 and S(x) undefined, is not consistent. So it seems that to restore (ii) in this case one would need to introduce a multi-valued search operator. One would then be operating with relations rather than functions and it would seem more plausible to take the inductive definitions as primitive rather than to base them on an artificial notion of computability or recursiveness cooked up to make (ii) true.

Let us consider now extending the theorem to structures other than 9. Theorem 3.3.1. (ii) will fail, so the inductive definitions to consider will be Fn - Ey + (F"). As Crilliot has shown in [ 11 the existential quantifier in the inductive definitions calls for a search operator in the recursion theory. One might expect to restore the theorem then, by replacing A: by Ey, and inter- preting %(F) as 'search-computable in F'. But t h s means that F must act on many-valued functions - i.e. relations, and the last sentence of the previous paragraph may be re-applie d .

Suppose now we take 92 as given, and seek for an appropriate class of inductive definitions. In particular we consider trying to find C so that for FE3C

(1) (C(F"))(-) = 92+(F) .

Grilliot [4] has shown that %+(F) is not closed under union, so C? cannot be closed under v. At first sight this seems to make the case hopeless, as one needs v to connect separate clauses. However, examination of the clauses in the inductive definitions ofD(F) and V ( F ) show that they are deterministic in the sense that if e.g., (z,?) E D then the relevant clause is determined by z . Indeed Kleene has chosen the definition of 'index' so that this shall be so. But, alas, there is still a non-deterministic feature. The clause for S4 (substitution) has the form

(2) (~ ,v ) ( ( (z~ , ; ) , , v )€ V A (z , ,U, ,V)ED)+(Z,G)ED

where z1 and z 2 are determined by projection functions from z . We cannot eliminate the 3 by counting down, since this uses a non-deterministic V.

294 R.O. GANDY

Compare (1) with

( 3 )

It seems almost impossible to imagine a criterion which would permit (2) but reject ( 3 ) . On the other hand (3) appears to lead one outside 92 +(F). To be more precise, what Grilliot shows is that there is no mthod uniform in F for presenting the union of two members of 92+(F) as a member of %?+(F). So one poses:

(3i < 2 ) ( ( z , i,;) E D ) -, ( z , uj E x .

Problem 10. Is there an F such that %?+(F) is not closed under union?

To sum up. For an arbitrary structure the classes (Fn ~ A; + (Fo))(")(Fn - Zy +(I?"))(=) promise to have nice closure properties, and be amenable to a variety of techniques. For !Jl when E is recursive in F E3c , the classes coincide with%?+(F). When E is not recursive in F the techniques cannot be applied and %?+(F) appear to behave very differently. (I think that all that is known is contained in Grilliot [ 6 ] ) . I t is not yet clear whether in this case %?+(F) should be regarded as merely pathological.

$4. Discussion

A number of important or interesting questions received scant or zero attention in 5 1-3. We indicate some of these by listing further problems. Then we discuss the possibility of applying the theory of inductive definitions to proof theory.

Problem 11. Find conditions which ensure that (em)( - ) = (C+)(").

Spector showed that for case (H) of 1.3. the equation holds for all relevant e. I had hoped that semantic tableaux would provide a proof for the case considered there; maybe they can be forced to do this. An entirely different approach would be along the lines suggested by Feferman (cf. 1.4.5.). One considers C and ecm) not just for a, but for a class of models for the theory of 21. For this class one can apply Lyndon's theorem to get Cm = C? +. One then has to work one's way back - via semi-invariant definability - to (e +I(") for itself.


The next problem is concerned with generalising the boundeness theorems of hyperarithmetic and hyperprojective theory. Let

Problem 12. Under what conditions is C?(<-) = C('")?

Moschovakis [4] shows, for arbitrary %, that the equation holds for C = Ah+. It is obviously not true (in general) if C= Cyt. The problem is important because when the equation holds one has a natural definition for 'finiteness' in &).

Problem 13. Give a direct proof that, for countable admissible A , (XI +/A)(") = (s - a; +/A)(").

Here 's - l l i 7 is 'strict niy as introduced by Banvise in [2] (i.e., formulae which can be put in the form (VXCA)cp with cp E Zl). Present proofs use the fact that the validity predicate is s - and then apply the completeness theorem for Lwl,w. I t is even possible that the above equation holds for uncountable A ; although then it is known that in general 1s - n;+l is much greater than I El+ I. If the method of semantic tableaux could be extended to s - n: it should provide answers to this and other problems.

In 9 1-3 we used languages which contained constants for all members of A . There are many cases in which this is much too crude and will prevent interesting distinctions from being made. (For some examples, see Hinman & Moschovakis [ I ] .) In [ 11 and [2] Moschovakis systematically allowed for the use of parameters only from a given B G A . If one uses the method of embedding (1.4.4.) then one would also restrict the use of parameters in the axioms of KPU. There is not exactly a problem here; just a line to pursue. Another very obvious line is to investigate what happens to the results and problems of 93 if one relativises to an 'F' of type higher than 2. One should then consider not only inductively defined subsets ofA ('I-sections'), but inductively defined subsets of pm(A). (See Grilliot [4] and [5] .)

ductively defined sets; namely that part of proof theory which is concerned There is one part of mathematical logic which is wholly concerned with in-

296 R.O. GANDY

with the study of formal proofs. At a first encounter it may look as if no kind of general study of the sorts of inductive definition used will be of much use here: one studies a particular formal system, and evolves methods which may be specific to it. And it could be that a theory of inductive definitions has little or nothing to contribute. One does not, after all, expect such a theory to tell one anything interesting about a particular inductively defined subset of N such as the prime numbers.

But I think recent work does suggest that there are certain patterns or forms of inductive definition waiting to be discovered. The intuitive definitions I have in mind are: (a) of cut-free (infinite) proof trees (see, for examples, Schutte [ I ] ) ; (b) of reducible or computable proofs (ranging from Gentzen’s original consistency proof to e.g., Martin-Lof [ 11); and their ordering relations. I t should be observed that even when the end product of the definition is in fact a recursively enumerable or recursive set, the inductive definitions used are usually nf (not necessarily monotonic - see, for example, Martin-Lofs definition of ‘computable’). What one might hope from a theory of forms is that inductive definitions of the same form would have the same closure ordinal, and would endow it with the same structure (e.g., by exhibiting the same pattern of principal sequences or by defining the same functions on it). The evidence for optimism is as follows. First, it is often easy to spot the irrelevant elements in a given definition, and also to discover which are the clauses which really ‘do the work’. Second, the same ordinals with, at least approximately, the same structure turn up in numerous different contexts (e.g. eO, r,,, F ( E ~ ~ + ~ ) ) . One has the feeling, often, that this is no accident, without quite being able to discern the cause. Third, the method originated by Bachmann [ I ] and since extensively elaborated (for example, in Pfeiffer [ I ] and Isles [ 11) establishes natural-seeming connections between various inductively defined sets of notations and sets (in particular the finite number classes) for which there is an obvious classification. (A sim- plification of Bachmann’s method, due to Feferman and Aczel, has been ex- plored, described and extended in Bridge [ 11 .) A further, and less arbitrary connection with a known classification is provided by Martin-Lofs conjecture (in [ I ] ) that the provable well-orderings of his formal system for n-fold iterated @+ inductive definitions have precisely those ordinals for which Pfeiffer et al. provided notations using the n + I-th number class. Now the closure ordinal of n-fold iterated KIP+ inductive definitions is precisely the n + I-st admissible ordinal, which is the ‘recursive’ analogue of H,.

(c) of ordinal notations


Martin-Lofs system and his normalisation procedure have a naturalness and transparency (as compared, say, with work on ni-analysis) which suggest that it should be possible to find the reason for the connection.

Lastly, I quote an example where a ‘form’ of inductive definition can actually be given, rather than merely hinted at. Richter [ 11 introduced the operation [al, G 2 ] ; this acts like a1 on sets which are not closed under a 1 ,

and like a2 on those which are. The idea can readily be extended to finite sequences: [ a l , a 2 , ..., @.,I. If the ai are ZT+, then the closure ordinal of the resulting @+ operation is < on. Schmidt, in her dissertation [ 11, has shown how cO can be characterised using transfinite sequences of Ey+ operations, labelled by previously introduced notations. The resulting ‘form’ is not ideal, since it does not cover other rIy+ inductive definitions which are known to close off at eO. But it suggests a promising direction for further investigation.

References

P. Aczel [ 11, Stage comparison theorems and game playing with inductive definitions

P. Aczel and W. Richter [ 11, Inductive definitions and analogues of large cardinals, in: (to appear).

Conference in Mathematical Logic, London ’70, Springer Lecture Notes in Maths.

H. Bachmann [ 11, Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungzahlen, Vierteljschr. Naturforsch. Ges. Zurich 95 (1950) 115- 147.

K.J. Barwise [ 11, Infinitary Logic and admissible sets, J. Symbolic Logic 34 (1969)

K.J. Barwise 121, Implicit definability and compactness in infinitary languages, in: The Syntax and Semantics of Infinitary Languages, Springer Lecture Notes in Maths. 7 2

255 (1972) 1-9.

226- 25 2.

(1968) 1-34. K.J. Barwise [ 31, this volume. K.J. Barwise, R.O. Gandy and Y.N. Moschovakis [ 11, The next admissible set, J . Sym-

bolic Logic 36 (1971) 108-120. E.W. Beth [ 11, Semantic construction of intuitionistic logic, Mededelingen der Kon. Ned.

Akad. v. Wet., new series 19 (1950) no. 11. J.E. Bridge [ 11, Some problems in mathematical logic (systems of ordinal functions and

ordinal notations) D. Phil thesis, Oxford, 1972. S. Feferman [ 11, Uniform inductive definitions and generalized recursion theory, ASL

meeting, Cleveland, Ohio, April 30, 1969. R.O. Gandy [ 11, General recursive functions of finite type and hierarchies of functions,

Annales de la Facult6 des Sciences de l’Universit6 de Clermont, Maths. 4 Fascicule (1967) 5-24.

C.E. Gordon [ 11, A comparison of abstract computability theories, Ph.D. dissertation, UCLA, 1968.

298 R.O. GANDY

R. Gostanian [ 11, The next admissible ordinal, Ph.D. dissertation, N.Y. University, 1971. T.J. Grilliot [ 11, Inductive definitions and computability, Trans. Amer. Math. Soc. 158

T.J. Grilliot [ 21, Implicit definability and hyperprojectivity, Scripta Mathematica, to

T.J. Grilliot [3] , Selection functions for recursive functionals, Notre Dame Journal of

T.J. Grilliot [ 4 j , Recursive Functions of Finite Higher Types, Ph.D. Dissertation, Duke

T.J. Grilliot [ 5 ] , Hierarchies based on objects of finite type, J. Symbolic Logic 34 (1969)

T.J. Grilliot [ 6 ] , On effectively discontinuous type-2 objects, J . Symbolic Logic 36

T.J. Grilliot [ 7 ] , Abstract recursion theory: a summary, this volume. A. Grzegorczyk, A. Mostowski and C. Ryll-Nardzewski [ 11, The classical and the w-

complete arithmetic, J . Symbolic Logic 23 (1958) 188-206. D. Isles [ 11, Regular ordinals and normal forms, in: Intuitionism and Proof Theory,

A. Kino e t al. (eds) (North-Holland, Amsterdam, 1970) 339-361. R.B. Jensen and C. Karp [ 1 1, Primitive recursive set functions, Proc. Symposium Pure

Math. Amer. Math. Soc. 13, Part I (1971) 143-176. P.G. Hinman and Y.N. Moschovakis 111, Computability over the continuum, in: Logic

Colloquium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 77-105.

S.C. Kleene [ 11, Recursive functionals and quantifiers of finite types 1, Trans. Amer. Math. SOC. 91 (1959) 1-52.

G. Kreisel 11, Model theoretic invariants, in: The Theory of Models, J . Addison e t al. (eds.) (North-Holland, Amsterdam, 1965) 190-205.

G. Kreisel [ 21, Some reasons for generalising recursion theory, in: Logic Colloquium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 139-198.

K. Kunen [ 11, Implicit definability and infinitary languages, J . Symbolic Logic 33 (1968) 446-451.

E.G.K. Lopez-Escobar [ 1 1 , An interpolation theorem for denumerably long formulas, Fundamenta Math. 5 8 (1965) 254-272.

D.B. MacQueen [ 11, Post's problem for recursion in higher types, Ph.D. dissertation, M.I.T. 1972.

P. Martin-Lof [ 11, Hauptsatz for the intuitionistic theory of iterated inductive definitions, in: Proc. 2nd Scandinavian Logic Symposium, J.-E. Fenstad (ed.) (North-Holland, Amsterdam, 1971) 179-216.

R. Montague [ 11, Recursion theory as a branch of model theory, in: Logic, Methodology and Philosophy of Science 111, B. van Rootselaar, J. Staal (eds.) (North-Holland, Amsterdam, 1968) 63-86.

(1971) 309-317.

appear.

Formal Logic 1 0 (1969) 225-234.

University, 1967.

177 - 182.

(1971) 245-248.

Y.N. Moschovakis [ 11, Abstract first order computability I , Trans. Amer. Math. SOC. 138

Y.N. Moschovakis [ 21, Abstract first order computability 11, Trans. Amer. Math. Soc.

Y.N. Moschovakis [ 31, Abstract computability and invariant definability, J . Symbolic

Y.N. Moschovakis 141, Elementary induction on abstract structures (North-Holland,

(1969) 427-464.

138 (1969) 465-504.

Logic 34 (1969) 605-633.

Amsterdam, 1973).


Y.N. Moschovakis [5], Hyperanalytic predicates, Trans. Amer. Math. Soc. 129 (1967)

A. Mostowski [ 11, Representability of sets in formal systems, Proc. Symposium Pure

H. Pfeiffer 111, Ausgezeichnete Folgen fur gewisse Abschnitte der zweiten und weiterer

R. Platek 111, Foundations of Recursion Theory, Ph.D. Dissertation, Stanford University,

W. Richter [ I ] , Recursively Mahlo ordinals and inductive definitions, in: Logic Collo-

249-282.

Math. Amer. Math. Soc. 5 (1962) 29-48.

Zahlklassen, Dissertation, Hannover, 1964.

1966.

quium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 27 3 -288.

G.E. Sacks [ 11, this volume. D. Schmidt [ 11, Topics in mathematical logic (characterisations of small constructuve

ordinals; constructive finite number classes) D. Phil. thesis, Oxford, 197 2. K. Schiitte 111, Beweistheorie (Springer, Berlin, 1960). R.M. Smullyan [ 11, Theory of formal systems (Princeton University Press, 1961). C. Spector [ 1 1 , Inductively defined sets of natural numbers, in: Infinitistic Methods

(Pergamon, Oxford, 1961) 97-102.

Note added in proof. Moschovakis has recently made a very considerable extension of the method of embedding. He has shown that ifA C Y is a 'nice' transitive set, and if C is any class of inductive operators overA satisfying certain rather weak closure conditions, then there is an admissible set A* such that

1 Note that the existential quantifer in N should also be exposed, and a bound (for 0 E s) assigned to it.

J.E.Fenstad, P.G.Xinmnn (eds.}, Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

INDUCTIVE DEFINITIONS AND REFLECTING PROPERTIES OF ADMISSIBLE ORDINALS

Wayne RICHTER The University of Minnesota

and

Peter ACZEL University of Oslo and Manchester University

Contents

Introduction

Part I. Reflecting properties 0 1. Summary of definitions and results 3 2. Elementary facts 8 3. Ordinal theoretic characterisations 0 4. The relative sizes of the first order reflecting ordinals 8 5. First oIder reflecting ordinals and the indescribable cardinals 0 6. Stability

p 7. First order inductive definitions, I p 8. Closed classes of inductive definitions p 9. First order inductive definitions, I1 010. Higher order inductive definitions, I 01 1. Higher order inductive definitions, I1

Part 11. Inductive definitions

Appendix. Proof of the coding lemma

References

Introduction

301

306 313 3 17 322 329 333

3 37 341 347 35 1 35 6

361

38 1

An operator or inductive definition (i.d.) r : P(w) +P(o) determines a transfinite sequence (I'g : E ON) of subsets of o, where I'h =

U {r(rE) : < A}. The closure ordinal I r ( of r is the least ordinal h such

301

302 W. RICHTER and P. ACZEL

that r(X) C r( Y ) whenever X C Y

= rA. The set defined by r is r" = rlrl. r is monotone if w. For monotone r we have

Monotone inductive definitions have long been used in logic and in particular in recursion theory. For example the definitions of the terms, formulas and theorems of predicate logic may be naturally formulated as monotone inductive definitions. More generally Post's production systems give a wide class of monotone inductive definitions, for defining sets of strings of symbols in a finite alphabet. These lead to a natural characterisation of the class of recursively enumerable sets of integers. All these inductive definitions have closure ordinal 5 w . But inductive definitions with larger closure ordinals may also be considered, and they determine notation systems for ordinals in the following way. For each x E r" let lxlr be the least ordinal X such that x E I-.*+'. Then (r", I I 1') is a notation system for the ordinal lrl. Note that because I I I' maps r" onto I rl, I I71 must be a countable ordinal. For example let A be the i.d.:

A(X) = { l} U { Z x : x EX} U { 3 . S e : V n [e](n) €X} ,

where [el is the e'th primitive recursive function, in a standard recursive enumeration of them. Then (Am, I I,.,) is a slightly modified version of Kleene's system of notations for the recursive ordinals, i.e. A" is a complete II! set such that I A1 = wl, the first non-recursive ordinal. Note that A is monotone. Certain monotone i.d.'s are basic to Kleene's definition of recursion in higher type objects, [91. Also monotone i.d.'s are extensively investigated in [ 131.

As w1 is a constructive analogue of the first uncountable ordinal, it was natural to try to f o r k l a t e constructive analogues for larger initial ordinals by constructing systems of notations for them. This led to the use of non-monotone i.d.'s. (See [ 141 and [ 151 .) An independent development led to the Kripke-Platek theory of recursion on admissible ordinals. These ordinals are a constructive analogue of the regular ordinals, the first two admissible ordinals being w and w

The main aim of this paper is to formulate constructive analogues for large regular ordinals, and to obtain notation systems for them using non-monotone inductive definitions.

ADMISSIBLE ORDINALS 303

Our results will be concerned with classes C of those i.d.'s that are definable in a certain way. Thus we say that the i.d. r is Jlz if { ( x , X ) E w XP(w) : x E r(X)} is definable by a n ; formula in the language of finite types over arithmetic. Similarly we define the classes of X k and A; i.d.'s. For example the i.d.'s involved in Post's production systems are all Zy when coded on o. The operator A, above, is an example of a lIy monotone i.d. We shal l write n",mon for the class of mono tonenk i.d.'s. Similarly for Ch-mon and A;-mon. Given a class C of i.d.'s we will be interested in I C? I = Sup { I rl : I?€ C} a n d I n d ( e ) = { X C w : X L m I?'= for some E C } . Herex<, Y means that X is many-one reducible to Y .

In many cases 1 C 1 can be compared with o(32 ) for a suitably chosen class 72 of relations on w . a(%) is defined to be the sup. of the order types of the well-ordering relations in72 . Thus it is well-known that o1 = o ( A Y ) = o ( A j).

We next list some of the earlier results on the ordinals of i.d.'s.

Proposition. (i) = I C ~ I = w ; (ii) (Spector [201) In(i)-mont = In , -monl= 1 wl;

(iii) (Candy, unpublished) I @ l = wl; (iv) (Richter, [ 161) In![ is a large admissible ordinal; e.g. much larger than

the first recursively Mahlo ordinal; (v) (Putnam, [I411 I A i I = 4 A i ) ;

(vi) (Gandy,unpublished) IEi-rnonI = w(A;).

We now summarise our results. 7r; is the least ordinal h such that (LA,€) sentence. u h is defined using X k sentences. For a precise reflects every

definition see 5 1 .

In general the characterisations of the closure ordinals of i.d.'s must be more complicated. I fA is a relation on ordinals let nk(A) be the least ordinal h such that (L,[A],E,A) reflects e v e r y n k sentence. Similarly for o z ( A ) . Let rk(r) = nk(A,) where A , = {(n,ol) : a E ON & IZ E F L Y } . Also let $(r) = o",(A,).


Theorem C. For m, n > 0 (i) /Ilk I = r$(F) where F is complete H k

(ii) IE$l = flm(r) where FiscompleteZk

The proofs of the above characterisations actually give much more information. In each case, as well as characterising I e I we may also characterise Ind (C). In the next result we use the notion of a “closed” class C . Each IlL and X k is a closed class form > 0 and every closed class C has a “-complete” element. See $8 for a definition of this notion.

Theorem D. I f e is a closed class 2 Ily, r is C-complete and X = I C I then (i) X is admissible relative to A , ;

(ii) X is projectible to w relative to A , ;

(iii) Ind (C) = {X C w : X is X-r.e. relative to Ar}. i.e. there is a A-recursive in A , injection f : h -+ w ;

When C? is “sufficiently absolute” then A , 1 X is X-recursive, so that the relativisation to A , may be omitted in the statement of Theorem D. This is the case when C i s l I i + l , X m + 2 , 1 1 i 0 or Xi.

Results along the lines of Theorems C and D have been recently obtained, independently, by Moschovakis. Moreover he has generalised them to classes of inductive definitions on arbitrary abstract structures.

to the ordinals of certain wellorderings. Part (i) has been independently obtained by Cenzer (see I S ] ) .

In the next result we locate the ordinals of inductive definitions in relation

Note tha tm+n>2isessent ia l in( i )as IA : (>_ l I l~ l>w1=w(A: ) .When m+n > 2 Sacks has shown in [18] that w(A$) is a stable ordinal, so that I A: 1 is stable. It might be conjectured from this that I A: 1 is at least admissible. But we have:


Theorem F. I A: 1 is not admissible.

The diagram in (ii) of Theorem E leaves open the order relationship

The next result, obtained independently by Aanderaa in [ I ] , gives us some between several pairs of ordinals of the diagram.

more information.

Theorem G. If m, n > 0 then IJIk I # IZ; I.

When m = n = 1 this result was first proved by showing directly that ni # 0:. But we do not know if n; # u k for n + m > 2.

The proof of Theorem G is symmetric be tweennk and Z k and hence gives no information on the relative magnitudes of the two ordinals.

This is explained by the following result of Aanderaa, which we state here for completeness. (See [ I ] .) PW(C) denotes that C has the pre-wellordering property. See [ 1 ] for a precise definition.

The following summarises what is known about when the pre-wellordering property holds.

Proposition. (i) PW (n i) and PW (Z: i), (ii) v = L implies PW ( x k ) f o r m > 2,

(iii) PD implies PW(n:,+,)and PW(Z:i,+2)for m > 0.

Here PD denotes the axiom of projective determinacy. I t follows that l f l i l< lZ : i l andlZ:il<ln:l.

This should be compared with the following. (See [21] for more details.)

Proposition. (i) w ( Z : ; : ) = ~ ( A ~ ) = ~ ~ a n d o ( n i ) = w f , wherea+isthe first admissible ordinal > a;

(ii) w ( n i ) = u ( A ; ) a n d w(A;)<w(Z:i)<o(Ai)+; (iii) v = Limplies ~ ( A L ) = up;) < o(Z; ) for m > 2.

3 06 W. RICHTER and P. ACZEL

These results lead to an improvement of Theorem E(ii) in certain cases. For example we have 0: = 1 X f 1 > In: 1 = 7 r i >: 1 A l l > w(n i ) > w(Ti ) = ~ ( b ; ) , Inil> I X i I >_ u i > o ( C i ) > o(ni) = w(Al ) = IA :I, and in;I 2 n; > ~ ( 2 ; ; ) .

Whether I C i l = C J ~ or between 7ri and I C i l or u i is not known.

The paper is divided into two parts. In Part I we give alternative characterisations of some of the reflecting properties, compare them with the reflecting properties for the indescribable cardinals, and investigate their relative magnitudes. See $ 1 for a survey of the definitions and results of Part I. This part makes no use of inductive definitions and may be read without reference to Part 11.

In Part I1 we prove the results stated in this introduction. Most of t h s part depends only on 8 1 of Part I, so that the reader mainly interested in inductive definitions can probably omit the other sections of Part I on a first reading.

In $7 we examine first order inductive definitions. In particular we give upper bounds to their ordinals, proving half of Theorem A. For the other half we need the construction introduced in $8. In this section we formulate the notion of a closed class of operators. The construction of the notation systems To and the associated coding lemma are the key to getting lower bounds for the ordinals of inductive definitions, and to proving Theorem D. The coding lemma is proved in the appendix. The proof of Theorem A is completed in $9. Theorems B and C are proved in § 10, while 3 1 1 has proofs of Theorems E, F and G. Many of the results in this paper were first announced in [ 21.

= 7 r i remains open. Also, the relationship

PART I. REFLECTING PROPERTIES

$1. Summary of definitions and results

In this part we shall study some of the classes of ordinals that will be used to characterise the ordinals associated with inductive definitions. These classes of ordinals will be defined in terms of certain “reflecting properties” closely analogous to those used in defining the indescribable cardinals of lianf and Scott. (See 171 and also LCvy’s [ 111 for a detailed discussion.) In order to bring out this analogy we shall start by considering the indescribable cardinals.

ADMISSIBLE ORDINALS 3 07

We shall use a perhaps excessively large language d: within which we can conveniently formulate al l our reflection properties. d: has the usual propositional connectives, and has variables and quantifiers for all finite types (variables of type 0 range over individuals, those of type 1 range over sets of individuals, etc.). d: also has a name (individual constant) for each set and a name (relation symbol) for each relation on sets. (We will use the same symbol for the object and its name.) In particular E will denote the membership relation between sets. The restricted quantifiers (Vx Ey), (3x Ey) are defined in the usual way. Formulae of d: may be classified according to their prenex form. When doing this we shall follow LCvy in ignoring restricted quantifiers that do not bound unrestricted quantifiers. A formula i sn ; (Xh) if it is logically equivalent to a formula in prenex form which first has m alternating blocks of type n universal and existential quantifiers starting with a block of universal (existential) quantifiers and then has quantifiers of types < n and restricted quantifiers. The allowance for restricted quantifiers is of course only significant when n = 0. If a Kl; formula cp contains no constants then we call it an;- formula. Similarly for Z;-.

If R 1, ..., R,, al, ..., a,, are the relation symbols and individual constants occurring in a sentence cp of d: let A such that a l , ..., a, € A and cp is true in the structure ( A , R , 1 A , ..., R , / A , a , ,...

cp denote thatA is a non-empty set

...) a,). We can now define the (weak) indescribables.

1.1. Definition. Let X C On and a E On. If cp is a sentence of d: then (Y reflects cponXif

(Note that On is the class of all ordinals and that we indentify an ordinal with its set of predecessors.) a reflects cp if a reflects cp on On.

sentence of 2. a is Kl; (E ;)-indescribable [on X ] if a reflects [on X ] every (Xh)

Some properties of the indescribable cardinals are summarised in the following theorems. Proofs of most of these m y be found in [ 1 I ] .

1.2. Theorem. a is H!-indescrfbable (Y > o is regular.

308 W. RICHTER and P.ACZEL

Let Rg = {a > o : a is regular}. The ordinal a is Mahlo on X if for evely f : a -+ a there is a /3 > 0 closed underf such that E X n a.

1.3. Theorem. (i) the following are equivalent a ) a is n:-indescnbable on X , b) a is L :-indescribable on X , c ) a = s u p ( X n a ) .

(ii) the following are equivalent a ) a is n! -indescribable on X b) a is -indescribable on X c ) a is Mahlo on X .

(iii) a isn,!,-indescribabZe on X e a is C~+l-indescribable on X . (iv) Zjn>Oorm>2(n>Oorm>3)tken aisnn,(Ln,)-indescribable

on X - a is II; (Z il)-indescribable on X 0 Rg.

Hierarchies of classes of large cardinals have been obtained by iterating such operators as L and M where for X 5 On:

L ( X ) = EX : 01 = s u p ( x n a)}

M ( X ) = {a E X : a is Mahlo on X } .

Iterations of an operator F are defined by transfinite induction on A:

F ~ ( x ) = X n n F ( F @ ( x ) . v<

The elements ofLh(Rg) are the (weak) A-kypen'naccessibles, while the elements of Mh(Rg) are the (weak) A-kyperMakZo ordinals. Let H 1 ( X ) = {a E X : a is ny-indescribable on X } and let Hn+2(X) = {a EX : a is 11;- indescribable on X } . Then by Theorem 1.3 H , = L and H 2 = M.

The relative magnitudes of the ordinals in H,(Rg) may be indicated by using the following diagonalisation of iterations:

F A ( X ) = { a > O : a E F " ( X ) }

1.4. Theorem (LCvy). f f n > 0 then Hn+l(Rg) CH,f(Rg), (Hf)A(Rg), etc.


Let us now turn to the strongly indescribable cardinals. These are defined using reflecting properties of the cumulative hierarchy of sets. Let R(a) =

UB<,P(R(/3)) for all a E On. (P(x) is the power set of x).

1.5. Definition. R(a) reflects cp on X C: On if

R(a) reflects cp ifR(a) reflects cp on On. a is strongly [on X I ifR(cr) reflects [on x] every nk (x:) sentence of L.

(X;)-indescribable

The properties of the notions of Definition 1.5 closely resemble those of Definition 1 . 1 . The strong @-indescribables coincide with the strongly inaccessible ordinals. For n > 0 an ordinal is strongly n; (Ck)-indescribable if and only if it is strongly inaccessible and isn; (EL)-indescribable. So, assuming the GCH, the two notions coincide when n > 0.

where Def(x) is the set of subsets of x definable in (x, E 1 x, I I ) ~ ~ ~ ) .

Let L, be the set.of constructible sets of order < a, (i.e. La= UB,.Def(Lp)

1.6. Definition. L, reflects cpon X if

La reflects cp if L, reflects cp on On.

If this definition is used as in Definition 1.5 the resulting indescribability notions may easily be seen to coincide with those of Definition 1.1.

In order to obtain the classes of ordinals that we are interested in we restrict the language L. Let -C, be the sublanguage of 6: obtained by only allowing E as a relation symbol.

1.7. Definition. a i s n ; @“,-reflecting [on X ] if L, reflects [on X] every n; (c;) sentence of L,.

Some properties of this definition are summarised in the following theorems, which should be compared with Theorems 1.2 and 1.3.


1.8. 'I'heorem. a is Il!-reflecting iff a is an admissible ordinal > a.

This result and Theorem 1.9 below w d be proved in 5 2. Let Ad = {a > w : a is admissible}. a E Ad is recursively Mahlo if for

every a-recursive function f : a -+ a there is an ordinal 0 > 0 closed underf such that (3 E X n a.

1.9. Theorem. (i) The following are equivalent a) a is n:-repecting on x b) a' is Z!-regecting on X c ) ( ~ = s u p ( ~ n a ) .

(ii) N is n$reflecting on X - a is recursively Mahlo on X.

(iii) (iv) If n > 0 or m > 2 ( n > Oor m > 3 ) then a isn; (Z",-reflecting on

is n:-rejlecting on x - a is I: ,O+l-reflecting on X.

X -a isnk (Zk)-reflectingon X 17 Ad.

As i t is often easier to work with ordinals rather than the constructible hierarchy the following characterisations will be useful. Let L, be the sublanguage of C that has relation symbols only for the primitive recursive relations on sets (see [8] fo1 the properties of this notion).

1 . lo. Theorem. a isn; (z:k)-reflecting [on XI i f and only i f a reflects [on X ] every n:, (c;) sentence of I,.

The pIiniitive recursive relations in the language L, are needed for reflecting propeities on ordinals in order to compensate for the richness of the €

relation for reflecting properties on the constructible hierarchy. Theorem 1.10 will be proved in 33.

Lh(Ad) is the class of )\-recursively inaccessible ordinals, while if RM(X) = (a E X : a is recursively Mahlo on X} then RMX(Ad) is the class of )\-recur- swely Mahlo ordinals. Let M,(X) = { a E X : a is @-reflecting on X } . Then M , = M , = L and M 2 = KM. The next result indicates the relative magnitndes of rhe vrdinals inM,l(Ad) and should be compared with Theorem 1.4.

I . l l .Theorem.Zfn> Othen

ADMISSIBLE ORDINALS

T h s will be proved in $4.

31 1

1.12. Definition. Let .”, (0;) be the least lI; (Z”,-reflecting ordinal.

By 1.9 n$ = IT: = w and $ = w1 are the recursive analogues of the first two regular cardinals. What can we say about T!? By 1.9 and 1.11 7r! is greater than the least recursively Mahlo ordinal, the least recursively hyper- Mahlo ordinal etc. In fact n! appears to be greater than any “reasonable” iteration into the transfinite of this diagonalisation process. When one thinks of a corresponding cardinal in set theory (with “recursively Mahlo” now replaced by “Mahlo”) the cardinal which comes to mind is the least ni- indescribable cardinal. We shall now try and justify the view that n:-reflection is the recursive analogue of H:-indescribability. The same ideas with some additional notational complexity provide an analogy between n:+2-reflection and nA-indescribability for all n > 0, but we shall concentrate on the case n = 1.

cardinals, is defined, as well as a recursive analogue of this class whose members are called 2-admissible. We then show that a cardinal is 2-regular if and only if it is strongly lI i -indescribable, and an ordinal is 2-admissible if and only if it is @-reflecting.

Certain properties of infinity can be stated in terms of fxed points of operations. For example K > w and K is regular if and only if: (1) for every f : K + K there is some 0 < a < K such that f ’fa C a. (We say a

If we modify (1) by requiring that the witness be regular, we obtain the Mahlo cardinals, etc.

An alternative way of modifying (1) is by using higher type operations on K . Let F : K~ .+ K ~ . F is K-bounded if for every f : K + K and 8 < K , the value F ( f ) ( l ) is determined by less than K values off. More precisely, F is K-bounded if

The analogy is obtained as follows. A class of cardinals, called the 2-regular

is a witness for f.)

V f 3 Y < Kvgk 1 Y = fE Y * F ( f ) ( U = F(g)(t)l .

0 < a < K is a witness f o r F if for every f : K + K ,

f “a C a * FCf)“a C a.


1.13. Definition. K > 0 is 2-regular if every K-bounded F : K~ -+ K~ has a witness.

1.14. Theorem. K is 2-regular iff K is strongly IIi-indescribable.

We now look at a recursive analogue of 2-regularity. Roughly speaking the following definition of 2-admissible is obtained by replacing in the definition of 2-regular, “bounded” by “recursive” and the functions by their Godel numbers. In the following definition we write {t}K : K -+ K to mean that {,$}K

is total on K .

1.15. Definition. to K -recursive functions if

(i) Let K E Ad and [ < K . { t}K maps K-recursive functions

(ii) Suppose {E}, maps K-recursive functions to K-recursive functions. a € K n Ad is a witness for .$ if t < a and { t} , maps a-recursive functions to a-recursive functions.

(iii) K E Ad is 2-admissible if every t; < K such that { t}K maps K-recursive functions to K-recursive functions has a witness.

1.16. Theorem. K is 2-admissible iff K is n $reflecting.

Theorems I . 14 and 1.16 will be proved in 3 5 . Certain classes of ordinals, defined in terms of reflecting properties, also

have characterisations in terms of stability properties. LetA <x; B ifA and B are transitive sets such that A S B and B cp of XE that only has constants for elements ofA. Kripke has defined the notion of an ordinal Q beingo-stable (see [lo]). His definition used his systems of equations for defining recursion on ordinals. For admissible f i he gave the following characterisation, which we shall take as a definition:

cp * A + cp for every Zy sentence

1.17. Definition. Q is fl-stable if a < /3 and L, i c y Lo.

When 0 is not admissible, this notion may well diverge from Kripke’s original one.


1.18. Theorem. a is JI:-reflecting ifand only i f a is at1-stable.

1.19. Theorem. For countable a, a is ni-reflecting ifand only ifa is a+-stable, where a+ is the first admissible ordinal > a.

These results will be proved in $6. Given A C nON all of our definitions and results wdl relativise to A . As we

Definition 1.6 is relativised by using (La [ A ] : a E ON) instead of shall need the relativisations in Part I1 we spell out exactly what this means.

(L, : a € ON). Here L,[A] = Up.,,DefA(Lp[A]) where DefA(x) is the set of subsets of x definable in (x, E bx, A f'x,a)aEx. The language L, must be replaced by the language &(A) which is L, with an added n-ary relation symbol to denote A . Definition 1.7 becomes: a is n",A)-reflecting [on X ] if L,[A] reflects [on XI every nk sentence of &(A). Similarly for X k ( A ) - reflecting.

Theorems 1.8 and 1.9 relativise in the obvious way. Ad must be replaced by Ad ( A ) = {a > w I a is admissible relative to A b a}.

The language Lp(A) is defined by allowing relation symbols for all relations primitive recursive in A . Most of the proofs relativise in a routine way.

5 2. Elementary facts

In order to prove our theorems we shall need to assume some familiarity with the notions of primitive recursive set function; admissible class, admissible ordinal and ordinal recursion on an admissible ordinal. We shall use [8] as our basic reference and will usually follow the terminology they use. We shall also need to refer to [6] when we use Jensen's notion of a rudimentary set function.

The notion of a primitive recursive function with domain M has been formulated for various classes M e.g. these notions turn out to be special cases of the following: F : M + V is primitive recursive if M is a primitive recursive function with domainM has been to M of a primitive recursive set function. In [8] a transitive prim closed class M is defined to be admissible i f M satisfies the X :-collection principle (there called Ly-reflection principle) which we shall formulate as follows:

mON and mV. As shown in [8] all

For every prenex C : formula 0 of L, if M I= Vx E a0 then


M + Vx EaOb for some b E M , where i f0 is 3 y 1 ... 3yk\k, with 9 E:, then O b i s 3 y 1 E b ... 3 y k E b q .

We shall find it more useful to use the characterisation in [ 6 ] .

2.1. Definition. The transitive class M is admissible if M is rud closed and satisfies c :-collection.

This definition is relativized by replacing Cy-collection by Cy(A)-collection, obtained by using L,(A) instead of L, , and adding the condition that a € M * A n a E M .

A relation R on a transitive set M is Zy on M if R is defined on M by a Cy formula of L,. A partial function with arguments and values in M is Ey on M if its graph is. We shall assume some familiarity with the closure properties of these relations and functions on an admissible M , as presented for example in [ 81. In particular we shall need the following:

2.2. Proposition (Definition by Ey-recursion). Let M be an admissible set. Let C b e a function such that G EM: M X M - + M a n d G PMis 72: onM. Let

Then F r M : M -+ M and F 1 M is Zy on M. Moreover the Zy definition of F r M depends only on the .E y definition o f G 1 M (and not on M ) .

Usually we will only be interested in F r M n ON. For the notion of an admissible ordinal a and a-recursion we shall follow

[ 81. An ordinal a is admissible if L, is admissible. f : na + a is a-recursive if it is ~7 on L,.

The following lemma will be useful and the proof will illustrate some of the techniques of a-recursion.

2.3. Lemma. If a > o is an admissible ordinal and f : a + a is a-recursive then there are arbitrarily large limit ordinals < a that are closed under f-

Proof. Let a > w be admissible and let f : a + a be a recursive. Define g : a -+ a by g(x ) = Max (x t 1, Supylxf(y)). Then g is a-recursive, x <g(x ) and f ( x ) < g ( y ) for x < y < a. Given yo < a let yn = gn(yo). Then


yo < y1 < ... < a and x 27, * f ( x ) 5 Y,+~. Let y = Sup,, 7,. Then y I a is a limit ordinal such that yo < y and y is closed underf as

So it only remains to show that y < a. For this we need 2.2. Let F ( x ) = G(x, F 1 x) where G(x,y) = g(z) if x is a successor ordinal,y is a function such thaty(x-1) is defined with value z < a, and G(x,y) = yo otherwise.

Then it is not hard to see that y, = F(n) for each n € w, and that as G r L, : L, X L, + L, and is Xy on L, it follows that F I' a is a-recursive and hence y = Sup,, y,, = Sup,< F(n) < a.

Proof of Theorem 1.8. Let a be KI:-reflecting. If a < a then La k l ( a €a). Hence there is a P < a such that Lp k l ( a €a); i.e. a w. Using Lemma 6 of [6] it is not hard to show that L, is rud closed for any limit ordinal a. Hence it remains only to show that L, satisfies Ey-collection. So let L, i= Vx E a 0 where 0 is a Xy formula of L,. Then by II$reflection there is a B < a such that Lp k V x E a 0 . Now if b = Lp € L, then La t= Vx € a @ as required.

Conversely, let a > w be admissible, and let cp be a @ sentence of L, such that L, k cp. We may assume that cp has the form Vxl ... x, 3yl ... y , ?Ir where ?Ir is Zoo. Hence L, + Vx, ... x, 3y0 where 0 is the E! formula 3y1 E y _.. gym €y+. For simplicity we shall just consider the case when n = l . I f p < a a n d a = L p thenL, k V x l E a 3 y 0 . H e n c e byzy-collection there is a b E L, such that L, 7 < a so that Vx, E a 3y E L,0. Let f(0) be the least such 7 < a. Then f. : a + a is a-recursive. Let Po < a such that every constant of 0 occurs in Lp,. Then by the Lemma 2.3 choose a limit ordinal (3 such that f lo < 0 < a and (3 is closed under f. Then we must have Lp the @ sentence cp.

Vxl E a 3y E b0. But b C L, for some

Vx, 3yB so that a reflects

In order to prove (iv) of Theorem 1.9 we shall need

2.4. Theorem. There is a KIg- sentence uo of L, such that the transitive class


M is admissible ifand only i f M 0,.

Proof. By Lemma 6 of [6] there are binary rud functions F,, ..., F, such that the classM is rud closed if and only if it is closed under F,, ..., F,.

By Lemma 2 of [6] there are Xt- formulae cpi(x,y,z) of L, that define the graphs of F, for i 5 8. So M is rud closed if and only i fM k 0, where O o is the Il!- sentence /Ii<, VxVy 3zcpi(x,y, z).

By Lemma 9 of [6] we may prove:

2.5. Lemma. There is a X y - formula Sat(x,y) of L, such that ifO(x) is a Z formula of L, with x as only free variable and a = r O(x)l then for all rud closed M , i f the constants of O(x) are in M then a E M and

M k Vx(O(x) +-+ Sat (a,x)) .

Using this lemma we see that the transitive rud closed classM is admissible if and only i f M O 1 where O 1 is then!- sentence VuVu[Vx E u Sat(u,x) --f 3zVx E u Sat (U,X)']. The theorem now follows if we let (5, be O o A 8,.

Proof of Theorem 1.9. (i) b) * a) is trivial.

c) =j b). Let a = Sup (X n a) and let cp be a I l y sentence such that La I= cp. If a l , ..., a, are the individual constants occurring in cp then a l , ..., a, E L, so that there is a /3 E X n a such that a l , ..., a, € L and

on X , by (iii).

tence 1(/3 € 0). Then La k cp, so that as cp is rI: there is a y € X f? a such that L7 k cp and hence /3 < y. Hence a = Sup ( X n a).

(ii) *. Let a ben!-reflecting on X. Then a is @-reflecting and hence by Theorem 1.8 a E Ad. Now let f : a+ a be a-recursive. Let O(x,y) be a Ey formula of L, that defines the graph off on L,. Then L, Vx 3y(O(x,y) v (1 On(x) A y = 0)). Hence there is a 0 E X n a such that Lp k Vx 3y(O(x,y) v (1 On (x) h y = 0)). Hence /3 > 0 is closed underf. So a is recursively Mahlo on X .

hence Lp t= cp, as cp is rIy. So a is rI; -reflecting on X and hence X 6 2-reflecting

a) * c). Let a be nt-reflecting onX and let /3 < a. Let cp be the sen-

(ii) +=. Let a be recursively Mahlo on X and let cp be a @ sentence of LE


such that L, k cp. As in the proof of Theorem 1.8 we will suppose that cp has the form Vxl 3yO where O is C: and define the a-recursive function f : a -+ a, and the ordinalPo <a. NOW letg(x) = Max(P0,x+l, f(x)). Theng : a + a is a-recursive so that there is a 0 E X (7 a such that 0 > 0 is closed under g. From the definition o f g it follows that 0 > P o is a limit ordinal and is closed under f so that Lp k Vxl 3yO. Thus a reflects cp as required.

be

Hence there is a p E X fl a such that Lp

(iii) is trivial. So for the converse let a be n;-reflecting on X and let cp such that L, I= cp. cp has the form 3x1 ... 3x,0(x1, ..., x,) where

O(xl, ..., xn) isn, . 0 So there areal, ..., a, E L, such that L, k O(al, ..., a,).

Lp I= 3x1 ... 3x,t7(xl, ..., x,). So a is C,+l-reflecting 0

B(al, ..., an). But then on X .

(iv) + is trivial a s X fl Ad CX. For the converse we use the I'I!- sentence oo given by Theorem 2.4. Let O o be a Zy- sentence expressing the existence of an infinite set. Then a E Ad tence of &. Now let a be n; @;)-reflecting on X and let be a n ; (E;) sentence of -C, such that L, t= cp. Then because of the restrictions on n and m cp A uo A O o is a n; (Z;) sentence of L, such that La I= cp A uo A O o . Hence there is a /?EX fl a such that Lp t= cp A uo A 8,. Hence 0 E Ad so that a reflects cp on X n Ad showing that a is II; (C;)-reflecting on X fl Ad.

L, uo h do , and uo A O o is a n $ - sen-

5 3. Ordinal theoretic characterisations

Let us call-an; (z;) *-reflecting [on XI if a reflects [ o n x ] every Ilk (E;) sentence of Lp. Our proof of Theorem 1-10 will be a little indirect, in that we first prove Theorems 1.8* and 1.9*, obtained from Theorems 1.8 and 1.9 by replacing 'reflecting' everywhere by '*-reflecting'. But first we need the following lemma.

3.1. Lemma. There is a bijective primitive recursive function N : ON + L such that if(Vx < a) zx < a then L, = N"a.

Proof. In Lemma 3.2 of [8] a primitive recursive bijection N : ON + L is obtained from Godel's primitive recursive surjection F : ON + L by successively removing repetions in the values of F. Examining their definition of N it is not hard to see that N'Ia = F'Ia for all limit ordinals a. If (Vx < a) 2x < a then either a = 0, a = w or a has the form a = ep. Clearly Lo = (3 = "'0. If


a = w or a = E~ then in [ 121 it is shown that L, = F'Ia and hence it follows that L, = N " a as a is a limit ordinal.

recursive in A , such that L, [ A ] = NA"a if (Vx < a) 2x < a. This result relativises to give a bijection NA : ON + L[A] which is primitive

3.2. Lemma. If a is n! *-reflecting then (i) a is a limit ordinal > w . (ii) a, b < a *a+b < a.

(iii) b < a =$ 2b <a.

Proof. First note that the graphs of primitive recursive functions are primitive recursive relations and hence are allowed in the language Lp.

hence u < /3 <a , thus a is a limit number. So a there is a limit ordinal < a ; i.e. a is a limit ordinal > w.

is trivial, If 0 < b < a and V x < b a+x < a then, as a is a limit ordinal, a P<a. Hencea+b= Supx. ,ba+x+1</3<a.

(iii) We prove that 2b < a by induction on b < a. If b = 0 then 2b =

1 < w <a. If 0 < b < a and V x < b 2' < a then by (ii) V x < b 2x+1 = 2 x + 2 x < a. Hence a + V x 3y [x < b + 2x+1 = y ] . So by reflection V x < b 2'+l </3 for some P < a. Hence 2b =

(i) If a < CY then a 'F 3x(x=a) so that /3 + 3x(x=a) for some /3 < a and Vx3y(x<y) . By reflection

(ii) Given a < a we prove that a+b < a by induction on b < a. If b = 0 this

Vx 3y [x<b+a+x+l=y]. So by reflection V x w and a is a limit ordinal so that as we have already observed L, is rud closed. Hence it suffices to show that L, satisfies Xy-collection. So let L, l= V x E u 3y\k(x,y,b) where q(x,y ,z) is Z:-. (We can assume without loss that there is only one existential quantifier 3y and only one constant b.) We must find c E L, such that

(*> L, V x e a 3 y E c q ( x , y , b ) .

Let R C 3 ON be the primitive recursive relation given by


Let a = N(ao), b = N(Po). By the previous lemmas La = N"a so that

where RE(a,P) -N(cw) E N ( @ ) . Hence, as a is Il! *-reflecting, there is a P < a such that P I= Vx(RE(x, ao) + 3yR(x,y,Po)) A Vx 3y (2* =y) . As Vx < 0 ( F < P), L@ = N"P so that

(*) follows if we let c = Lp E L,.

3.4. Theorem 1.9*.

Proof. This follows the same pattern as the proof of Theorem 1.9 and so will be omitted. In the proof of (iv) we need the next lemma, which replaces Theorem 2.4.

3.5. Lemma. mere is a I$- sentence u1 of .Cp such that a is admissible ifand only ifa b ul.

Proof. Let us assume that then!- sentence uo of L, given in Theorem 2.4 is in Prenex form with Z 8- matrix \k (x l , ..., x k ) . Now let R(al , ..., ak) - /= \k(N(al), . . . ,N(ak)) for al, ..., ak E ON. Then R is a primitive recursive relation. Let B o be obtained from uo by replacing \k(xl, ..., x k ) by R ( x l , ..., x k ) . Then if L, ="'a

a /= 0 0 - L, I= 0 0 .

Hence by Lemma 3.1 we can let ul be B o A V x 3 y

We can now turn to the proof of Theorem 1.1C

2x = y ) .

(i)-(iii) of Theorems 1.9 and 1.9* yield the theorem in the cases Ilk (m 5 2 ) and Ek (rn 5 3). By Theorems 1.8 and 1.8* and (iv) of Theorems 1.9 and 1.9* the remaining cases need only be proved when a E Ad and X E Ad. With these restrictions the remaining cases will follow from:


3.6. Lemma. Let n + rn > 0

that for admissible a (i) For each n; sentence 8 of Lp there is a Ilk sentence OE of L, such

(ii) For each l3; sentence 8 of LE there is a n ; sentence O p of Lp such that for admissible a

Using this lemma let us conclude the proof of Theorem 1.10. Let n > 0 or m > 2 and let a benh-reflecting on X . Let 8 be a n ; sentence of Lp such that a F 8. Then 0, is a n; sentence of L, such that La F OE as a is admissible. Hence there is a 0 E X n a such that Lo + OE. As X C Ad, /3 is admissible so that 0 F 8 . Hence L, reflects 8 . Similarly if (n > 0 or m > 3) and a is E; - reflecting on X and 8 is a E; sentence of Lp then 1 8 is a n ; sentence of X p so that l(1 O), is a X; sentence of LE and the argument is as above. The proof of the converse implications is exactly similar using (ii) of the lemma instead of (i).

hoof of Lemma 3.6. (i) By thestability Theorem 2.5 of [8] we may easily associate with each

primitive recursive relationR a Xy- formula pR(xl, ..., x,) of L, such that for admissible a and a l , ..., a, < a

Now let 8 be a sentence of L p . If 8 contains individual constants for sets that are not ordinals, then a F 8 can never hold, so let Og be (1 E 0). Otherwise define 8, as follows. First replace each constant for an ordinal 0 by a constant for N ( 0 ) . Then replace each occurrence of a relation symbol R(sl, ..., s,) in O by pR(sl, ..., s,). Then for admissible a it is clear that


Now if 8 isn; and n > 0 then 8, is alsoII2 and so we can let OE be 8* .

is in prenex form. So it has the form of an alternating sequence of m blocks of universal and existential type 0 quantifiers followed by a II! formula \k(xl, ..., xk) . Now \k(xl, ..., x k ) is built up from primitive recursive relations and ordinals using the boolean operations and restricted quantifiers. Hence there is a primitive recursive relation R and ordinals P1, ..., P1 such that for all (Y

If 8 is n; (m > 0) then we have to be more careful. We may assume that O

" k W"1, ..., "k) -" i= R(P1, ..., P I , "1, ..', "k) .

Now define OE as follows: If m is even, replace 9 ( x 1 , ..., x k ) in 8 by qR(N(P1), ..., N(P1), xl , ..., x k ) and if m is odd, replace \k(xl, -.., x k ) in 8 by 7 q ~ , ~ ( N ( @ ~ ) , ..., N(P1) , xl, ..., x k ) . Then OE is IIk and has the desired properties.

sets, then L, k 8 never holds so we can let 8, be (0 = 1). Otherwise define O 0 as follows. First replace each individual constant for the set a by the constant for ordinal ar such that N(ol) = a. Then replace each occurrence of s E t in 8 by R,(s, t ) , where RE(a,P) e N ( a ) EN(@. (When proving the relativised version of 3.6 there may be occurrences of an atomic formulaA(sl, ..., sn). These must be replaced by RA(s , , ..., sn) where RA is the relation primitive recursive inA such that RA(al, ..., an) e A ( N A ( a l ) , . . . ,NA(a,2)).) Clearly for admissible ordinals a

(ii) Let 8 be a sentence of L,. If 8 contains constants for non-constructible

Now if 8 i s n k with n > 0 then O 0 is also II; and hence we can let 8, be O O . If 8 isn; with m > 0 then we must again be more careful. We can assume that 8 is in prenex form with a sequence of quantifiers followed by a II! formula \k(xl, ..., x k ) . Now % determines a primitive recursive relation R and ordinals P I , ..., such that for all a

Now define 8, by replacing \ k ( x l , ..., x k ) in 8 by R(P1, ..., P1, xl, ..., x k ) . Then O p is a rlk sentence of L, satisfying the lemma.


We conclude this section with a characterisation of admissible ordinals that will be useful in the appendix. We state it in relativised form.

3.7. Theorem. Let A be a relation on ordinals. The ordinal (3 is admissible relative to A (3 if and only if for all a < (3 and all R C ON that is primitive recursive in A i f

then there is a < A < /3 such that

V x < h 3 y < AR(a,x ,y)

Proof. Note that this characterisation uses a restricted form of KIs *-reflection. Hence it is only necessary to observe that this special form is sufficient for the proofs of 3.2 and 3.3.

54. The relative sizes of the first order reflecting ordinals

In this section we shall need some m r e results about ordinal recursion on an admissible ordinal. Iff is a partial function on the admissible ordinal a then f is a-partial recursive if the graph off is definable on L, by a Xy formula of fE.

As in Theorem 4.4 of [8] we may prove:

4.1. Normal Form Theorem. For each n 2 0 there is a primitive recursive relation T, and there is a primitive recursive function U such that i f a is admissible and f is an n-aly a-partial recursive function then there is an e < a such that for a l , ..., a, < a

Moreover e depends only on a Zy formula of L, that defines the graph o f f on L,. If this formula contains no constants then e < a. e is called an a - index o f f .


Note the uniformity in this theorem. For example i t follows that if F : ON -+ ON is primitive recursive then there is an e < w such that F I‘ a is a-recursive with a-index e for all admissible ordinals a.

Let us write {e},(al, ..., a,) for U(p,yT,(e, a l , ..., a,,y)). It will be useful to allow n = 0.

The next result is a uniform generalisation of Kleene’s S - m - n theorem.

4.2. Theorem. For each m > 0 there is a primitive recursive function S , suck that for all admissible ordinals a if e, a l , ..., a,, al, ..., a, < a then {el,(al, ---,a,, ~ 1 , --, a,)= {S,(e,al, .-., am)l,Ja1, --, a,>-

This theorem may be proved roughly as follows: Iff is an mtn-ary a- partial recursive function whose graph is defined by the x: formula O(x1, .*.,xrn,xm+l, ...>xm+n ) on L, then for al ... a,< a Xa, ... or, f ( a l , ..., a,, al, ..., a,) is also a-partial recursive, with graph defined by the C: formula O(al, ..., a,, x l , ..., x,) on La. Now S, is chosen so that if e is the index off determined by O(x,, ..., x,+,) then Sm(e,al, ..., a,) is the index of ha,, ... ..., a,f(al ... a,, al _.. a,*) determined by O(al, ..., a,, x l , ..., x,). We leave a detailed definition of S, as a primitive recursive function independent of a to the imagination of the reader.

We now use Theorem 4.1 to define universal KI:+l and E:+l formulae of 1,. For each n 2 Olet Cl(xo, ..., x,) be 3yT,(xo, ..., x , , y ) and let C,+l(xO, ..., x,) be 3yIl,(x0, ..., x, , y ) form > 0, where KIm(xo, ..., xk) is 1 Xm(xO, ..., xk). Clearly C,(xo, ..., x,) is a G, 0 formula of L, and Il,(xo, ..., x,) is a n , 0 formula of L, for each m > 0, n 2 0.

Let us call two formulae of 1, O,(xl, ..., x,), 02(x1, ..., x,) equivalent on a i f f o r a l l e l , ..., a,<&

(Y k Ol(al, ..., a,)-a l=62(a1, .-.,a,>.

4.3. Lemma.Let m > 0. Z fq(x l , ..., x,) is a Ek- (n:-)-)ormula ofL, then there is an e < w such that q(x l , ..., x,) and E,(e,xl, ..., x,)@l,(e,xl, ..., x,)) are equivalent on evely admissible ordinal.

Proof. This is by induction on m. Note that then: case follows from the


EL case by taking negations. I f m = 1 and e(xl, ..., x,) is a Ly- formula of L,, then, using the stability Theorem 2.5 of [8], we may find a Ey- formula cp(xl, ..., x,, x,+]) of LE such that for admissible a and a l , ..., a,, p < a

But cp(xl, ..., x , , ~ ) defines the graph of an a-partial recursive function on each admissible ordinal with index e < o independent of a. Hence if a is admissible and a l , ..., a, < a then

Hence U(xl, ..., xn) is equivalent to Xl(e,xl, ..., x,) on admissibles. Now suppose that the result has been proved form > 0 and let

cp(x,, ..., xn) be Z:+]. Then we may assume that it has the form 3y ... 3yk U(y I , ..., y k , xl , ..., x,) for some n, formula B(y 0 ..., yk , X I , ‘.., X,).

Now let G be the graph of a primitive recursive function mapping k-tuples of ordinals one-one onto the ordinals. Then cp(xl, ..., x,) is equivalent on every admissible to 3y@’(xl, ..., x,, y ) where U’(xl, ..., x,, y ) is the rIL formula

VYl ..’ VYk(G011, . . . ,YkJ)+eB(Y1, .-. , .Yk > X I > ...>X,))’

By induction hypothesis there is an e < w such that O‘(xl, ..., x,,y) is equivalent to ll,(e, x l , ..., x,, y ) on every admissible. Hence q(xl, ..., x,) is equivalent to Em+l(e,xl, ..., xn) on every admissible.

4.4. Corollary. I f X 5 Ad then for n > 0

Proof. By Theorem 1.10 a E M n ( X ) if and only if a E X and for every n: sentence cp of L,, a + cp * (3P E X n a)P + cp. By Lemma 4.3 and Theorem


4.2 if cp is an: sentence of L, then there is an ordinal a such that cp is equivalent to n,(a) on every admissible. The corollary now follows when X C Ad.

Below we shd be concerned with operators F on classes of ordinals that have the following properties.

4.5. (i) F ( X ) C L(X) (ii) X C Y =$ F ( X ) E F(Y)

(iii) h < a E F ( X ) 3 a € F ( X f l @,a]) where(h,a] = { p : h < f l l a } . I t follows from (iii) that for all A

F ( X ) E F(X n ( h , ~ ] ) U (h+l)

where ( h , ~ ] = (0 : h < p}.

properties, then so does F A for h > 0 and also F A . Examples of such F are L, M , H, , RM, M,. Moreover, if F has these

4.6. Definition. If F satisfies (i)-(iii) above and n > 0, then F is JIE-preserving if there is a primitive recursive function f : ON + ON such that if X =

{a E Ad : a k n,(a)} then a) F ( X ) = {a E Ad : a I= n,(f(a))} b) M,(Ad) E X U p *M,(Ad) C F ( X ) U p for all p E ON.

and

0 4.7. Lemma. For n > 0, M, is II,+l-preserving.

Proof. If X = {a E Ad : a + n,+,(a)} then by 4.4 aEM,(X) if and only if a E Ad & a I= [n,+,(a) & Vx 3y(n,(x) - R ( a , x , y ) ) ] where R is the primitive recursive relation such that R(a, b, 0) -0 k n,(b) & 0 E Ad & /3 n,+,(a). So by 4.3 M J X ) = {CY E Ad : a l=n,+l(e,a)} for some e < w. Now i f f = h x S l ( e , x ) thenfis primitive recursive and

Now let M,+l(Ad) C X U p and let a €M,+,(Ad), We must show that a EM,(X) U p. If a < p, then we are done. Otherwise CY E X so that a E Ad and Q k n,+,(a). Now suppose that a i= n,(b). Then CY t= n,(b) A n,,,(a).


As a is flf+l-reflecting on Ad there is a 0 E Ad n a such that 0 k n,(b) A

KI,+](a). Hence P E X f~ a and 0 I= Il,(b). Thus we have shown that a EM,(X).

4.8. Lemma. If F is HE-preserving, then so is F A

Proof. Let F be @-preserving and letf be a primitive recursive function such that F({a E Ad : a n,(a)}) = {a E Ad : a k n,(f(a))}. Our first aim is to find a primitive recursive function g such that for admissible a and a, c E ON

So let 8(x1 ,x2 ,x3) be the formula

where R = { (u , u) : f ( U ( u ) ) = u } is primitive recursive. Clearly this is n:- so that f?(x, , ~ 2 , ~ 3 ) is equivalent on admissibles to KIn(eo, ~ 1 , ~ 2 , ~ 3 ) for some eo < w . By a uniform version of the second recursion theorem on admissible ordinals there is an e < w such that {e},(a,x) * S3(eo,e,e,x) for a,x < a and admissible a. Now l e tg = ha,xS3(e,,,e,a,x). Then on admissibles n,(g(a,c)) is equivalent to n,(eo,e,a,c) which is equivalent to O(e,a,c). Hence for admissible a

so that (1) is proved. Let

F@’(X) = {a > p : a E FO(X)}

Our next aim is to show that for all 0 E ON


This will be proved by induction on /3. Let X = {a E Ad : a /=II,(a)). By induction hypothesis, for b < /3 < a

So that (2) is proved. Now we shall find a primitive recursive functionf' such that

(3) FA({& E Ad : a k n,(a)}) = { a E Ad : a kn,(f'(a))} .

L e t X = { a E A d : a kn,(a)}.The formulaVxVy[g(z,x)=y+nn(y)] i s a n:- formula so that there is an e l < w such that for admissible a and a E ON

But F A ( X ) C X C Ad so that F A ( X ) = {a E Ad : a k n,(f'(a))}where f' = hxSl (e l ,x ) . So (3) is proved.

It now remains to show that if X = {a E Ad : a k n,(a)} and


M,(Ad) C X U p thenM,(Ad) E F A ( X ) U p. So let X, p satisfy the above assumptions. We first show that for all /3 E ON:

This will be proved by induction on /3. By induction hypothesis, if b < /3, then

M,(Ad) C F b ( X ) U Max ( p , b + 1)

C F ( b ) ( X ) U Max (p, b + 1) .

But as F isn:-preserving, by (2), if b < 0, then

M,(Ad) C F(F(b)(X)) U Max (p, b + 1)

C F(Fb(X)) U Max ( p , B + 1) by 4.5 (iii) .

Hence

Hence (4) is proved and now if a €M,(Ad) then if a < p we are done. Other- wise, by (4) Q € n,,.Fp(X) so that a EFA(X) Up. ThusM,(Ad) C FA(X) u P.

We can now prove Theorem 1.11. I f F is n,+, 0 -preserving, thenM,+l(Ad) C F(Ad) as Ad = {a E Ad :

a previous two lemmas.

n,(e,)} for some eo < o. Hence Theorem 1.1 1 follows from the

4.9. Remark. I f Y is a primitive recursive class of ordinals such that Y C Ad and Ad is replaced by Y in Definition 4.6, then the proofs of the previous two lemmas still hold so that we get that for n > 0:


55. Reflecting ordinals and indescribable cardinals

In this section we will prove Theorems 1.14 and 1.16.

5.1. Lemma. If K is 2-regular then K > w and K is regular.

Proof. Let K be 2-regular. I t suffices to show that every g : K + K has a witness. For a given g, let F : K~ + K~ be defined by F ( f ) (f) = g(f(0)) for all f : K + K

and all t < K . F is clearly K-bounded. Let a be a witness for F. We show a is a w i t n e s s f o r g . L e t p < ~ andf : K + K such t h a t f ( f ) = p f o r a l l ( < ~ . T h e n f " a C a and hence F(f)"a C a. Thus

Hence g"a C a.

5.2. Proof of Theorem 1.14. K is 2-regular iff K is strongly n:-indescribable. We show

(b) K is strongly inaccessible (a) K 2-regular* & i (c) K is Hf-indescribable

3 (d) K is strongly IIi-indescribable

=+ (a).

We first show (a) =+ (b). Let K be 2-regular. Since K is regular it remains to s h o w h < ~ *2 '<~ .Supposeno t . LetX<K a n d 2 ' 2 ~ . Le t rmap 2'onto K . Define F : K~ + K~ by

r(fF A) if f"XE 2 ,

0, otherwise,

for f < K . F is K -bounded since F ( f ) (f) is determined by values off on h < K .

Let a be a witness for F. It is clear that a can be chosen 2 2. Let g : X + 2 such that r(g) > a and letf : K -+ K so thatf r X = g andf(t) = 0 for f 2 A. Thenf"a c 2 C a. Since a is a witness for F, F( f ) (O) <a. But


which is a contradiction. To show (a) * (c) let p be a Il: sentence of d: such that K k p. We must

find a 0 < (Y < K such that (Y

mappings of On X On onto On, and K , L be the associated pairing functions (cf. LCvy [ 1 I]). We first switch from set quantifiers to quantifiers of binary relations which are characteristic functions, and then switch to the language with unary function quantifiers instead of set quantifiers. In this language with the aid of P, K , L we can put formulas in a normal form (cf. Rogers [ 171, where this is done for formulas of second-order arithmetic). Thus there is a quantifier-free formula Q such that K k Vf3EQ(f, E) and for every a 5 K

which is closed under P,

p. Let P be one of the standard bijective

Furthermore, Q can be chosen so that in Q there is no nesting off(i.e. no terms of the formf(f( ...))). For a givenfand E the truth or falsity of Q ( f , t ) is determined by the values off for finitely many arguments and the answers to finitely many questions about membership in the relations appearing in Q. Since there is no nesting off the finite set of arguments off needed depends only on { and the relations in Q but not onfitself. Thus,

where

Hence, since K is regular,

where

Let h(P) = p q . C(P,q). Then h : K + K and for allf,g : K + K ,


( 1) f r h (0) = g r h (0) 3 [vt I P. QV, t ) - ~ ( g , a. Let G : K~ -+ ' K so that (2) and let F : K~ -+ K~ so that F ( f ) ( P ) = G(f) for all 0 < K . F is K-bounded since

C ( f ) = pu [P"u X u< u & 35 5 U . Q ( f , E) & h ( t ) 5 u ] ,

Let a be a witness for F and letf : a + a . We show 3 t < a. Q ( f , E ) . Let g : K + K so that g Fa =f. Theng"a C a and F(g)"a E a since a is a witness. Let & = pc. Q(g , t ) . Then 6 , h(6) 5 G(g) I F(g) (0) < a by definition of G. Thus from (l),

and hence pt . Q(f,E) = 6 < a. Thus a under P (by (2) ) , a + cp.

show (d) *(a). Let K be strongly ni-indescribable and F : K~ + K~ be K -

bounded. We show that F has a witness. Let

Vf 3 [ Q ( f , t ) and since a is closed

A proof that (b) & (c) *(d) appears in L6vy [ 111 p. 217. It remains to

Then X C R(K) . Note that (f 1 y, t , ~ ) E X * F ( f ) ( t ) = 7). Since F is K-bounded,

Since K is strongly n -indescribable there is an a < K such that

i.e. 0 < a < K and

We show a is a witness for F. Let f : K -+ K such that f "a C: a. Since f 1 a E (ya andf l a 1 y = f r y for y < a, we have from (3)

( 3 ) " a v t < a 37,q < a wr 7, t , ~ ) E X n ~ ( 4 i .

V [ < a 377 < a F ( f ) ( $ ) = 7, i.e. F( f ) "a E a .

332 W . RICHTER and P. ACZEL

5.3. Remark. In the proof of (d) * (a), the assumption that F is K-bounded cannot be eliminated. For each K it is easy to define an F : K~ + K~ which is not K-bounded and has no witness.

5.4. Proof of Theorem 1.16. K is 2-admissible iff K is n!-reflecting.

sive functions. We show t has a witness. By hypothesis, Suppose K is @-reflecting. Let {$IK map K-recursive functions t o K-recur-

By using the T predicate this is equivalent t o

0 The sentence on the right is equivalent t o a @ sentence cp(t), Since K i s n 3 - reflecting (and hcnce II!-reflecting on Ad) there is some a E K n Ad such that a /= cp($). But by the definition of p($) this implies functions to a-recursive functions and hence a is a witness for $.

Now suppose K is 2-admissible and let cp be a @ sentence of X p such that K /= cp. We show that K reflects cp. For simplicity we assume that cp is of the form Vx 3yVz $(x,y,z) where $ is a Z;: formula with constants less than K .

Let a be admissible so that all constants in $ are less than a. We introduce certain Gijdel numbers of a-partial recursive functions which, by the uniformity of the Normal Form and S- m - n theorems, can be chosen t o be independent of the particular choice of a. First choose a < a so that

maps a-recursive

Let g be a primitive recursive ordinal function such that


and let t < w be a Godel number (independent of a) ofg. Then from (4),

{t} , maps a-recursive functions to a-recursive functions.

Since K + cp, by ( 5 ) {t IK maps K-recursive functions to K-recursive functions. Since K is 2-admissible there is an a E K n Ad which is a witness for E . But then by (9, a I= cp.

5.5. Remark. The definition of 2-admissible given here is equivalent to the definition which appears in [2]. The full definition of n-admissible is given in [21.

8 6. Stability

In this section we prove Theorems 1.18 and 1.19. Note that if A < x y B and A s C 5 B then A <zy C. It follows that if a is

P-stable and a < y < P then a is y-stable. Hence the weakest stability property for an ordinal a is that of being a t 1-stable. 1.18 implies that even this weakest stability property determines ordinals with rather strong reflecting properties. But first we need:

6.1. Lemma. If a is a+l-stable then a is admissible.

Proof. Let L, k Vx €alp where cp is a Xy formula 1,. Then La+] VxEacpb whereb=L,EL,+l.HenceL,+l k 3 z V x E a c p Z . I f a i s a + l - stable then L, k 3zVx Eacp'. Hence L, satisfies Zr-collection. The lemma now follows, as a is clearly a limit ordinal so that La is rud closed.

Proof of 1.18. a is HA-reflecting if and only if a is a+]-stable.

for sets in L,, such that Let a beIIA-reflecting. Let cp be a Z: sentence of&, with constants only

I= cp. We may assume that cp has the form


3x1 ... 3x,,8(xl, ..., x,) where 8(xl, ..., x,) is@. So let a l , ..., a, E

such that 8 l(uo), ..., 8,(v0) of LE, with constants in L,, such that ai = { b E La : L, i= 8,(b)} for i = I , ..., n. Let 8’ be obtained from @(al , ..., a,) by first replacing every occurrence of ai E s by 3y E s Vz (z Ey ++ z E a j ) and then replacing every occurrence of s € a j by e,(s) for i = 1, ..., n. Clearly 0’ is a HA sentence such that L, i= 8’ . As a is HA-reflecting there is a 0 < a such that Lp i= 8’. Now ifai = { b E Lp : Lp k 8,(b)} then a; E as we may assume that x1 actually occurs free in 8(xl, ..., xn) so that the constants of Oi(uo) are also constants of 8’ and hence are constants for sets in Lp. It follows that L, + O(a;, ..., a;) and hence L, k cp.

Theorem 2.4. Let \k(x) be a Ey formula of ZE that defines L inside each admissible class A. Hence Vx\k(x) expresses V = L and the transitive models of oo A Vx\k(x) all have the form Lp for some admissible p. Now let cp be a HA sentence of L, such that L, k cp. Then L, cpl where cpl is cp A uo v VxJl(x) is also an; sentence of L,. Hence k 3x(trans(x) A +or’) where cpy’ is obtained from cpl by restricting all quantifiers to x , and trans(x) is Vy ExVzEy(zEx) .Asa isa+1-s tab le i t fo l lows that La/= 3x(trans(x)Acp(iY)). Hence there is a transitive set in L, that satisfies cpl . But this must have the form Lp for some admissible 0 and Lp k cp. It follows that a reflectscp, so that cy is nh-reflecting.

that result in 6.4. Some of the ideas in [4] will be basic to our proof. For a transitive setA letA+ be the smallest admissible set such thatA € A + . I f S C A we say that S i s H k (E.”,) overA if S= {a € A : A k cp(a)} for some II; (E;) formula cp(x) of XE. Theorem 3.l(a) of [4] states that ifA is a countable transitive set closed under unordered pairs then for S C A, S is H i overA if and only if S is Zy over A’. The proof of this result in [4] may be made to yield the following formulation which gives us the extra information we shall need.

t= 8 ( a l , ..., an). As L,+1 = Def(L,) there are HA formulae

Conversely let a be a+l-stable. Let uo be the H!- sentence given in

We now turn to the proof of 1.19. In fact we shall prove a generalisation of

6.2. Theorem. formula cp+(uo, u l , ..., v n ) of LE having the same constants as cp(ul, ..., u,) such that for every non-empty countable transitive set A and every admissible setBsuch t h a t A E B , i f a l , ..., a n E A t h e n

(i) Ifp(ul, ..., un) is a formula of& then there is a Ly


(ii) If ip(ul, ..., u,) is a Zy formula of L, then there is a FIf formula cp-(ul, ..., u,) having the same constants as cp(ul, ..., u,) such that i f A is an infinite transitive set containing the sets whose constants occur in cp(ul, ..., u,) then f o r a I , ..., a,EA

Proof. We shall require some familiarity with the infinitary languages LB for admissible B. See for example [3].

(i) Let ip(ul, ..., u,) be a n t formula of L,. We may assume that it has the form V X , ... VXmO(ul, ..., u,) where B ( q , ..., u,) is a FIh formula of L, with extra relation symbolsX1, ..., X,. Given a non-empty transitive setA we may define the infinitary sentences q 0 ( A ) and \kl(A) as follows: q o ( A ) is AuEAVy(y € a - VbEu(y=b) ) and \kl(A) is V x VaEA(a=x). Then the models of *o(A) A \kl(A) are all isomorphic to ( A ,E r A a l , ..., a, € A then (1)

e(al, ..., a,)) E B i.e. it is a sentence of L,.

q ( u o , u l , ..., u , + ~ ) and x(uo) , such that i fA, B, a l , ..., a, are as above then if b € B (2) B k \k(A,al, ..., a,, b ) iff b = (\ko(A) A q l ( A ) -+ @(al, ..., a,)), and if b is countable then (3) B completeness theorem for countable infinitary sentences (see Theorem 2.7

Hence if

A + cp(al, ..., a,) iff \ko(A) A \kl(A) -+ @(a,, ..., a,) is logically valid. Note that ifA E B where B is admissible then (*,,(A) A \kl(A) +

Now it is a routine matter, using [3] to find Z: formulae of LE

x(B) iff b is a logically valid sentence of LB. (3) follows from the

of ~ 3 1 ) . \k(uo, ..., u,+,) may be chosen to have the same constants as cp(ul, ..., u,),

while x(uo) may be chosen to have no constants. Now let cp+(uo, ..., u,) be 3u,+l(\Ir(uo, ..., u , + ~ ) A x(u,+l)). The result

follows from (1)-(3) using the fact that (*&I) A * , (A) -+ e ( a l , ..., a,)) is countable i f k is countable.

(ii) Let cp(ul, ..., u , ) be a Zy formula of L,. Let KP be the theory of admissible sets, as formulated in [3]. Then by 3.3 of [4], ifA is a transitive set


and Y3 is an end extension o f A U { A } that is a model of KP then % is an end extension o fA+. Hence if a l , ..., a, € A then A+ k O(al, ..., a,) iff % model of Kp that is an end-extension o f A U { A } .

KP has an elementary subsystem 8' 4 8 that is an A-model of KP of the same cardinality as A , assuming that A is infinite. Every such A-model 23' is isomorphic t o ( A , E ) for some E C A X A . Then there is an f : A + A and a € A such that

O(al , ..., a,) for every A-model % of KP where an A-model of KP is a

Now by the downward Lowenheim-Skolem theorem every A-model % of

(a) ( A , E ) is a model of KP (b) f : ( A ,€ A ) E (a,, E r a,) where aE = {b € A : hEa) ( c ) b, 5 aE for all b € a E . It follows from the above tha tA+ k O(al, ..., a,) iff ( A , E ) k 6 ( f ( a l ) , ...

...,f( a,)) for all E C A X A , f : A + A a n d a € A such that (a) &(b) & (c). I t is now a routine matter to find the required Hi formula obtained by

forinalizing the right-hand side of the above equivalence.

6.3. Definition. An admissible set A is ni-reflecting i f A and a is transitive) for all n], sentences cp of L,.

cp * 3a € A (a cp

The following is a generalisation of 1.19.

6.4. Theorem. The counkzble admissible set A is n ;-reflecting if and only if A < ~ O A+.

Proof. Let A be a countable admissible set that is H:-reflecting. Let cp be a C y sentence of L,, with constants only for sets in A , such that A + k cp. Let T

be Vx 3y(x E y ) . Then, by (6.2)(ii) with IZ = 0, A k= cp- A T . Hence a k cp- A T

for some transitive a € A . I t follows that a is an infinite transitive set such that a /= cp ~. By (6.2) (ii) a+ Hence A < z y A + .

Conversely, let A < q A+ and let cp be a ni sentence of LE such that A /= y . Then by (6.2) (i) with n = 0, A' cp+(A). Hence A+ k cp, where cpl is the Cl) sentence 3x( t rans(x) ~cp'(x)). But cp, only has constants for sets in A , so tha tA + cpl 1.e. A k y'(a) for some transitive set a € A . AsA is countable so is a, so that by (6.2)(i) a + cp. ThusA is ni-reflecting.

cp. But as a+ C A and cp is E 7 it follows that A k cp.

In order t o obtain 1.19 we need:


6.5. Lemma. L, is ni-reflecting iff a is n:-reflecfing.

Proof. Let a be Hi-reflecting. If cp is a rI! sentence of L, such that L, i= cp then Lp k cp for some 0 < a. But now a = Lp is a transitive element ofA such that a k q. Hence L, is Hi-reflecting.

occurring in the proof of 1.18. If cp is a r1: sentence such that L, cp then L, i= cp A u. Hence there is a transitive set a E L, such that a /= q A u. But a = Lp for some 0 < a. So Lp k cp for some p < a. Hence a is nf -reflecting.

every ordinal a.

Conversely, let L, be nf-reflecting. Let u be the fl! sentence uo A Vx\k(x)

Now 1.19 follows from 6.4 and 6.5 when we observe that (La)+ = La+ for

PART 11. INDUCTIVE DEF€NITIONS

57. First order inductive definitions, I

We begin by considering inductive definitions which are either recursive or closely related to recursive inductive definitions. These very simple cases illustrate some of the principles used in characterizing the closure ordinals of more complicated inductive definitions.

7.1. Definition. For any inductive definitions ro, rl let

Let r = [r0, rl]. In constructing the transfinite sequence ( F a : a E On) one repeatedly applies Po until closure under r0 is reached, (i.e. until a h is reached such that r0(rh) C r’); then rl is applied once; then ro is repeatedly applied until closure under Po is reached, etc. rl is applied only when closure under Po is reached. Note that if ro(I”) C r’ then r(r’) = rl(r’). I [Po, rl] I is the least X such that both ro(r’) E I” and r,(r’) 5 FA.

For any recursive relation R and inductive definition r, the truth or falsity ofR(n, r’) is determined by the answers to a finite number of questions about membership in r‘. For limit h, the answers to these questions are the same as the answers to the same questions about membership in I’E for suitable large $ < X. Hence for recursive R and limit h,


7.2. Proposition. (i) IKI:l = w , (ii) I [n:,n",i = w 2 , I [ I I ! , H O , , ~ ~ , I I = w 3 , etc.

Proof. Hence,

(i) Let r En:. Then for some recursive R , n E F(X) --R(n, X).

~1 E r(rw) - ~ ( n , rw) *R(n , rE) for some l < w , by (1)

+ n E r(rt) c rt+I c rw .

Thus r (YW) C r W and hence I rl I w. To show In! I 2 w, let r o ( X ) =

{ O} U {( 1 ,x) : x E X } . Then ro E @. 0 E r " and I 01 = 0; if n E rr and In I = $, then ( 1 , n ) E rg and I ( l , n ) l = f + 1. Thus I Po I 2 w.

(ii) Letn E r O ( X ) - R o ( n , X ) a n d n E r l ( X ) - R l ( n , X ) whereRo and R are recursive. Then as in the proof of (i) we have:

( 3 ) Iflimit A then ro(rh) c r h and hence r(rh) = rl(rh). Now let n E I ' (P2) . We show n E P2. Since limit 02, n f rl(rW2) by

(3) , i.e.R,(n, rW2). Then by (2) there is some [ < w2 such that V6 < w2. f 5 6 * R l ( n , r'). Since the limit ordinals are cofinal in w2 there is some limit 6, $ 56 < w 2 , and henceRl(n, I"). Thus n E I',(r').,BUt since limit 6, r1(r6) = r(r6) and hence n E r(r6) C E I'w . Let r0 be as above and let rl(X) = ( ( 2 , ~ ) : x EX}. Let r = [r0, I?,]. It is easy to show that i r1>w2.

7.3. Remark. If R is L so that 7.2 still holds if Xy replaces n! everywhere.

w3 < ... < I@l. We have

then we still have (1) and the first equivalence in (2)

Note thatn!S [llt,ll!] C [II!,ll:,II:] C. . . I ' Iy .Thusw<w2<


7.4. Theorem(Gandy). I @ I = IZiI = ol.

As I Z! I 2 IXIy I 2 I ny-rnonl 2 w1 we have one half of the theorem. For the other half we will use the next two lemmas. These will also be used for getting upper bounds for other classes of first order inductive definitions.

7.5. Lemma. Let r En:. Then (I'E : r; < h) is unifarmly E on L, for h E Ad. Hence for h E Ad l? is Ey on L,.

Proof. If h E Ad and x S o such that x E L, then r ( x ) is fit on L, as it is defined by a formula with quantifiers restricted to w < A. Hence r ( x ) E L, as r ( x ) C w. So if G ( x , y ) = U { r ( y ' z n a) : z E x ) then G r L, : LAX L,+ L, Moreover G r L, is uniformly Ey on L, for h E Ad. Let F(x) = G ( x , F 1 x ) . Then by 2.2 F r LA : L, + L, and is uniformly Ey on L,. By an easy induction we see that I ' E = F ( [ ) for all t E ON, so that (I'E : t < h ) = F h is uniformly Xy on L, for h E Ad. Hence I" is Ey on L, as x E r CJ ( 3 ~ < X ) X E rs.

7.6. Lemma. Let (I'E : r; < h) be Ey on L, where h E Ad. Let R be recursive. Then

VxR(n,x , FA) * 3 ~ < hVxR(n,x, rE) .

Proof. Suppose VxR(n,x , r') where h E Ad. Then for eachx, V z < xR(n,z , FA). Since h is a limit, by ( 2 ) , V z <xR(n , z , I") for some ,$ < A. Let f ( x ) = & < hVz < x R ( n , z , r"). Then f : w + h is A-recursive. As w < h a = Sup,,, f ( n ) <h. I t remains to show that VxR(n,x , F a ) .

Case 1. Limit a. Suppose that for some z o , 1 R(n , zo , ra). Then there is some t < a such that l R ( n , z O , rS) whenever ( 5 6 <a. Since limit a, there is some x > z o such that t < f ( x ) < a and hence1 R(n , zO, F f ( x ) ) . But then by definition off, V z I x R ( n , z , l? f (x) ) . In particular R(n,zo , F f ( x ) ) which is a contradiction.

Case 2. Not limit a. Then since f is non-decreasing there is some y such that for all x >_y, f ( x ) = a. But by definition off this clearly implies VxR(n,x , Fa).

We can now complete the proof of Theorem 7.4. We must show:

340- W. RICHTER and P. ACZEL

Proof. Let r be E!. Then

iz E r(X) - 3yVxR(n ,y ,x ,X)

for some recursive R . Then

n E r(rwl) - v x ~ ( n , y , x , rwl) for some y <a,

=. V X R ( ~ , ~ , X , r5) for some < ol, y < w,

- n ~ r ( r c ) ~ r w l .

Hence r ( r w l ) C r W 1 so that IFIS ol. A different proof appeared in [2]. The present proof is due to Grilliot.

As in the definitions of [@, @], [@, @, @] etc. we may define [C,, C,] , [C,, C,, C z ] etc. for any classes C,, e,, ... of i.d.'s.

Proof. The first inequalities in (i) and (ii) are trivial, so we turn to the last inequalities.

(i) Let r = [ I-,, r,] where ro E E; and rl € E:+l. Then r' E nh and hence

(4) first that n is even. Then for some recursive R

: E < h) is Xy on L, for h E Ad, so that by the proof of Lemma 7.7 If h E Ad then r0(rh) C rh and hence r(rh) = rl(rh). Suppose

a E Q ( X ) IJ 3 x , v x , ... 3x,R(a,x, , ... ) x , , X ) .

Hence by (2) and (4), if A E Ad then


for some X z + 2 formula Q(u) of E, that is independent of h E Ad. Now let K be the least element ofM,+,(Ad). Suppose (I E r (rK) . Then as

K E Ad it follows from (5) that L, there is a h < K such that h E Ad and L, P Q(u)- Hence by (5) a E r(rh) E rK. Thus r(rK) C r K and hence lrl IK.

Q(u). As K is Z:+2-reflecting on Ad

If n is odd then for some recursive R

(I E r,(x) - 3x,,Vxl ... VX,R(U,X,,, ..., x,, X )

Hence using (2) and (4) again, if h E Ad then

for some X : + 2 formula Q(u) of .& that is independent of h E Ad. The rest of the proof is as before.

where Po E Z!, rl E X:+l and r2 E X k + l . The proof of (i) shows that

Then as in (5) if h EM,+l(Ad)

for some x i + 2 formula Q(u) of L, that is independent of h EM,+,(Ad). The rest of the proof is as in (i).

(ii) This follows the same pattern as the proof of (i). Let r = [ro, rl, r2]

(4') If h EM,+l(Ad) then [ro, rl](YX) C rx and hence r(rh) = r2(rh).

(5 ' ) (I E r(r9 - L, k= e ( a )

In the next section we will prove results which will enable us to reverse the inequalities in this lemma.

!j 8. Closed classes of inductive definitions

In this section we formulate the notion of a closed class C. The results in this section will enable us to give characterisations of IC I and Ind (C) for many of these classes.

8.1. Definition. f : A 5, r if

342, W. RICHTER and P. ACZEL

(a) f is a recursive function and { f (e )} is total for all e ; (b) if {el : X I, Y then { f ( e ) } : -L ( X ) I, r ( Y ) .

A 5, r iff : A 5, for some f . A II r is defined similarly.

8.2. Theorem. If A 5, r then A m 5, I'" and I A I _< I PI. Similarly, with 5, replaced by I

This is an immediate consequence of the following:

8.3. Lemma. If A 5, r there is a recursive function g such that for all a , g : A ~ S , ra.

Proof. Let f : A 5, r. By the recursion theorem there is an e such that {e} = { f ( e ) } = g, say.g is total since { f ( e ) } is. We show by induction on Q

thatg : ha I, P. Suppose

{e} = g : A @ 5, I'p and hence g = { f (e )} : A(A p, 5, r(I'p)

for all f l <a. Then,

8.4. Definition. r is C-complete if r E C and e = { A : A 5, r}.

8.5. Theorem. I f r is C-complete then I e I = I I ' l and Ind (C) = {X : X 5, rm}.

Proof. Let r be C-complete. As E C, IC I > I rl and Ind(C) 2 { X : X 5, rm}. I f A E (2 then A <, r and hence by 8.2 I A I 5 I rl and Am 5, r". Hence I C I 5lri and Ind (C) E {X : X 5, rm}.

8.6. Theorem. There is a l l~+l-complete operator. Similarly for X z + l

Proof. We shall need the following folklore result, which is well-known when


n = 0 or n = 1, but is equally true for larger n.

8.7. Proposition. There is a universal IT&+l operator. Similarly for L&+l.

By this we mean a n & + l operator

We will show that r islI&+l-complete. Let Al(X) =

such that every n&+, operator A has the form A(X) = r,(X) = {XI ( a , x ) E r ( X ) } for some a E W .

{ ( e , x ) : x E A({e}-'X)}. When n > 0, A 1 is easily seen to hen",,, and hence has the form I?, for some u E W . Now let f be a recursive function such that { f ( e ) } ( x ) = (u , (e ,xD. Then f : A 2, I'.

When n = 0 we must be more careful as A1 may not be rIi+l . We will define a rI:+l operator A 2 such that if { e } is total then ( e , x ) E A1(X;

( e , x ) E A2(X). Thenf : A 5, r if we let A 2 = r,. Let rp(X,x) be formula defining r. By separating out positive and negative occurrences of X in q(X ,x> we may write the formula as e ( X , o - X , x ) where O(X, Y , x ) contains only positive occurrences of X and Y . Then

Now if m is odd let

A2(X) = { ( e , x ) : e({e)-'X, {e} - ' (W-X) ,x) )

and if m is even let

0 Then in each case A 2 isn,,,.

8.8. Definition. is closed if (a) There is a C-complete operator;

(c) Every recursive operator is in C! . (b) r1,r2e e * r l u r 2 , r l n r 2 ~ e ;

The following result is now trivial.

8.9. Theorem. n&+l and X & + l are closed.

,344 W. RICHTER and P. ACZEL

In order to obtain characterisations of I C I and Ind(C) for closed classes we will need a method for constructing notation systemsW = ( M , 11) which is more general than that mentioned in the introduction. We shall first give an example which bears some resemblance to Kleene’s systems of notations for the constructive ordinals. We define a transfinite sequence of sets (M6 : [ E ON). In the definition la1 =&(a hx[b](x,X) is the b’th function primitive recursive in X in a recursive enumeration, uniform in X , of the functions primitive recursive in X .

Mo = 0

Ma+l = M a u { 0) u {C 1 ,a, b ) : a E M , & Vx[b] ( X d y U l ) EM,}

MA = US<AME for limit h

= EON^^.

Note that the definition ofM has the appearance of a set inductively defined by an i.d. But the situation is complicated by the fact that the definition ofM,+, depends not only on the previously definedM,, but also on (Mi,, : a EM,). Given any sequence Mt : [ E O N ) we use the following notation.

M = U { M E : [ E O N }

l ~ l = p [ ( a E M ~ + ~ ) for a € M

IMI = Sup{lxl : x E M }

M,* = {(x,y) : x , y EM, & 1x1 I ly I}

M * = u{M,* : (YEON}.

for (YE ON

I f X c o l e t 9 ( X ) = { x : ( x , x ) E X } a n d X < , = { y : ( y , x ) E X & ( x , y ) $ X } . Then clearly Ma = 9(M,*) andMIUI =(Mi),, fora E M a .

Hence the definition ofM,+l above may be written

M,+1 = M a u @(M,*)

where


Notice that 0 is Hy. We will see below that M * is inductively defined by a l l y i.d. We now generalize the above procedure to an arbitrary 0.

8.10. Definition. For any i.d. 0,%@ = (M', 11) is defined by:

Mo = b

Ma+l =Ma U 0(M,*)

M,=U{M,:a<h} iflimith

M @ = u{M&: EON}.

AlthoughM' does not have an obvious inductive definition we show that M* does.

8.1 1. Definition. For any i.d. 0, 0, is defined by 0, (X) = { ( x , y ) : x E T(X) & y E 0 ( X ) \ qx)} u { ( x , y ) : x,y E 0 ( X ) \ S(X)}.

8.12. Remark. For closed C? note that 0 E C * 0, E C.

8.13. Lemma. For all a 0: =M,* and hence 10,1= lM'l and @:=M*.

-

- - -

Proof. Note thatM,*+l = M i U 0, ( M i ) . The result now follows by induction on a.

-

8.1 4. Equivalence Theorem. If C is closed and 0 is C -complete then I C I = I M @ I and Ind ( C ) = { X C w : X I, M'}.

Proof. If C is closed and 0 E C it follows from 8.12 and 8.13 that IC121M'I andInd(C)>{XCw:X<,M@}asO,EC - and x E M @ -(X,X)€@:.

-

For the converse it is sufficient to prove the following.

8.15. Lemma. I f I? 5, 0 then there is a recursive function f such that for all a f : r" 5, Ma; hence r" <, Me and I r ls IM' I.


Proof. Letfo be a recursive function such that { f o ( e ) } ( x ) = ( {e}(x) , {e} (x) ) . S o {e}--' 7= { f o ( e ) } - l . Let f l : r Srn 0. Then for total {a}, r{a}-' =

{f, (a)}--' 0. Then

Now choose e such that {e} = { . f l fo (e)} and l e t f = {e} . Then rf- '7= f--'0. Hence F f - ' ( M , ) = r f - ' S ( M * ) = f - 'O(M,*) ,

f-'M,+, . It follows by induction on Q that F a = f - lM,; i.e. f : I"y 5, M,,

so thatf-lM, U r ( f - l M 0 ) = f - ' M , Uf-'@(M,*) = f- 7 ( M , U @ ( M z ) ) =

In 57 we have seen how to prove, for certain C , that 1 e I I K where K is the least reflecting ordinal of a certain kind. In order to. show that I C I = K we will choose a 'good' notation system% = (M,II) such that IMI I I CI and show thathf has the required reflection property. A s M is a set of notations for the ordinals < / M I , statements about ordinals< / M I can be rewritten as statements about M . The reflection property for [MI will then follow from closure properties of M . The Coding Lemma below gives a formulation of this rewriting process for X: statements.

8.16. Definition. A notation system 311 = ( M , 1 1 ) isgood i f% = %' where O ( X ) = Z ( X ) u Q ( X ) and E(X) = {0} u { ( ] , a , b ) : a E 7 ( X j &

V x [ b ] ( x , X , , ) E 7(X)} U { ( 2 , a , b ) : a E 7 ( X ) orb E 7(X)}, and @ ( X ) is alwaysdisjoint f r o r n ( O } ~ { ( I , a , b ) : a , b E w } U { ( 2 , a , b ) : a , b E o } .

I f LX is good then an ordinal X 5 /MI is W-good If Z ( M l ) ClzJ,. Thus [MI i sm-good, but usually there will be%-good ordinals< IMI.

8.17. Coding Lemma. Let 311 = (M, 1 1 ) be a good notation system and let T , = {(x,(u) : a: E On & x EM,}. Then

(i) b'very W-good ordinal is in Ad ( T , 1. (ii) For evely Ly- forinula q ( u l , ..., u n ) of .Cp(Tm) there is aprimitive

recursive function h such that for every %-good ordinal X

a , , ..., a n E M h & X t=p( la l l , ..., IanI)-h(al , ..., a n ) E M A

(iii) I f X is %-good then for X E w X is h-r.e. in Tm 1 X - X <,, M A .


This lemma will be proved in the appendix.

8.18. Corollary. L e t W , T,, be as in the lemma, and let f (a) = pn [n E M & In 1 = a] for a < IMI. Then for each W-good ordinal X f I’ h : X + o is a X-recursive in T , I’ X injection.

Proof. f ( a ) = pn[(n , a+l) E TT & (n,a) 4 T,]. Hence f 1 1 is A-recursive in T,, 1 X form-good A. I t is clearly an injection.

8.19 Theorem. Let C 2 @ be closed and let r be C-complete. Let h = I C I a n d A r = { ( n , a ) : a < h & n E r a } . T h e n

(i) X is admissible relative to A , ; (ii) X is projectible to w relative to A , ;

(iii) Ind ( C ) = { X C w : X is A-r. e. in A , }. Proof. Let 0 ( X ) = X ( X ) U @(X) where @ ( X ) = ( (3 ,a ) : u E F(X)}. Then as E En: E C and e is closed it follows that 0 E C. Also Xx(3,x) : r ( X ) < ,0 (X) for all X and hence f ’ : F 5,0 where f : r <, F and f ’ is a recursive function such that {f’(e)}(x) N_ ( 3 , {f(e)}(x)). Hence 0 is C- complete. Now 7?2 = 311’ is a good notation system and by the Equivalence Theorem I C I = IMI and Ind (C) = { X E o : X 5, M}. Hence by the Coding Lemma and its corollary the theorem follows as long as we replace A , by T, I\ h. I t only remains to show that A and T , 1 h are X-recursive in each other. But by 8.15 there is a recursive function h such that h : F a 5, M& for all a. Hence (n , a ) E A , - (h (n), a) E T,, 1 , so that A , is 1-recursive in T , I A. For the converse note that as F is C-complete and 0, E C it follows that g : OF 5, r for some g. Hence g : M,* Srn r” for all a, b y 8.3 and 8.13. s o

showing that T , 1 X is A-recursive in A, .

3 9. First order inductive definitions, I I

We are now ready to characterise the ordinals of first order inductive definitions.


9.1. Theorem. (i) I Hy I is the least element of Ad; (ii) I [@JI,"] I is the least element ofM,,+l(Ad);

(iii) I [ny,Il~,rl~]l is the least element ofM,,+l(M,+l(Ad)), etc.

9.2. Remark. By 7.8 this theorem is also true when each n: is replaced by 0

z k + l .

Before proving the theorem we derive some immediate consequences.

9.3. Corollary. in,"i = J~;+,I = n'n+l 0

hoof. In: l=Icyl=w=n(: by 1.9(i) ,7.2and7.3.1nyl=IZ!l=wl=n2 0 by

= o

1.8 and 7.4. Forn > 1 [ny,n:] =H; and by 1.9(i~)M,+~(Ad) =M,+I(ON). Hence InfI = lZ,,+ll = 7 ~ : + ~ by 9.1 and 9.2. By 1.9fiii) ?r: = 0 for alln.

9.4. Corollary. I [ny, n 00, Il: ] 1 is the least recursively inaccessible limit of recursively inaccessible ordinals, etc.

the least recursively hyper-Mahlo ordinal, etc.

(i) J [ny, Fl! ] I is the least recursively inaccessible ordinal;

(ii) 1 [ny, n y ] 1 is the least recursively Mahlo ordinal; I I l l y , f l y , ny] I is

We now turn to the proof of the theorem. By 7.8 it only remains to prove:

9.5. Lemma. (i) Q E Ad for some Q I In:/; (ii) ~ E M , , + ~ ( A d ) f o r s o m e ~ < l [ I ' I ~ , K I ~ ] l ; (iii) ~ € 1 Z ~ ~ + , ( M ~ + ~ ( A d ) ) f o r s o m e o l _ wl. To give a direct proof let 311 = 311' where 0 = Z, and let a = IMI. T h e n W is a good notation system so that (YE Ad, by the Coding Lemma. As 0 E I I y , so is 0, so that Q = I@,[ 5 IKIyl.

(ii) First assume that n isodd. Let% = W' where a E 0 ( X ) - a E Z(X) v [z(x> c T(X) &a E Q;(x>~, Q ; ( X ) = ~ ( 3 , e ) : e E +,(T(X))) andaE@, , (X)~(Vx ,EX)(3x2EX) ,..(Vx,, EX)[a] (x l , ..., x , , )EX. Here Axl , ..., x,, [ a ] ( x l , ..., x,) is the a'th n-ary primitive recursive function in a recursive enumeration of them.

An easy argument shows that 0, = [&,(@A)<], so that 0, E [@,IT,"]

(i) This folIows from Theorem 7.4, whose proof assumed the result

- - -


as (a;)< Ell:. 7?2 is a good notation system so that by the Coding Lemma a = IMI E Ad. Let cp be all:,, sentence of L, such that a k cp. We may assume that cp has the form

where \k is a ET- formula of L, and cl, ..., ck E M . By the Coding Lemma there is a primitive recursive function h such that for all%-good ordinals h

Now choose e E w such that

Then it follows that form-good h

Hence as LY k cp and a i s m -good

Now if h = I(3, e ) I then h < a , X ism-good, and hence admissible, and (3,e)E@A(Mi)so that h i=cp.ThusaEM,+I(Ad) a n d @ = lO,l<_l - [lly,ll:]I, as required.

When n > 0 is even the proof is as above except that

and the ll:+, sentence cp now has the form

V X 1 3 X * ... 3X, l * (X ] )...) X,,ICll ,..., I C k l )

with 9 , c l , ..., ck as before.


In case n = 0 let @.,(X) = X. Then as before a = IMI 5 I [fly, n,"] I and a € Ad. In order to show that (Y EM,(Ad) we must show that a is a limit of admissibles. So let f l < a. Then f l = la1 for some a E M . Then ( 3 , a ) E M as Z ( M * ) C M . Let A = I (3 ,a) l <a. Then h ism-good and hence admissible, and f l = lal<h. SoaEMI(Ad) .

(iii) Let % =W@ where

a E O ( X ) - a E E ( X ) v [ Z ( X > S 9 ( X ) & a €cpA,(X)]

v [ Z ( X ) u cpL(X) c 9(X) & a E @AI(X)]

where @L1(X)= ( ( 5 , e ) : e E Q n ( 7 ( X ) ) } and a,, (ii) 0, = [Z<,(@&,(cpA1)5] E [ny, l l i , lT ,"] so tha t&= IMI =

are as in (ii). Then as in

10,17 - I [ncn;,n;li. As in the proof of (ii) we may show that a EM,+,(Ad). Moreover we may

show that l(5,e)l EM,+l(Ad) whenever (5 ,e) E M . Hence using once more the argument in the proof of($ we can show that aEMn+,(Mm+l(Ad)).

The next result characterises Ind(C) for certain classes of first order inductive definitions.

9.6.Theorem.Ife isany ufthecZassesn(i', [lly,n,"], [ I l ~ , f l ~ , f l , " ] , e t c . and h = IC I then there isa T'E C such that h = lrl and Ind(C) =

{X C w : X I, r-} = { X C w : X i s A-r.e.}.

9.7. Remark. This result also applies to the classes n,"+, and to the classes obtained from the ones considered by replacing each KI: by ZE+l.

Proof. The proof has the same form in each case. We illustrate with C = [@, n,"]. In the proof of 9.5 (ii) a good notation system W = W@ is defined such that !MI 5 h and \MI EMn+l(Ad), But h is the first element of M n + , ( A d ) b y 9 . 1 . H e n c e I M I = X . S o i f r = 0 < E C t h e n h = I r I . By thecod- i n g k m m {X 5 w : x is A-r.e.} S {X c w : X%, rm}, asM I, M * = r". . Hence { X C o : X is A-r.e.} 5 Ind (C) as r E C. A E C implies that A" = Ah is A-r.e. by 7.5. Hence Ind (C) 5 {X 5 w : X is h-r.e.}, proving the theorem.

For completeness we conclude this section with the following easily proved result.


9.8, Theorem. (i) Ind(rI!)={XEo: Xisr.e.1. (ii) Ind(X7) = { X E w : Xis a “recursive”union of arithmetical sets}.

5 10. Higher order inductive definitions, I

In the previous section it was shown that the closure ordinals of certain classes of first-order inductive definitions are reflecting ordinals of a pre- scribed form. In this section we obtain corresponding results for higher type inductive definitions. The techniques are similar to those of $87 and 9.

In Lemma 7.5 it was shown that for r E ~ A , (rt : t; < A) is uniformly Zy on L, for h E Ad. For other r this need not be the case and this makes the characterisation of In; 1 for m , n > 0 somewhat more difficult.

In the following lemma the class A is some relation on ordinals.

10.1. Lemma. Suppose m, n > Oand r En;. Let X be a class of limit ordinals greater than 0, and K E X be nr(A)-rejlecting on X . if (I“ : < X) is uniformly Z: A on L,[A] for X E X , then I rls K . Similarly with n F ( A ) and nr replaced by Z T ( A ) and Z T, respectively.

Proof. Let I’ E nr. Then for some JIr formula q(y, Y ) of L, , for all Y C o

n € r ( Y ) w w l=q(n,Y) .

Let $o(z) be the formulaz E w , and $k+l(Zk+l) be V Y k [ Z k + l ( Y k ) IC/k(Yk)l Then each $k+l is an : formula (in the constant a). Let q*(n,X) be obtained from cp by restricting each quantifier of type k to G k . Then q* E n; a n d f o r h > w a n d Y L u ,

Let q*(y ,Y ) be Q Z . $ ( Z , y , Y ) where Q Z is a sequence of quantified variables of appropriate type for a prenex n: formula and $ is X:. Then for AEX, (1) s E r ( r h ) ~ L h [ A 1 k QZ.$(Z ,s , rx )

La [ A ] I= Q z V$ E On 3 6 E On [ t; _< 6 A $ ( Z , s, r6)] - L a @ ] I= QZV$ E On 36 E O n 3 y [ ( < 6 ~ R ( h , y ) ~ $ ( Z , s , y ) j - LAMI I= PdS) >


where R is a X A formula of L,(A) (independent of h EX) such that for 6 < A, Y EL,[AI

and cpI(s) is the sentence appearing immediately above it in ( I ) . Clearly cpl is a nr formula of &(A). Now let K E X be nr(A)-reflecting and s E r ( rK) . We show s E r K . L, [ A ] k q l ( s ) by (1). Since cpl is a nr formula of Lc,(A) there is some h E X n K such that L,[A] k cpl(s). Hence by (l), s E r(rh) 5 P.

10.2. Definition. For a given i.d. r let A , ( x , y ) -y E On & x E ry . Let n,"(r) be the least n?(A,)-reflecting ordinal; similarly for u,"(r).

10.3. Theorem.Letm, n > Oand r be complete 11;. Then Inrl= n,"(r). Similarly 1 z; r I = OF( r) i f r is complete z: ," .

Proof. We prove this for rIr. The proof for Er is similar. Let r be complete 11;. For A € Ad@,) a n d t < A , I+= {x E o : A , ( x , ~ ) } E L,[A,], by IIg-separation. And since fo ry E L, [A,],

i t follows that (r" : t < X) is uniformly Ey on L,[A,] for h E Ad (A r). LettingX= Ad(A,) and K = nF(r) in Lemma 1, we see that IrIrl I n,"(r).

To show Inrl2nr(r) it suffices to show Inrl isn;(r)-reflecting. Let O b e a s i n theproof0f8.19andlet31i'=~M,ll)=~~~.By8.14IMI=I~~l. If T, and h are as in the proof of 8.19 then

Since h is recursive, the predicate h ( x ) = y is Ey on L, and hence X! on L,[T,] for X > w. Hence A , is X: on LJT,] for h > w. So if cp is a 11," sentence of &(A,) there is a n ; sentence cp* ofX,(T,,) such that for h > w Lh[Ar] 1 cp- L,[Tlm] l= cp*. Hence it suffices to show that IMI isrI,"(T,). reflecting. This will follow from the next lemma. Let ~ ( u , , ..., up) be an;- formula of Lp(T,) with the indicated free variables.


10.4. Lemma. There is a IIF i.d. \k such that for%-good hand cl, ..., cQ E M X

Roof. This will be in five parts. Assume throughout that A ranges over%- good ordinals.

(1) If R is primitive recursive in T , then by the Coding Lemma there is a primitive recursive function h, , independent of A, such that for a l , ..., a, E M

In particular

( 2 ) Let 9 ( Y , X ) if and only if X E w and Y E w X w is the graph of a

(i) x , y E Q & h=(x, y ) E X * x = y , and ( i i )yEX* 3x E Q h , ( x , y ) E X .

It should be clear that Q is arithmetical. Moreover Q(Y,M, ) holds if and only if Y is the graph of a bijection f : w is a bijection; w S A. Hence 3 Y Q ( Y , M X ) .

formula of Lp

bijection f : w Q C X such that

Q E MA such that Y* = Ax If(x) I

(3) I fR is primitive recursive in TclK let B R ( Y , X , x l , ..., x k ) be the Zy-

Then if S ( Y , M X ) and ~ 1 , ..., ak E o

(4) Let p*(Y ,X ,u l , ..., ua) be obtained from q ( u l ,..., u a ) by replacing every atomic formula R ( x l , ..., x k ) by 6,( Y , x , x l , ..., x k ) . Then p* is a l37- formula of Lp, and if S ( Y , M h ) and a l , ..., up E w then


( 5 ) q ( X ) may KIOW be defined to be the set of (c l , ..., c,) such that c , , ..., < f: 9tX') arid tor all Y such that S( Y , 9 ( X ) ) and all a ,, ..., ap,

b , , .,b,, 11 A , 5 f ~ l ( Y ( a f , b l ) ~ ~ k ( h ~ , ~ ~ ) E S ( X ) ) t h e n

Then 9 is a Kl? i.d. that satisfies the lemma.

We L a n now ccmplete the proof of the theorem. Let /MI k cp where cp is a fl; sentence of fr,( T m). We must find A < /MI such that h cp. cp must have the f o r m p( (a l I, .., l a y ] ) foi s o m e I l r - formula of Lp(Tm) anda l , ..., au EM. Lxet 111 be the 1.d. given by 1,ernmu 10 4. Then as 9 is11," and I' is complete 11: there 19 a g q ( X ) :<,1, I1(X) for all X . Let a = g ( ( a l , ..., a&). Then for ?X-giiOd x

I n gciicral wl: c.inirot expect (ha1 II1,Tl = n,"' or In,"'[ = u r for m,n > 0. I i i ort lcr t o use 1,eiiima 10.1 to show that Inil I n : , for example, we would w,mt to h o w that for I'EH!, X E L gu,i~aritee that I" = I'(@) belongs to 1, 1 or even to L. hi the case of 11; and Z 1 , Iiowever, we can do better by making use of a result due to Barwise, Lat idy m d Moschowkis, formulated here in Theorem 6.2.

f \ P ( o ) * r (X) E L I . But there is no "3 n3

n3

1 ct I n be the class of recursively inaccessible ordinals.


10.6. Lemma. If' I' is n or Z i then (Y6 : E < h ) is uniformly E: on I,, for h E In.

Proof. It is sufficient to show that if h E In then (i) x E L, =$ r ( x n w ) E L,, and (ii) { ( x , y ) E L, x L, : r(x nu) = y ) is E: on L, uniformly for A E In, as

Let r be n:. Then by 6.2(i) there is a Ef- formula cp+ of %, such that if we may then carry through the proof of Lemma 7.5.

A , B are admissible sets and x E A E B then

But if h E In and x E L, then x E Lp for some admissible p < h and dso p+ <x. s o

12 E r ( x n W ) - L,,+ t= P + ( L ~ , ~ , X ) .

Hence r ( x n w ) is Zy on Iir+ so that r(x n w ) E Lp++l C L,, proving (i). To prove (ii) let uo be the n2- sentence of 2.4. Then if x , y E L, and

h E In, r(x nu) = y if and only if there are transitive sets A , B E L, such that [A,Bare transit ive&xEAEB&A + u o & B ~ u o & y ~ w & V i z E o (B i= cp+(A,n,x) - n Ey)]. The expression [...I can be defined by a X:- formula \E(A,B,x,y) of l,, independent of h, so that r ( x n u ) = y - L, l= 3a 3b\E(a,b,x,y), proving (ii).

If r is Zt then the proof is as above except that p+ is now n r .

Proof. In{l>n; and IE! l2a i followsfrom 10.5.Note thatbyTheorem 1.9 n:, a: E In, T; is nf-reflecting on In and a: is Xi-reflecting on In. Hence by Lemma 10.6 we may use Lemma 10.1 with n = rn = 1, X = In and A = 0 to get In:I< ni and I E:l I u i .


8 11. Higher order inductive definitions, II

Recall from the introduction that a(%?) = Sup{l<l : < €%well-orders a subset of w } , where ] 0. We will prove Theorems E, F and G of the introduction.

Proof. ofA C w. I t suffices to find r E A; such that 1 rl = I<[. Let

(i) We first show that [A; 12 a(A",. Let < be a A; well-ordering

As n > 0 r is clearly A;. By induction on a, (X I< is the order type o f x in the well-ordering<. Hence l r l = [<I.

L e t r € A k and

= {x € A : Ix I< < a} where

We next show that w(A;) 2 / A ; I. The technique here is implicit in [ 141.

Then Q is a univalent system of notations for the ordinals less than IF/. Let

The < is a well-ordering and

@ ( S , X , Y) - S is a (strict) well-ordering of Y & Vk @ Y(Xk = $9) &

= 1 rl. It suffices to show that < is A;. For X C w X w let X , = { y : y X k } . I f S 2 w X w and Y C w let

vk E Y ( X , = u {r(xl) : / S IC) ) t~ v k , i ~ Y ( ~ # Z = . X , # X J &

w,< w X,) c u,< X , .


Then as m,n > 0 and m + n > 2 @ is A;. Clearly @(S,X, Y) if and only if S is a well-ordering of Y of order type Irl such that X , = rIk's for k E Y , and x k = fork 4 Y . Hence if

so that Q is A;. As

it follows that < is A ;. be complete Z; such that a 4 I'T. Let r , (X) = {a} for all

X and let = [rl,r2]. Then E A;+l and I rl = lr,l t 1 = I X;l f 1. Hence 1 A$+lI > I X k I ; 1 A",,1( > ln&l is proved in the same way.

X; sentence p of lp logically equivalent to

(ii) (a) Let

(b) is just 10.5. For (c) let <be all; well-ordering. Then there is a

(1) 3 f Vk VZ [k < !2=' f ( k ) < f(Z)] .

Then A k cp A 2 1-4 and hence 1<1 is not X; -reflecting The proof that r z 2 w(E;) is similar to the above, interchangingll; and X; throughout and replacing (1) by

(d) is trivial.

Remark. We do not know of any cases where equality holds in (c). Note that


T; 5 Ink I < ?rk+l. Thus ?rk and Ink I are not too far apart. Similarly for uk and IXLI.

1 1.2. Theorem. I A i I is not admissible.

Proof. We shall use the following fact extracted from 57.10 of [19].

Proposition. There is a I l i relationJC o X w X ?(o) such that I' is A: if and only if r ( X ) = F,(X) = iy : J ( n , y , X ) ) for some n < w.

11.3. lamma. I f h is recursively inaccessible then Z: on L ~ .

: n < o & 5 < h ) is

Proof.By 10.6,aseachI', i s l l : , ( r i : . $<h) i sZy o n L k f o r a l l n < o . B u t as r, is IIi uniformly in n , and the proof of 10.6 is uniform, (I'i : 5 < A ) may be seen to be Ey on L, uniformly in n giving the lemma.

11.4. Lemma. I A i I is a limit of admissibles.

Proof.Leta<IA;I.Then the re i sa I 'E A: suchthata<lI ' l . Let O ( X ) = S ( X ) U { ( 3 , a ) : a € r ( X ) } , whereZisas in8 .16 .ThenW =me i s a good notation system so that by the Coding Lemma ]MI is admissible. But as I '<,Oitfollowsfrom8.15 t h a t I I ' l ~ l M I . B u t l M I = l O < I - and,asI ' is A:, so is 0 and hence 0,. Now let

-

A(X) = @<(a U fx : O,(X) 2 X & x E a} .

ThenAis A; and lA1=lO<l+ l . H e n c e I M I < I A l 5 l A ~ I , s o t h a t a <_ 1 M 1 < I A 4 1 and 1 MI isadmissible proving the lemma. Note that we have

- -

shownthatl~l<IA~IforallI'E A:.

We can now prove the theorem. Suppose that h = 1 A: I is admissible. Then by the previous lemma it is recursively inaccessible and hence by Lemma 11.3 (I'i : n < o & g < h ) i s X y onL, .Letf(n)=Ir , I forn<w.Thenf : o + h is A-recursive, as


But A = 1 A i l = Sup,<,)r,) = Sup,<,f(n), contradicting the admissibility of A.

We conclude this section by showing that under very general conditions I el # 11 e I . We also obtain a related spectrum result.

11.5. Definition. If C is a class of inductive definitions then the spectrum sp(c) of e is {iri : r a?).

11.6. Definition. A closed class e is V-closed if Zy 5 C? and 1'E C implies I'l E C, where

11.7. Theorem. If e is V-closed then I C I 4 sp (1 e ) and in particular i e i # i i e j .

Note that the last inequality fotlows'because I lCI E s p ( 1 e). If I e I < I1 C I the theorem implies sp (1 C) is not an initial segment of ordinals. m, n > 0. In particular we get

is not V-closed. However, n;,,, ilk, and Zk are V-closed for

11 -8. Corollary. If m, n > 0 then Ink 1 # I Xk 1.

We turn to the proof of the theorem. Let C be V-closed. Let I' be C- complete and A E 1 e. It is sufficient to show that I A I f 1 1 ' 1 as 11'1 = I CI. Let O(X) =


As e is first order closed 0 E C . As r <-,,, 0 it follows that 0 ise -complete and hence by Theorem 8.14

Lemma 1. IF1 = (MI.

Lemma 2. For a I IMI,


Lemma 3. For a < IMI and i < 4 , ( 4 , a ) € M4a+i - a E A&. In particular, A Q I,M4a.

Proof. We use induction on a.

( 4 , ~ ) E M ~ ~ + ~ - 3 v [ v t 3 < 4 a + i & d ( a , v ) ]

-3p<a. 3 j < 4 q a , 4 p ti)

e3 3P<a.aE A A D

- a € A Q .

Since I I'l= IMI it suffices to prove:

Lemma 4. I A I f IMI.

Proof. Suppose I A I = IMI = a. We get a contradiction by showing I A I <a. SinceaEM*(6,a)EM&laI+ 1=1(6,aTI,amustbe alimitordinaland hence a = 401. By definition of a, A Aa E Aff . Hence by Lemma 3,

V x [ x E A ( { a : ( 4 , a ) E M a } ) * ( 4 , x ) € M Q ]

By Lemma 3 ,

V X [ X E A(Ap) *x € A @ ] ,

andhence I A l < p < a .

Appendix. Proof of the Coding Lemma

SA.1. Acceptable ordinal systems.

that if % is a good notation system, then 9R has these closure properties. We begin by discussing certain closure properties on systemsw and show


A . l . Definition. Let 'h?= ( M , 1 1 ) be any ordinal system and B C w. W is B- restricted productive if there is a primitive recursive function p such that for every n E w ,

(i) Vx[{n ) (x ,B)EMl ~p(n)EM$Ip(n)12sup{I(n}(x,B)I + 1 : x E w } ; (ii) p ( n ) E M ~ V x [ { n } ( x , B ) E ~ . p is called a B-restricted productive function for%'.

The closure condition (i) is analogous to the closure of infinite regular cardinals with respect to mappings from smaller ordinals. (ii) is a technical requirement which ensures that there are not extraneous notations inM.

A.2. Definition. m is acceptable if there are recursive functions j , 0 and a primitive recursive function p such that

uniforrnly in a, where J is the complete

%.

if x 4 f ig. We say that 3tI is acceptable in terms of j , p , 0.

(i) j : M 5, M and for a E M , if l j (a) I 5 a then J(MIQI) is recursive in M, i.d. J ( X ) = { x : 3yTX(x , x , y ) } ;

(ii) if a E M then An. p ( a , n ) is an Mla,-restricted productive function for

(iii) a E M v b E M * a &3 b E M & inf { l a I, I b I ) 5 la v b I , where Ix I = I MI

We next show that there are functions j , p , @ such that if CLT is a good notation system and h ism-good then W A is acceptable in terms of j ,p , 8

A.3. Lemma. Let 311= ( M , 1 1 ) be a good notation system. If a E M then J (Mla , ) is recursive in M , for all a 2 la I + 2 , uniformly in a.

Proof. Let Q! >_ ( a ( + 2 and let e be a recursive function such that

Note that 1 $ M . It suffices to show there is a recursive functionf such that for all x, x 4 J(Mlal) f (a,x) E M a . Now

x ~ J ( ~ l u l ) - V t 7 ~ ' ~ ' ( ~ , X , t ) - vt[e(a>X)l(t,Ml,,) = a EMl,l+I - ( 1 , a , e(a,x)) E Ma .


Thusletf(a,x)= ( l ,a ,e(a,x)) .

that for all%-good X, j : Mh 5, M , and for a EM,, [ @ ) I = la1 + 2. Let el be a recursive function such that for all a, t [el(a)]( t ,MIO,) = a . Then for

To show that 9Z! satisfies A.2 (i) it remains to find a recursive function j so

%-good A,

a E M , + Vt[e I (n ) ] ( t ,Mlo l )=a E M ,

~( l ,O ,e l ( a ) )EMh&l( l ,O ,e l ( a ) ) J= lal t 1 .

Also a 4 M A 4 ( 1, 0, el@)) $MA. Thus let j (a) = ( 1 , 0, e (( 1,0, el (a)))).

A.4. Lemma. There are functions j , p , 0 such that i f % i s a good notation system and X is %-good, then 3n, is acceptable in terms of j , p , @ .

Proof. I t remains to find v andp. I t is easy to see that a 0 b = (2,a, b ) has the desired property. To find p 3 let e be a primitive recursive function such that for any n and X E w , the range of ht[e(n)]( t ,X) equals (0) U the range of At . {n}( t ,X) . Then for a EM,,

and if Vx{n}(x ,MIuI) EM, then

Thus it is easy to see that we can choosep(a,n) = ( l ,a ,e (n) ) .

In view of Lemma A.4, to prove the Coding Lemma it suffices to prove the following:

A S . Theorem. Let W = ( M , 11) be acceptable in terms of j , p , 8 . (9 IMI E Ad(Tm );

(fi) For evev Xi- formula cp(ul, ..., un) of LP(Tm), there is a primitive recursive function h such that


Furthermore, h is completely determined by the functions j , p , 8, a member uo E M , and the formula q.

(iii) Let F be an ordinal function which is I M 1-partial recursive in T, . Then there is a recursive function k such that for al , ..., a, E M , if F(lall, ..., la,l) isdefined then k(al , ..., a,)EMandF(lall ,..., la,/)< Ik(al, ..., a,)l.

(iv) X E w is I MI -r.e. in T, i f f X 5 ,,, M.

The remainder of the appendix is devoted to the proof of Theorem A S . Suppose % = ( M , 1 1 ) is acceptable in terms of j , p , 8. Let uo E M and luol = 0. In Lemmas A.6-16 below the reader should observe that the functions described are either independent of the particular acceptable system or are completely determined by j , p , @and uo. (In Lemma A.8 the choice of e is independent of %; in Lemma A.9 an index of h can be found as the value of a recursive function of the indices of the f i , gi which is independent of%.)

A.6. Lemma. There is a recursive function +M such that: (i) a E M & b E M 3 a tM b E M & la tM b I > max { l a

(ii) a tM b E M * a E M & b E M . b I};

Proof. Let e be a recursive function such that

andle ta t M b = p ( u 0 , e ( a , b ) ) .

3A.2. ‘MI-recursion. We next define a class of partial number-theoretic functions based on%.

These functions behave very much like the functions partial recursive in the type 2 functional E of Kleene [9], where for f E w w ,

0 if 3 r [ f ( t ) = O ] ,

1 otherwise.


Using these functions we are able to carry out computations involving 5% which are needed to show that IMI is admissible. The following definition by schemata of the predicate {z}%(x) =y parallels the corresponding definition by Kleene [9] of the partial recursive functionals of finite types. As described in [9] this definition by schemata may be viewed as a transfinite inductive definition. The essential difference here is that there are an infinite number of starting functions in the case SO. Thus the characteristic function of each Ma for a < IMI is given outright.

..., y m , respectively.

In the following, x a n d y are abbreviations for xl, ..., x, andy

0 if x EMla,, for each a E M ; i 1 otherwise,

S0.a ( (0 , ~ , a ) } ~ ( x ) =

SI. { ( l ,n )}%(x)=xl + 1 ;

~ 2 . { ( 2 , n , q ) j C " [ ( x ) = q ;

~ 3 .

S4. { ~ 4 , n , a , b ) } 3 " ( ~ ) ~ { a } " ( { b } 3 " ( x ) , x ) ;

{(3,n)}* (XI = x1 ;

{(5,n+l,a,b)}m(0,x)- {u}*(x) {( 5, n + 1, a, b )Im ( y + I , x) = {bjM (y, { ( 5 , n +I , a, b)Iw ( y , x), X) ;

s5. ( S6. {(6,n,k,a)}- (x) = {a}* (xl) ;

where x1 is obtained from x by movingxk+l to the front.

s8. {(g,n,a)jx (x) = E(ht . {a}w(t,x)) ;

where both sides are undefined if for the given x, At . {a>im ( t , ~ ) is not total.

S9. {(9,n+rn+l,rn)}*(z ,~,y)= { Z } ~ ( X ) ;

Since we are defining only partial number-theoretic functions there is no S7 clause. {z}" is called them-partial recursive function with index z. {zp is%-recursive if it is everywhere defined. Note that if z is the index of an %-partial recursive function, then ( z ) ~ is the number of variables of the function. It is easy to prove the standard theorems of recursive function


theory with the exception of the normal form theorem. In particular the Kleene S--m-n theorem, the Kleene second recursion theorem and the theorem on definition by cases are proved exactly as in 191. Thus we have:

A.7. Lemma. For each m 2 1 : there is a primitive recursive function S m ( z , y I , ...,y,) such that if f ( y l , ..., y , , ~ ) isan %-partial recursive function with index z then for each fixed y ,, .. . , y , , Sm(z, y , , ..., y , ) is an index of f ( y l ,..., y m , x ) a s a func t iono fx .

A.8. Lemma (Second recursion theorem). Given any %-partial recursive fiinction f ( z , x ) an integer e can be foutid such that { e } m ( x ) = f ( e , x ) .

A.9. Lemma. I f f n , J ~ , g a r e m -partial recursive then the function

is Cm -partial recursive.

A.10. Lemma. I f f is %-partial recursive, then so is p y [ f ( x , y ) = 01.

A.1 I . Remark. I t is clear from scheme SO and the fact that the %-partial recursive functions are closed under composition, primitive recursion and the p-scheme, that each function partial recursive in some M,, where (Y < [ M I , is

[ a ] = { Z ) ~ ( ( U ~ ) , ..., ( a ) ( z ) l A l ) and let D = { ( z , a ) : { zym [a] is defined}. The inductive definition described by schemata S&S9 associates with each ( z , a ) E D an ordinal as follows. I(z,a)l'm = 0 if ( z ) ~ is 0, 1, 2, or 3, that is if {z}w((a)o, ..., (a)tz), -,) is defined by one of SO-S3. In case S4, letting

-partial recursive . I t is convenient to deal with functions ofjust one variable. Let { z } 'M

a = (a)() > -.'I (a)(,)l L 1 >

{z}" [a] = { z j m ( a ) = {by'({cfm(a),a)= { h f M i({cYrn ~ a l , a ) ]

for some b, c. Thus let


The cases S5, S6, and S9 are similar. For example in case S9,

Thuslet I(u,(z,a,y))lq = I(z,a)l'm + 1 . In case S8 we have

{z)" [a] = {z)* (a) = E(ht. {b}M (t,a)) = E(ht. {b)* [(t ,a)])

for some b. In this case we define

The ordinal function 1 1% makes possible proofs by induction. The following lemma and corollary are a generalization of the fact that a function recursive in E is actually recursive in0, for some a < w1 (where Ois from Kleene P I ).

A.12. Lemma. There are recursive functions f and g such that (i) ( z , a ) E D o g ( z , a ) E M ;

(ii) If (z,a) E D then for all x,

Proof. The recursive functions f and g are defined siniultaneously by the Kleene second recursion theorem of ordinary recursive function theory. (ii) and (i) in the direction * are then proven by induction on l(z,a)Icm . (i) in the direction quire an elaborate definition of f andg involving a number of auxiliary functions arising from applications of the S-m-n theorem of ordinary recursive function theory. Instead of this we give an informal description suppressing explicit reference to most of the auxiliary functions. We begin by assuming (z,a) E D and show in case Si how f (z,a) andg(z,a) must be defined in this case so that they satisfy (ii) and (i) in the direction *. Then we show in S i that iff (z,a) and g(z,a) are defined as in S i theng(z,a) E M implies <z ,a ) ED. The reader familiar with the Second Recursion Theorem will have

is proven by induction on I g(z,a)l. A rigorous proof would re-


no trouble in showing, if desired, that there actually exist such recursive functionsfandg. Case SO. (z,a) E D and

0 if (a10 -fl(z),l 7

= i 1 otherwise.

Thus let g(z,a) = ( z ) ~ and choosef(z,a) so that for all x,

Case S’O. z = (0 , l , ( ~ ) ~ ) andf(z,a) = ( z ) ~ E M . Since (z)2 E M , {z}“ [a] N

{(O, l , ( ~ ) ~ ) } ~ ( ( a ) ~ ) is defined by clause SO in the definition of { }” .

We consider in detail cases S8 and S4. Case S8. (z,a) E D and

The definition of f a n d g is trivial in cases Sl-S3, and easy in cases S5, S9.

o if 3t.{bYm [ ( t , a ) ] = 0 ,

1 otherwise. {z}” [a] = E(Xt.{b}% [ ( t , P ) ] ) =

where b = ( z ) ~ . By the Induction Hypothesis,

and for all t , x,

Since 3n is @restricted productive we can find u E M such that for all t , Ijg(b,(t ,a))l< IuI. (mre precisely u = p(uo,e) where for all 1, {e}( t ,M,uol )= jg(b,(t ,a)) . e of course depends ong, a and b.) Then by A.2.(i),


where these reducibilities are uniform in a, b, t. Then from (2), (3) we can find a u (depending on a, b) so tfiat for all t , {b}% [ ( t , ~ ) ] = {u}(t,Mlul). Then choose w so that

Let g(z,a)=j(u). Then choose f ( z , a ) so that for all x,

It follows from (4) that f(z,a) satisfies the desired equation. Case S'8. z = ( 8 , n , b ) andg(z,a)EM. S inceg( z ,a )=j (u ) , j (u )EM; since j : M 5, M , u EM; since u = p(uo, e) E M we have by A.2 (ii), for all t , {e}(t,MIUol) = jg(b,(t ,a)) EM and hence g(b , ( t , a ) ) EM. Also by A.2(i),(ii), for all t.

Hence by the Induction Hypothesis, for all t , ( b , ( t , a ) ) E D, i.e. {b}% [ ( t ,a>] is defined. Hence {z}~ [a] --E(Xt. {byx [( t , a ) ] ) is defined, i.e. ( z ,a ) ED. Case S4. (x,a> E D and {z>" [a] = {b}% [ ( { c } ~ [a] , a ) ] . Let d = ({c>" [a] ,a) . T h e n ( c , a ) E D a n d ( b , d ) E D , and I(c,a)lT, l(b,d)lrK < l(z,a)l". By the induction hypothesis,

g(c,a) E M and g(b ,d ) E M ,


and for all x,

We begin by showing how to choose g(z,a) so that g(z,a) E M and

By (5) we can find a u (depending on a, b,c) such that for every x,

By A.2(ii), p(g(c,a),u) E M and

Letg(z,a) = p(g(c,a),u) tMjg(c,a). It is clear from (8) thatg(z,a) satisfies (6). To find f (z,a) observe that by (6) and A.2(i),

uniformly in b. d ; and

(10) Mlg(c,u)l S J ( % c , a ) l ) i t M i g ( z , a ) l '

uniformly in c,a. From (5), (9) we can find u, w, y so that for all x,


where w is obtained by eliminating d in the previous equation by referring back to the definition of d and then using (lo), andy is obtained by using (10). Thuslet f(z,a) =y. CaseSY4 . z=(4 ,n ,a ,b )andg(z ,a )EM. Since

and henceg(b,d)EMand Ig(b,d)l <Ip(g(c,a),u)l< Ig(z,a)I.Againby the induction hypothesis, {b]" [d] is defined. Since {z}" [a] = {byx [d], { z } ~ [a] is defined.

X C nw is said to be 92-r.e. if X is the domain of an W-partial recursive function; X is%-recursive if the representing function of X is%-recursive.

A.13. Corollary. Let XC w. (i) X i s 92-r.e. i f f X & M ;

(ii) X is%-recursive iff X 5, Ma for some a < I MI; (iii) If h is %-partial recursive there is a recursive function k such that for

all a,

h(u)EM*k(u)EM& Ih(u)IS Ik(u)l ;

(iv) If h : w +M and h is %-recursive, then Sup { I h(x) I : x E w } < I MI

Proof. defined. So i fg is as in Lemma A.12 then n E X O g ( z , (n) ) EM. Thus X 5, M . Now suppose h is a recursive function such that h : X 2, M and choose e so that {e}%(n)- ( (0 , I,h(n))}%(O). Then,

(i) If X is %-r.e. then there is a z E w such that n E X - { z ) ~ ( n ) is

n E X h ( n ) E M - ( (0 , l,h(n))}%(O) is defined - (elTK (n) is defined.


Thus X is %-r.e. (ii) i t follows from Remark A . l l that i f X is recursive inMa for some

(Y < / M I , then X is 9Zrecursive. For the other direction it suffices to show that each total function { z } ~ is recursive in Ma for some a < IMI. By A.12, for all n,

Choose e so that for all n, {e}(n,MlUo,) = jg(z,n) and let c =p(uo ,e) . Then c E M a n d I c l > Ijg(z,n)l for alln.Hence for alln,Mlg(z,n)l <tMI,I uniformly in n. Thus there is an e l such that for all n ,

{el}(nJqcl) = {f(Z> ( n ) ) m M , g ( z , ( n ) ) l ) '

Then from (1 l), {z}" is recursive in MICI.

{ ~ } ' ~ [ a ] =h(a)EM;hence byA. l2 ,g (z , a )EMandfo r allx, (iii) L e t h = {z}" , a = ( a ) , andk(a)=p(g(z,a), f(2,a)). F o r h ( a ) E M ,

Thusk(a) E M and

(iv) Let k be as in (iii). Then for allx, k ( x ) E M and Ih(x)l < Ik(x)l. Choose e so that for all x, k(x) = {e}(x,MlUol). Then p(uo,e) E M and for A x , Ih(x)l<Ik(x)l<lP(uo,e)I.

5A.3. Selection. Using @and the fact that by SO every Ma is W-recursive uniformly in a

notation for a , given a E M v b EM it is possible to decide %-recursively whetherlal<lbl orIbI<IaI.(Recallthatifa@MthenIaI=IMI.)More precisely:

A.14. Lemma. There is an %-partial recursive function d such that: ( i ) a E M & la l< Ib l*d (a ,b )=O; (ii) Ib l< la1 *d(a,b)= 1.


Using A.9 it is easy to see that d is 772-partial recursive. If a E M & la I 2 Ib I thenh(a,b) E M a n d la1 2 la v bl < Ih(a,b)l; Hencea EMlh(a,b)l & b $Mlal and so{kh(a,b)}'lr(a)= O& { I ~ ( a ) } ~ ( b ) = 1. Thusd(a,b)= 0. Similarly, if Ibl < la1 t h e n h ( ~ , b ) E M a n d e i t h e r a E M ~ ~ ( ~ , ~ ) ~ & b EMlaI ora$Mlh(a ,b) l and hence by the definition of d, d(a, b) = 1.

The following argument is similar to Gandy's unpublished proof of the existence of selection functions associated with functionals of type 2.

A.15. Lemma. ljrzere is an 5%-partial recursive function u such that if A t . {z} ( t ) is total then

3t { z } ( t ) E M * u(z) is defined & {z} (u(z)) E M .

Proof. Letg be the recursive function of A.12 and let e be obtained from the second recursion theorem (Lemma A.8) SO that

e is found by using the recursion theorem in a manner similar to the proof of Kleene [9, XVI]. Le ty be the least t such that {z}(t) EM; equivalently,y is the least t such that {e}m(t , z) = 0. We show by induction on x that {e) 'M(y-x,z)=x for O<x<y .Th i s i s true i f x = Osincein thiscase {e}' (y -x, z) = {elm ( y , z) = 0 = x. Suppose x > 0. Then by the induction hypothesis {e)cm(y-(x-l),z) =x-1. In particular, since {e} (y-(x-l),z) is defined, g(e , (y- (x-l),z)) EM. Since also {z}(y-x) $ M , we have d({z} (y-x) ,g(e , (y - (x - l ) , z ) ) ) = 1. This implies by (12),

m


Settingx = y this gives {e}m(O,z) = y . Thus it suffices to let u(z) = {e}m (0 ,z ) .

A.16. Corollary. Let Q Cn+' w be%-r.e. Then there is an W-partial recursive function XxvyQ(x,y) of n variables such that for all x ,

Proof. Let Q ( ~ , y ) - { z } ~ ( x , y ) is defined. Then

Q(x ,y ) (j {Sm(z, x)]" ( y ) is defined

-g(S tyz ,x ) , (y , ) E M .

Let e be a recursive function such that {e(x)} (y) = g(Sm(z,x), (y) ) . Then

Thus let vyQ(x,y) N ue(x).

The following lemma summarises some of the properties of %-r.e. relations.

A. 17. Lemma. %-r.e.

(i) I f f is m-partial recursive, then the relation f ( x ) = z is

(ii) I j Q ism-r.e., then the function

z if Q W ) ,

undefined otherwise , f (XI =

is %-partial recursive.

are W-1.e.

sal and existential number quantification, and inverse images by %-recursive functions.

(iii) The relations y E M & 1x1 < l y l ; y E M & 1x1 I l y l , ~ E M & ly I 51x1

(iv) The W-r.e. relations are closed under conjunction, disjunction, univer-

Proof. Suppose j is W-partial recursive. Let


(2,1,0) if x = z ,

0 otherwise, e(x , z ) =

andg(x,z) = {e(f(x),z)}*(O). Theng is %-partial recursive and

g(x,z) is defined e e ( f ( x ) , z ) ) = (2,1,0)

- f ( x ) = z .

To prove (ii) let Q(x) -g(x) is defined, and letf(x) = O - g ( x ) + z . To prove(ii i)wehaveyEM& Ixl<lyl-{(O, l , ~ ) } ~ ( x ) = O;thenuse(i). The other relations in (iii) are handled similarly. To prove (iv) we consider just the cases of universal and existential quantification. Suppose Q(x,y) is 7Z-r.e. By (ii) the function

ism-partial recursive. Then

To treat existential quantification, let Q(x,y) - {z}”(x,y) is defined. Then by A.16,

3 ~ Q ( x , v ) -Q(x,vQ(x,v)> - {z}“ (x, vyQ(x,y)) is defined.

$ A.4. Proof of Theorem A5.

A.18. Lemma. There is an 9?i-partial recursive function g such that if z E M

and {e}M ( u , x ) is defined for each u E MI,, then


Proof. Le tg (e , z , x ) = vyQ(e , z , x , y ) where

A.19. Lemma. For each ordinal function Fprimitive recursive in T, there is an %-partial recursive function h, such that if a l , ..., a , E M then h,(a,, ..., a,) E M and

Proof. We shall use the characterisation of the ordinal functions primitive recursive in K K given by the following schemata (see [S]):

0 if x E M ,

1 otherwise (i) F(x , y )=

(ii) F ( x ) = x i

(iii) F(x) = 0

(iv) F(x) = x + 1

(v) F ( x , y , u , u) =

x i f u < u ,

y otherwise.

(vi) F(x,y) = G(x,H(x),y)

(Vii) W , y ) = G(H(x),y)

(viii) F ( z , x ) = G(sup,,,F(u,x),z,x) .

For (i) we first need an %-recursive function k such that I k(n) 1 = n for all n. Let k ( 0 ) = u o and


Then k has the desired property. Now in case (i) let lull = 1 and

A.20. Lemma. (i) I MI is closed under functions primitive recursive in Tw . (ii) Let R C On be primitive recursive in Ten. Then

is 371 -r. e.

Proof. a l , ..., u, EM&R(la l J , ..., la,l)}andFbe therepresentingfunctionofR. Then using A. 19,

(i) is an immediate consequence of A.19. Let A = {(al , ...,.a,).

(al ,..., ~ , ) E A -al ,..., a, EM&F(Iall ,..., la,l)= 0

-al ,..., a, E M & h F ( a l ,..., a,)EMI .

ThusA = nM n h$M1 which is%-r.e. using A.17.


A.21. Lemma(proof of A.5.1). IMI EAd(T%).

Proof. By Theorem 3.7 it remains to show that if R sive in T, ,a< [MI and

3 0 n is primitive recur-

then there is a < A < ]MI such that

(14) Vx < h3y < hR(a,x,y) .

Suppose ( 13) holds. Let Ic I = a and f (x) = vy [ y EM & R( Ic I, Ix I, I y I)]. Then f ism-partial recursive by A.20. Le tg be the %-recursive function defined by: g(0) = c and

Then Ig(n)l < Ig(n+1)1 and 1x1 < Ig(n)l * If(x)I 5 Ig(n+1)/. Let h = Sup,,, Ig(n)I .a<h< IMI by A.l3(iv). We show that h satisfies(l4).

( 14) then follows from ( 15) and the definition off.

A.22. Lemma(proofofA.S(ii)).IfR c n l M I is ]MI-r.e. in T, then thereisa primitive recursive function h such that

a l , ..., a, E M & R( I all, ..., I a, I ) - h(al, ..., a,) E M .

Proof. Let R be such that

R(a

MI -r.e. in Tm . Then there is a primitive recursive relation S


It follows that A is W-r.e. Hence by A.13 there is a recursive function h , such that A = hi1(,M). It remains to find a primitive recursive h. By the S-m-n theorem for ordinary recursive function theory there is a primitive recursive function such that, lettinga = a l , ..., a,, {g(a)}(x,Mluol) = hl(a) for all x. Let h(a) = p(u,,, g(a)); then h is primitive recursive and

Remark. The above proof is the only place where we use the fact that p is primitive recursive instead of just recursive.

A.23. Lemma (proof of A.5 (iii)). Let F be [ MI-partial recursive in T q . Then there i s a recursive function k such that feral, ..., an EM, i fF(lal1, ..., la,\)

isdefined then k(al ,..., a, )EMandF( la l I ,..., la,/)< Ik(al ,..., a,)l.

Proof. By the normal form theorem relativised to Tm , the graph of F is I MI- r.e. in Tq ; hence by A.22 there is a recursive function h such that

Let f ( a ) = vy[y EM& h(a l , ..., a,,y) EM]. Thenfis %-partial recursive. Hence by A. 13 there is a recursive function k such that

Then for a EM, if F(lall, ..., la,/) is defined


A.24.Lemrna(proofofAS(iv).XCois [MI-r.e. in T% i f fX<,M.

Proof. M is IMI-r.e. in T% since n EM- 301 < / M I . Tq (n,.). Hence if X<,M, X is also ]MI-r.e. in 27%. Now suppose X is [MI-r.e. in T, , and let k be an W-recursive function such that Ik(n) I = n for all n (see the proof of A. 19). Then using A.22 there is a recursive function h such that

n EX- Ik(n)I E X

- h k ( n ) E M

X is the inverse image of t heW -r.e. set M under the W-recursive function hk and hence i sV-r .e . by A.l7(iv). HenceXS, M b y A.I3(i).

Remarks. from that Gven in [ 161. The requirement t ha t j (called there g ) be a many-one reduction of M to M is necessary for the proof of A. 12 and its omission was an oversight in [ 161. The other change is the requirement that p be primitive recursive instead of recursive, and as mentioned above this is only to ensure that the function h of A S is primitive recursive.

(ii) AS(iii) is not used elsewhere but it appears to be of interest in its own right and its proof comes naturally from our construction. It was used in [ 161 in an earlier proof of some of our results but is not needed in our present formulation.

(iii) The method we have used in proving the Coding Lemma is to utilize techniques from the well-developed theory of recursive functionals of type 2. In particular, the crucial results needed about W-recursion, namely the Boundeness Theorem (A.13 (iv)), and Theorem A. 15 on the existence of selection functions are proved by standard methods from the theory of recursive functionals of type 2. On the other hand the theory of recursive functionals may be regarded as a part of the theory of inductive definitions. This suggests that an ultimately simpler and more elegant proof of the Coding Lemma in a more general setting can be provided within the “pure” theory of inductive definitions.

(i) The definition of an acceptable ordinal system differs slightly


References

[ 11 S. Aanderaa, Inductive definitions and their closure ordinals, this volume. [ 21 P. Aczel and W. Richter, Inductive definitions and analogues of large cardinals, in:

Conference in Mathematical Logic, London '70 (Springer, Berlin, 197 1) 1-10. [ 3 ] J. Barwise, Infinitary logic and admissible sets, J. Symb. Logic 34 (1969) 226-251. [4] J. Barwise, R.O. Gandy and Y.N. Moschovakis, The next admissible set, J. Symb.

[5] D. Cenzer, Analytic inductive definitions, Abstract, Notices, A.M.S. 703-E2, Vol. 20

[6] K. Devlin, An introduction to the fine structure of the constructible hierarchy,

[7] W.P. Hanf and D. Scott, Classifying inaccessible cardinals, Notices of the A.M.S.

[8] R.B. Jensen and C. Karp, Primitive recursive set functions, in: D. Scott (ed.)

Logic 36 (1971) 108-120.

(1973) pA376.

this volume.

8 (1961) 445.

Axiomatic Set Theory, Proceedings Pure Math. 13 (Amer. Math. SOC. Providence, R.I., 197 1) 143-176.

[9] S.C. Kleene, Recursive functionals and quantifiers of finite types I, Trans Amer.

[ 101 S. Kripke, Transfinite recursion, constructible sets and analogues of cardinals, in Math. SOC. 91 (1959) 1-52.

Lecture Notes prepared in connection with the Summer Institute on Axiomatic Set Theory, held at Los Angeles (1967).

Theory, Proceedings Pure Math. 13 (Amer. Math. SOC., Providence, R.I., 197 1)

[ 121 Th.A. Linden, Equivalences between Godel's definitions of constructibility, in:

[ 111 A. Gvy, The sizes of the indescribable cardinals, in: D. Scott (ed.) Axiomatic Set

143-176.

J.N. Crossley (ed.) Sets, Models and Recursion Theory (North-Holland, Amsterdam, 1967) 33-43.

[ 131 Y.N. Moschovakis, Elementary Induction on Abstract Structures (North-Holland,

[ 141 H. Putnam, On hierarchies and systems of notations, Proc. Amer. Math. SOC. 15

[ 151 W. Richter, Constructive transfinite number classes, Bull. Amer. Math. SOC. 73

[ 161 W. Richter, Recursively Mahlo ordinals and inductive definitions, in: R.O. Gandy

Amsterdam, 1973).

(1964) 44-50.

(1967) 261-265.

and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) 273-288.

(171 H. Rogers, Jr., Theory of recursive functions and effective computability (McGraw-

[ 181 G . Sacks, The 1-section of a type n object, this volume. [ 191 J.R. Shoenfield, Mathematical logic (Addison Wesley, Reading, Mass., 1967). [ 201 C. Spector, Inductively defined sets of natural numbers, in: Infinitistic Methods

[21] H. Tanaka, On analytic well-orderings, J. Symb. Logic 35 (1970) 198-204. [ 22) S.C. Kleene, On the forms of the predicates in the theory of constructive ordinals,

Hill, 1967).

(Pergamon Press, Oxford, 1961) 97-102.

11, Amer. J . of Math. 77 (1955) 405-428.

PART IV

AXIOMATIC APPROACHES AND GENERAL DISCUSSION

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., 1974

ON AXIOMATIZING RECURSION THEORY

Jens Erik FENSTAD University of Oslo

Generalized recursion theory can be many different things. Starting from ordinary recursion theory one may e.g. move up in types over w, or look to more general domains such as ordinals, admissible sets and acceptable structures. Alternatively, one may want t o study in a more general setting one particular approach to ordinary recursion theory, thus e.g. try to develop a general theory based on schemes or fured point operators, or work out a general theory of inductive defmability , or develop in a suitable abstract setting the various model theoretic approaches such as representability in formal systems or invariant and implicit definability.

The approach of this paper is axiomatic. This is nothing new. Of previous axiomatic studies of recursion theory we mention Strong [ 141, Wagner [IS], and Friedman [4]. Our interest in the axiomatics of generalized recursion theory was more directly inspired by Moschovakis [ 101, and any one familiar with his “Axioms for Computation Theories” will soon see our dependence upon his work.

Our objective is two-fold: First to contribute to the discussion and choice of the “correct” primitives for axiomatic recursion theory. Second to indicate new results, partly proved, partly conjectural, within the (modified) Moschovakis framework.

First one general remark on axiomatizing recursion theory. This may in itself be a worthy objective. Through an axiomatic analysis one may hope to get a satisfying classification and comparison of existing generalizations (technically through “representation” theorems and “imbedding” results). And one may,perhaps, also obtain a better insight into the“concrete” examples on which the axiomatization is based. But it is not clear - and some disagree - that the field is at present ripe for axiomatization. Hence we are approaching our topic in a tentative manner.

385

386 J.E. FENSTAD

As Moschovakis in [ 101 we take as our basic relation

which asserts that the “computing device” named or coded by a acting on the input sequence u = ( x l , ..., x n ) gives z as output.

Let 0 denote the set of all computation tuples (a, a,z) such that the relation {a}(u) = z obtains. I t is possible to write down axioms for a computation set 0 which suffice to derive the most basic results of recursion theory, say up through the faed-point or second recursion theorem.

However, many arguments seem to require an analysis not only of the computation tuples, but of the whole structure of “subcomputations” of a given computation tuple. Now computations, and hence subcomputations, can be many different things. And in an axiomatic analysis of the variety of approaches hinted at in the opening paragraph of the paper it would be rash to commit oneself at the outset to one specific idea of ‘computation’.

In [ 101 Moschovakis emphasized the fact that whatever computations may be, they have a well-defined length, which always is an ordinal, finite or infinite. Thus he proposed to add as a further primitive a map from the set 0 of computation tuples to the ordinals, denoting by la , u,z lo the ordinal associated with the tuple (a, u, z ) E 0.

We shall add as a further primitive a relation between computation tuples In t h s paper we shall abstract another aspect of the notion of computation.

which is intended to express that (a’, u ’ ,zf) is a subcomputation of (a, u,z), or, in other words, that the computation (a, u,z) depends upon (af, u’,zf). The basic axioms will state that the relation is transitive and wellfounded.

Remark. In our approach we have chosen functions and computations rather than sets and inductive definitions as basic notions. We have also chosen to exhibit the codes for the computations directly in the axioms rather than tried to develop the theory in a more “coordinate-free’’ or invariant manner. I t is, perhaps, still an open question which will be the most “useful” way to organize generalized recursion theory into a theory.

ON AXIOMATIZING RECURSION THEORY 387

The rest of the paper wdl be divided in four sections. In Section 1 we give the basic definition of a computation theory. In this we follow Moschovakis closely, making the modifications necessary due to our use of the subcomputation relation < instead of the length concept.

In Section 2 we list some basic facts about computation theories. This is in all essentials a repetition of material from [lo], but is included for the convenience of the reader. In this part of the theory there does not seem to be much difference between the length concept and the subcomputation relation. The important thing is that both allows us to carry over to the abstract setting certain results proved in the “concrete” examples by transfinite induction on associated ordinals, or, alternatively, by a course-of-value induction on “subcomputations”. The basic result here, due to Moschovakis in the axiomatic setting, is the first recursion theorem. The section concludes with the definition of regular computation theories. Such theories have selection operators, hence we have a reasonable theory for the computable and semicomputable relations. And for this class of theories we can introduce an adequate notion of finiteness, a set being finite if we can computably quantify over it.

In Section 3 we discuss the problem how to strengthen regularity. Of several possibilities we have chosen to emphasize two: One is the idea that a “computation” should be a finite object in the sense of the theory; the other is that the theory should satisfy the prewellordering property. The first we can express by requiring that for every (a, u, z) E 0, the set

is finite in the theory. One formulation of the other says that the set

is computable in the theory (where [la, u,zll is the ordinal of S(a,u,z)). In Section 4 we move beyond the “normality” conditions discussed above,

representation theorems and imbedding results being the main themes. Our exposition will be sketchy for several reasons: One is that a complete development would be too long, - another and more important one is that several of the results are still in a preliminary stage.

In conclusion I would like to acknowledge my great debts to Peter Aczel and Peter Hinman, who patiently have explained many results and methods

388 J.E. FENSTAD

of general recursion theory to me, and to Johan Moldestad and Dag Normann, who have with great enthusiasm participated in the investigations reported on in this paper.

1. Computation theories: basic definitions

In this section we give the basic definitions using the subcomputation relation as primitive notion.

Definition 1 . A computation domain is a structure

\u = ( A , C , N , s , M , K , L ) ,

where A is the universe, N CI C C A and ( N , s 1 N ) is isomorphic to the nonnegative integers. Cis called the set of codes. M is a pairing function on C, i.e.

a , b E C

M(a,b) = M(a', b') E C

iff M(a, b) E C , and

implies a = a' A b = b'

K and L are inverses to M , i.e. they map C into C and

c = M(a, b) E C iff a = K(c) A b = L (c) .

To facilitate the presentation we introduce some notational convention (following Moschovakis [lo]). We use x, y , z, ... a, b,c, ... i, j , k , ... u, 7, ... u, 7 or (u , 7 ) denotes the concatenation of sequences. And as usual lh (u) = the length of the sequence u. A computation tuple is any sequence (a, u,z) such t h a t a E C a n d l h ( a , u , z ) 2 2.

for elements in A . for elements in C. for elements in N . for finite sequences from A .

Definition 2. The system (O,<) is called a computation structure on the domain % if < is a transitive relation on the set of computation tuples and 0 is the wellfounded part of <.


Thus (a, o,z) E 0 iff the set

is wellfounded with respect to the relation <. Note that if (a, u,z) E 0 and (a’, uf,z’) < (a, u,z), then (u’, u’,zf) E 0.

Note: We have built into our definition the convention that something which looks like a computation, i.e. an arbitrary computation tuple (a, o,z), is not a computation if and only if its “subcomputation tree” contains an infinite descending path. In practice this may not always be so, but if an attempt at a computation stops after a finite number of steps without giving a bonafide computation, we can always start repeating ourselves in some suitable way so as to obtain an infinite descending path.

As in [ lo ] we shall make use of the notions of partial multiple-valued (pmv) function and functional. We recall some notations: f(4 -+ z

f(o) = g(o) f(4 = z

f S g iff VaVz[f(o)+z * g(o)-+z]. A mapping is a total, single-valued function.

natural number n we can associated a pmv function {a}: in the following way:

iff z Ef(o). iff z is one value off at u. iff Vz [f(o) + z iff g(o) + z ] . iff f(o) = {z}. iff ~ ( o ) + z A V U [ ~ ( U ) + U *u=z ] .

Let (O,<) be a computation structure on %. To every a E C and every

{a}:(u)-+z iff I h ( o ) = n A ( a , o , z ) E O .

Definition 3. Let (O,<) be a computation structure on %. A pmv functionf on A is @-computable if for some .f€ C

We call i a @-code o f f and write f = ti}”, where n is the number of arguments off.

390 J.E. FENSTAD

A pmv functional on A

maps pmv functions onA and elements ofA into subsets ofA (including the empty subset, 8). cp is called continuous if

Definition 4. Let (@,<) be a computation structure on the domain '21. A pmv continuous functional cp on A is called @-computable if there exists a (p E C such that for all e l , ..., eQ E C and all u = (xl, ..., x n ) from A , we have: a) cp({el};l, ..., {eQ>;Q, (I) + z if t + l g n ( e l , ..-, ep, 0) + z . b) If cp({el};', ..., {e&:Q,u) + z , then there exist pmv functionsgl, . . . ,g,

such that

ii. For all i = 1, ..., Q , ifgi(tl, ..., t,i) -+ u, then i . g l L tell:', ..., gQC teQ>:Q andcp(g1, - . . ,g~,,u)-+z.

Note : This is the first essential use of the suncomputation relation. For a motivation of the definition of O-computable functional, see [ 10, p. 2091.

Definition 5. Let (@,<) be a computation structure on the domain 91. to,<) is called a computation theory on \21 if there exist @-computable mappings p l , ...,pI3 such that the following functions and functionals are @computable with @-codes as indicated and such that the iteration property holds:

11-VIII. Similar to I and state that the following functions are O-computable: Identity function, the successor function s, the characteristic functions of C and N , the pairing function M and the inverses K and L .


X-XII. Similar to IX and state that the following functionals are @-computable: Primitive recursion, permutation of arguments, point evaluation.

XIII. Iteration property: For all n,m ~ 1 3 ( n , m ) is a O-code for a mapping Sz(a ,x l ,..., x,)suchthatfor alla,xl , ..., x , E C a n d a l l y l , ..., y , € A :

(ii) If {a};+”(x,, ..., x,, y (9 tal;+m(x,, ...,x,, Y l , ... ,r,> = {S#,Xl, ..., X,>l(Jq, ... >Y,>.

..., y,) + z , then

Remarks. 1 . The missing parts of the definition can be found in [ 10, pp. 205-2061.

2. If we drop the primitive <, the rest can be stated using 0 alone, and we arrive at Moschovakis’ notion of a pre-computation theory. This part seems to contain the basic core (i.e. “pre-Post” theory) of any systematization, including the fixed-point or second recursion theorem.

3. To every tuple (a,u,z) E 0 there is associated an ordinal Ila, u,zll = the ordinal of the set S(o,o,zl. 0 with this ordinal assignment is a computation theory in the sense of Moschovakis.

2. Computation theories: basic facts

2.1. Inductive generation of theories and equivalence. Let ‘?l be a computation domain and let

f = f i ,..., f, and cp=cp1,..-,cpk

be sequences of pmv functions and continuous pmv functionals on A . It is possible to construct a theory PR [f,cp], the prime computation theory generated by f and cp, which in the following precise sense is the least computation theory which makes all the functions f and functionals cp(uniformly) computable.

Definition 6. Let (@,<) and (Of,<’) be computation theories on the same domain ‘u. We say that 0’ extends 0, in symbols,

392 J.E. FENSTAD

(dropping, as we usual do, the explicit reference to < and < ’) if there exists a @’-computable mappingp(a,n) such that p(C,N) C N and such that for all n-tuples u, all a E C and z E A i. (a, u,z) E 0 iff (p(a,n), u,z) E 0’.

ii. I f (a, u,z) , (af, u’,z’) E 0 and (a, u,z) < (a‘, u’,z’), then

If 0’ 5 0 and 0 5 O r , we say that 0 and 0’ are equivalent and write 0 - 0‘. (p(a,n), 0 , ~ ) <’ (p(a’,n’), of ,z ’ ) .

Remark. It seems that Moschovakis’ motivation for his version of the notion of equivalence (see [ 10, pp. 217-21 81) is even more appropriate for the present version.

As in [ 10, p. 2181 we have the following result which justifies the claim made above.

(i) Let (0,<) be a computation theory on V l , and let f and cp be sequences ofpmv functions and continuous functionals on ?I. Then

and if H is any other computation theory on 91 such that 0 <H and f are H- computable and cp are uniformly H-computable, then

Remark. There are some difficulties in carrying over (iii) [ 10,p. 2191 to the present frame. We mention this since this is the only example of a result in [ 10, 3 3 1-81 which has not had an immediate counterpart. (The difficulty is that if we pass from 0 to H via a map p and then back to 0 via a map q , the ordinal of (a, u , z ) is less than the ordinal of(q(p(a,n), n) , u , z), but the former is not necessarily a subcomputation of the latter.

2.2. The first recursion theorem. The theorem was proved by Moschovakis in the axiomatic setting. The proof carries immediately over to the present set-up.

Theorem. Let te, <) be a computation theory on 8 . Let cp( f, x) be a


@computable continuous pmv functional over A . Let f ' be the least solution

of

V x € A : cp(f,x)= f ( x ) .

Then f* is @-computable.

The theorem is particularly important in discussing the relationship between recursion theory and inductive definability. I t corresponds to the fact, and can sometimes be used to show, that I;, inductive definitions has a I;1 minimal solution.

2.3. Selection operators. We first give the definition of O-semicomputable and @-compu table relations.

Definition 7 . The relation R(o) is @-semicomputable if there is a @-computable pmv function f such that

R(o) iff. f (o )+ 0 .

The relation R(o) is @-computable if there is a @computable mapping f such that

R(o) iff. f ( o ) = 0 .

The existence of a selection operator seems to be necessary in order to prove some of the basic facts about O-semicomputable and @-computable relations, such as closure of O-semicomputable relations under 3-quantification and disjunction, and also to prove that a relation R is @-computable iff R and 1 R are O-semicomputable.

Definition 8. Let (@,<) be a computation theory on 'ZI. An n-ary selection operator for (@,<) is an n t l - a ry @-computable pmv function q(a, o) with @-code 4 such that (i) If there is an x such that {a},(x, a) + 0, then q(a, cr) is defined and

(ii) If {a} ( x , a) + 0 and q(a, a) + x , then vx [q (a, u) + x * {a} ( x , 0) + 01.

(a,x,o,O)<(G,a,o,x).

394 J.E. FENSTAD

Remarks. [ 10, p. 2251.

rather than V x [q(a, u) + x * la} ( x , u) + 01, which is necessary when working with pmv objects.

1. For a motivation of (ii), equally valid in the present case, see

2. By an oversight Moschovakis [ 10, p. 2551 only required {a}(q(a, a), a) + 0

2.4. The notion of finiteness in general computation theories. Any good general approach to recursion theory must embody a suitable notion of “finiteness”. We repeat the basic definitions from [ 10, pp. 230-2331.

Definition 9. A computation theory (@,<) on % is called regular if (i) C = A .

(ii) Equality “x =y” is @computable. (iii) (@,<) has selection operators.

Definition 10. Let (@,<) be a regular theory on ’u, and l e tB C A . By theB- quantifier we understand the continuous pmv functional EB(f) defined by

0 if 3 x E B [ f ( x ) + O ] .

1 if V x E B [ f ( x ) + 11 .

The set B is called @-finite with @-canonical code e , if the B-quantifier Es is @-computable with @-code e.

We refer the reader to [ 101 for a list of five properties of this particular notion of finiteness which may justify the claim that it is “natural”. But it would be premature to conclude from this that the present version gives all

the properties of ordinary (i.e. true) finiteness necessary for the combinatorial arguments of e.g. degree theory.

3. Extending regularity

In this section we will discuss the problem of how to extend regularity. Of several possibilities we have chosen to emphasize two.


A. The idea that a “computation” is a finite object in the sense of the theory seems to play an important role in many arguments of recursion theory. “Computation” is not a primitive of our system, but a perhaps satisfactory approximation consists in requiring that a computation tuple depends on only a finite number of other tuples, i.e. for every (a, u,z) E 0, the set

is finite (uniformly in (a, u, z)) in the sense for Section 2.4.

Note: A more complete technical statement would require that there is a 0- computable mappingp(n) such that for each (a, u,z) E 0 {p(n)},(a, u,z) is a @-canonical code for S(,,,,z). And since we are dealing with sequences of arbitrary finite length, we assume some suitable coding convention.

Remark. A computation theory in the sense of Moschovakis is a pair (0, )lo ), where 11 is a map from 0 into the ordinals. Using a length-function it seems to be difficult to capture the idea that a “computation” should be a finite object in the sense of the theory.

As a first approximation one could consider the set

But one cannot outright say that this set, which is the set of all computations with smaller length, is a finite object in the theory. Indeed, if this requirement ismade, the set of natural numbers necessarily will be finite in 0.

Some restriction must therefore be added and Moschovakis proposed to compare computations of equal length, i.e. he required the finiteness of the set

I t is possible to proceed on this basis, and it may have some advantage later on (see Section 4), but it is not in my opinion a satisfactory conceptual analysis of the motivation behind normality (see [ 10, p. 2331).

B. The prewellordering property is another important tool in general recursion theory. One formulation is as follows. Let ~la,u,zll denote the ordinal of the

396 J.E. FENSTAD

set S,,,,,,,, (a, u,z) E 0. The theory (@,<)has the prewellordering propery if there exists a @-computable function p ( x , y ) such that if either x E 0 or y E 0 then p(x,y) is defined and single-valued, and whenevery E 0, then

in other words, the set {x 1 x E 0 A ((x ( 1 <_ ( ( y 11) is @-computable, uniformly iny .

We shall make some remarks on the relationship between the finiteness of the sets S(a,u,zl and the prewellordering property. I t is convenient to introduce some terminology.

Definition 11. Let (0, <) be a computation theory on a domain ’ZI. 1. (El,<) is called p-normal if it is regular and has the prewellordering

2. (0, <) is called s-normal if it is regular and the sets S(a,o,zl are (uniformly)

3 . (0, <) is called strongly normal if it is both p-normal and s-normal.

property (see B above).

@-finite for ( a , u,z ) E 0 (see A above).

Remarks. the sets in (*) above are uniformly @-finite.

normality, and strong normality lead to essentially the same class of theories, viz. the so-called “Spector theories” (see Section 4.1).

1. “Normality” in the sense of Moschovakis [ 101 requires that

is @finite, normality in the sense of Moschovakis, p - 2. I f the domain

(i) S-normality implies a “weak” form of the prewellordering property: We can define a @-computable pmv function q(x,y) such that if

x,y E 0, then q(x,y) = 0 iff IIxII I llyll .

This is so since we can computably quantify over finite sets, hence have the following recursion equation


(ii) If we in addition assume that the relation < is O-semicomputable, then we have the prewellordering property. In this case we have the following recursion equations for the function p(x,y): (a) (b)

p(x,y) = 0 if V x ’ E S, 3y’[y’ <y A p(x’,y’) = 01. p ( x , y ) = 1 if 3x’[x‘ < y A Vy’ E Sy p(x’, y’ ) = 11.

Note : Since < is defined for all tuples of the form (a, u,z) and O is the well- founded part of <, the assumption that < is O-semicomputable is rather problematic. However, the following argument may add some plausibility. An arbitrary computation tuple (a , u,z) may or may not represent a “true” computation. But as soon as we are given a set of instructions a and an input sequence u we should be able to start “generating” the “subcomputations”, and this is what the O-semicomputability of < is intended to express. This argument seems to carry force in the case of single-valued computations. I t is in the case of multiple-valued computations that we would have difficulties in extending a relation <on O to a relation < on all tuples (a , u , z ) such that the extended relation is O-semicomputable and such that O is the well-founded part of the extended <.

(iii) If (0, <) is an s-normal theory over a 0-finite domain 91 , we have the prewellordering property in several important cases, e.g. if O is the recursion theory associated with some total type-2 functional F(assuming 2E I F ) , or the recursion theory associated with the type of partial type-2 functionals considered by Hinman [6], or the “quantifiers” of Aczel [ 11. Since (see 4.1.1) every s-normal(@, <) is equivalent to a theory PR [ F ] for some continuous partial F , it remains an interesting problem to determine those theories PR [ F ] for which we have the prewellordering property.

(iv) For infinite s-normal theories (O,<) on a partially ordered domain

4.2.5. If we drop the finiteness assumption 4.2.5, a counterexample can probably be constructed.

,I) we have the prewellordering property under the assumptions 4.2.3-

(v) P-normality does not imply s-normality. “Ordinary” recursion theory over w can be constructed in such a way as to provide a counterexample.

398 J.E. FENSTAD

4. Beyond normality

Moving beyond normality there is one fundamental distinction to make: Either a theory (@,<) has a domain which is finite in the sense of the theory, or its domain is infinite.

4.1. Theories over finite domains. In analogy with Moschovakis [ lo] we call a p-normal (or, which amounts to essentially the same - see the remark following Definition 1 I in Section 3 - strongly normal) theory (@, <) on a domain 8 a Spector theory if the domain A is @-finite. For such theories one can prove a great number of results which were originally established for hyperarithmetic and hyperprojective theory (see Moschovakis [9] and [ lo], - a detailed exposition within the axiomatic set up can be found in Vegem [ 161).

Hyperarithmetic theory over w is the theory of recursion in 2E, hyperprojective theory is a generalization of t h s to more general domains. How different is an arbitrary Spector theory from recursion in some functional over the domain, i.e. what kind of representation theorems do we have for Spector theories?

We state some results. But first a few terminological remarks. In many cases it is convenient to assume that the search operator v is computable in 0, where v ( f ) = {x I f ( x ) -+ 0). It is known from [ 10, p. 2661 that if 0 has a selection operator, then 0 is weakly equivalent to @[v], hence, for reasons detailed there, we may as well work with @[v], which allows us greater freedom in defining pmv objects.

For convenience we also introduce the following notations: sc(@) = the set of all @-computable relations on '21. en(@) = the set of all O-semicomputable relations on 9 ( .

(The notation en (O), the envelope of 0, is taken from Moschovakis [ 1 11 .)

4.1.1. For any s-normal (0, < ) on a domain '21, there exists a continuous partial functional F on 91 such that

0- PR [F] .

This result is jointly due to J. Moldestad and D. Normann, and the proof uses the uniform finiteness of the sets S(a,o,z). D. Normann has also adapted the


main result of Sacks [ 121 to show

4.1.2. For any Spector theory over w there exists a total functional F such that

sc (0) = sc ( F ) (and en ( F ) C en (0))

Remark. D. Normann actually proves a more general result: Let M be a countable admissible set which satisfies local countability and A O-dependent choice, then: (i) M = L & ~ ) , for some generic class K E O(M) = M n On; (ii) M is K- admissible, and (iii) O(M) is the least 0 such that L f is K-admissible. - 4.1.2 is an immediate corollary. (Note that the somewhat complicated hierarchy for type-2 recursion used by Sacks [ 121 is avoided in this approach. I t may be essential for the k-section result.) We expect that Normann’s result can be adapted to admissible sets with urelements, yielding a generalization of 4.1.2 to arbitrary countable domains.

In Moschovakis [ 1 11 we find a counterexample to show that 4.1.2 cannot be lifted from sections to envelopes:

4.1.3. mere exists a Spector theory 0 over w such that en (0) f en (F), for all total type-2 functionals over w .

The problem remains to characterize those Spector theories which are equivalent to prime recursion is some total type-2 functional over the domain. This is, of course, only one step toward a full classification of Spector theories.

In this connection a Gandy-Spector theorem or, better, a normal form theorem for the class en (0) may be of interest. I t is not at all clear how to formulate such theorems in a sufficiently general form. Our proposal for a “weak” version:

Definition 12. Let P ( X , u) be a second order relation over 3 . P(X , u) is called @computable with indexp if whenever {e}@ is the characteristic function of some set X , E sc (O), then

0, if P(X,, u)

1, ow. {PIo(e, 4 +

400 J.E. FENSTAD

We say that the theory 0 has the weak Gandy-Spector property if whenever R € en (@), there is a @-computable P such that

R(o) iff. (3X€sc(@))P(X,o)

4.1.4. Evely Spector theory has the weak Gandy-Spector property.

This has been proved by J . Moldestad. I t remains an interesting problem to find other “natural” kinds of normal form theorems which can be used to characterize certain classes of Spector theories.

Remark. The original or “strong” Candy-Spector theorem for hyperarithmetical theory provides a P which is first order with respect to the language adequate to describe the domain %.

We shall comment on one more topic. The connection between inductive definability and hyperprojectivity was investigated by Grilliot [ 51 . His results was adapted to the present frame by Moldestad IS]. Let Ind(A) denote the class of relations which are inductive in some operator of class A, i.e. reducible to some I’”, where r € A.

4.1.5. Let 0 be a Specto-r theory on \u and R a sequence of @-computable relations on BI.

(i) Ind(X2(R)) 5 en(@). (ii) 0- PR [R,= ,v ,E] , i fandonly ifInd(Z2(R))=en(@).

Here the implication from right to left in (ii) goes beyond Crilliot [5], and gives a certain characterization of the “minimal” Spector theory on a domain yl. A general result would give a necessary and sufficient condition on 0 for the validity of the equation Ind(X,(R)) = en(@).

4.2. Theories over infinite domains. In Moschovakis [ 101 the case of infinite domains is particularly satisfying. Any Friedberg theory in his sense is a recursion theory generated in a “natural” way from an admissible prewellordering of the domain.


In our case the situation is more complicated. Our definition of normality does not automatically give us a prewellordering of the domain. In order for us to proceed we must add extra assumptions on the domain. The goal will be to abstract the “natural” or “minimal” recursion theory associated with an admissible set (possibly with urelements, - for this concept see Barwise [2]).

Remark. The more complicated situation in our set up is perhaps not a too serious disadvantage. There seems to be recursion theories over infinite domains (i.e. infinite in the sense of the theory) on which there is no natural associated prewellordering (see [7]). And there seems to be admissible sets where the prewellordering associated with the rank function is not admissible in the sense of [ 101. (Let A be an admissible set such that A is uncountable but A n On is countable.) Such examples should not be denied their proper existence in an axiomatic analysis of computation theories.

We admit at once that we do not yet claim to have a “good” definition of computation theories over infinite domains. We shall make some preliminary suggestions, and hope that further work will lead to a “correct“ analysis.

Let (@,<) be an s-normal theory on the domain \zI. The basic intention is to express that there is a suitable correspondence between the complexity of the domain 2l and the complexity of the computations in 0. Some partial ordering of the domain seems to be necessary in order to code whole computations in a “natural”, i.e. order preserving way into the domain. We make the following proposal:

4.2.1. mere is a @-computable partial ordering 5 of A such that the initial segments of 5 are well-founded and (unifomiy) @-finite.

A weak way of expressing the correspondence between 0 and the domain is to assume:

Remark. A more refined analysis would probably postulate the existence of some kind of “coding”-function:

4.2.3. mere is a @computable mapping K : 0 -+A such that if we set

402 J.E. FENSTAD

K * (x) = { w E A I w 5 K (x)}, then (i) K*(x) is @-finitefor each x E 0;

(ii) x ES, implies K*(x) C K * ( ~ ) ; (iii) A = U,,@ K*(x).

Note that 4.2.2 follows from (iii) in 4.2.3 and that 4.2.3(i) is already implied by 4.2.1.

assumption. As a first approximation we propose: In addition to the coding process we seem to need some sort of “decoding”

4.2.4. There i s a @-computable nzuppingp(n) such that {p(n)},(u, U , Z , w) = 0

i f f ( a , (7, z ) 0 A [la, (I, z 1 1 = Iw I.

In other words, the relation (a , u,z) E 0lw’ is @-computable. The prewellordering associated w i t h 5 is easily seen to be uniformly 0-

computable. I f we strengthen this to

4.2.5. {w E A I Iw I I Iwol} is uniformly @-finite in wo € A ,

we arrive at the class of “Friedberg theories” in the sense of Moschovakis 10, 5 101. And as he shows these are exactly the class of computation

theories associated with admissible prewellorderings. I t remains to isolate the properties which characterizes the recursion theories associated with admissible partial orderings, i.e. with arbitrary admissible sets.

A topic of central importance is the relationship between theories over finite and infinite domains. The basic example here is the relationship between hyperarithmetic and meta-recursion (or L,,-recursion) theory. This was generalized in Barwise, Candy, Moschovakis [3] (- see also Barwise [ 2 ] ) .

In our context we are looking for a theorem which states that finite theories can be imbedded into infinite ones. The idea behind is simply this. A computation theory on an infinite domain can be a “good” recursion theory, in particular, if we have a suitable correspondence (codinddecoding) between the domain and computations over the domain, and if the semi-computable relations are exactly the Xl -definable relations over the domain. The imbedding theorem should say that we can “enlarge” a finite theory to a “good” infinite theory. And, as a possible application, we would expect that fine structure results for, say, the semi-computable relations of the given finite theory could


be obtained by “pull-back’’ from the enlargement. The motivating example is again various results for IT: sets obtained via meta-recursion theory.

We state the following preliminary version of the imbedding theorem:

4.2.6. Let (@,<) be a Spector theory on a domain %. I t is then possible to construct

(i) a domain ( y!* ,I* ) where ’u * extends ‘u and<* is a well-founded

(ii) a relation R on (a *,I* ), and (iii) an “infinite” theoty (O*, <* ) on (% * ,<* >, such that

partial ordering on (11 *,

(a) <* is @*-computable and initial segments of<* are uniformly

(b) R is @*-computable and @*-semicomputability equals Z,(R)-

(c) a subset R 5 A , where A is the domain of ‘u, is O-semicornputable

O* -finite,

definability over ?l*, and

i f and only i f R is @*-semicornputable.

Remark. The result and method of proof is clearly inspired by Barwise, Candy, Moschovakis [3] and it uses in the construction of a* the recent developed theory of admissible sets With urelements, see Banvise [2] and his forthcoming lecture notes. There is also some recent work of P. Aczel and Y. Moschovakis which overlaps with the present result. Both Aczel and Moschovakis work in the context of a Spector class (which is equivalent to being the envelope of a Spector theory) and their goal is to relate these classes to the “next admissible ordinal” (Aczel) or “next admissible set” (Moschovakis). Both Aczel and Moschovakis carry their analysis a step further than the above result, showing the existence of a unique, minimal “next” structure. From this also follows the existence of a unique minimal “infinite” extension @* of a given Spector theory 0.- We have deliberately used the word “infinite theory” to describe O*, since it is a bit unclear at the moment how strong properties we can en- force on @*; it will satisfy the properties 4.2.1-4.2.5. We also expect that c

can be strengthened to assert the equivalence between 0 and @* 13, but some details remain to be sorted out.

4.3. One further goal is to push the analysis of “computation” so far as to establish the domain of validity for the priority arguments. Only then can we claim to have a reasonably complete axiomatic analysis of generalized recur-

404 J.E. FENSTAD

sion theory. This is very much an open field.

Remark. The concept of a Friedberg theory does not seem to be entirely adequate, see Simpson’s paper “Post’s problem for admissible sets” in this volume. On the positive side see the appendix to that paper and also Simpson ~ 3 1 .

References

[ I ] P. Aczel, Representability in some systems of second order arithmetic, Israel Jour.

[2] K.J. Barwise, Admissible sets over models of set theory, this volume. [ 31 K.J. Barwise, R. Gandy and Y.N. Moschovakis, The next admissible set, J . Symbolic

(41 H. Friedman, Axiomatic recursive function theory, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 1971) 113- 137.

[ S ] T. Grilliot, Inductive definitions and computability, Trans. Amer. Math. SOC. 158

[6] P. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. SOC.

[7] P. Hinman and Y.N. Moschovakis, Computability over the continuum, in: R.O.

Math. 8 (1970) 309-328.

Logic 36 (1971) 108-120.

(1971) 309-317.

131 (1968) 526-543.

Gandy and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 1971) 77-105.

IS] J . Moldestad, cand. real. thesis, Oslo 1972 (in Norwegian). 191 Y.N. Moschovakis, Abstract f i s t order computability I , Trans. Amer. Math. SOC. 138

(1969) 427-464; and 11,138 (1969) 465-504. [ 101 Y.N. Moschovakis, Axioms for computation theories - first draft, in: R.O. Gandy

and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 197 1) 199-255.

[ 11 ] Y.N. Moschovakis, Structural characterizations of classes of relations, this volume. [ 121 G.E. Sacks, The 1-section of a type n object, this volume. 1131 S. Simpson, Admissible selection operators, Notices Amer. Math. Soc. 19 (1972)

[ 141 H.R. Strong, Algebraically generalized recursive function theory, 16M Jour. Res.

[ 15) E.G. Wagner, Uniform reflexive structures: on the nature of Godelizations and

[ 161 M. Vegem, cand. real. thesis, Oslo 1972 (in Norwegian).

A-599.

Devel. 12 (1968) 465-475.

relative computability, Trans. Amer. Math. SOC. 144 (1969) 1-41.

J.E. Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., I974

DISSECTING ABSTRACT RECURSION

Thomas J. GRILLIOT The Pennsylvania State University

1. Introduction 2 . Syntactic description of recursion 3 . Semantic description of recursion: implicit definability 4. Semantic description of recursion: quantifier form 5 . Completeness of C, E, S, A , I schemes 6 . Extending the notion of finiteness

1. introduction

We have two prototypes for this discussion: recursiveness on the natural numbers and hyperarithmeticalness on the natural numbers. The latter differs from the former in that a quantification-over-w scheme is admitted. Crudely speaking, to say that F is recursive in Gl, ..., G,, means that F is computable or can be combinatorially generated from the structure ( w ; 0, s, =, G,, ..., GJ, where 0 and s are the usual zero and successor function of w. (Note: 0, s,= are sufficient to specify the structure of w as we know from the investigations of Peano.) Similarly, to say that F is hyperarithmetical in Gl, ..., Gn means that F can be generated combinatorially from the structure ( o ; O , s,=,G1, ... ..., G,) with the aid of an oracle that can test quantification over w . I t is natural to replace the structure ( w ; O,s, =, G,, ..., G n ) by an arbitrary structure (A;Gl, ..., G,) where A is a set and G,, ..., G, are functions or predicates on A . This is precisely what our investigation is all about: to find out what recrusive- ness on an arbitrary structure means. Unfortunately, the matter is not so easy in that the structure of the natural numbers is fairly unique with respect to other structures. Consequently the abstract study of recursiveness and hyperarithmeticalness may tend to be prejudiced by preconceptions based on usual recursiveness and hyperarithmeticalness on the natural numbers. However, with a little investigation one can isolate these potential prejudices and discover that they seem to be five in number, which we denote by C(constant),

405

4

406 Th.J. GRILLLOT

E (equality), S (search), A (for all), I (for infinitely many).

w because it is finitely generated by 0, s. In general, however, an arbitrary set need not be so combinatorially generated, and hence one cannot argue convincingly that constant functions are ips0 fact0 recursive.

E-scheme: equality relation should be recursive. This scheme holds for w because 0, s generate w , s is one-one and 0 can be distinguished from successor numbers. The general situation is quite different. One cannot, for example, convincingly argue that equality on real numbers is ips0 facto recursive.

S-scheme: unordered search operator should be recursive. The situation of w is particularly nice in that one has not only an unordered search operator but even an ordered one (the so-called p-operator). This is because w is finitely generated by 0, s. An alternative formulation of the S-scheme is that semirecursive predicates are closed under existential quantification. Again one cannot argue on purely combinatorial grounds that an unordered search operator is ipso facto recursive.

A-scheme: the universal-quantifier operator should be “recursive”. This certainly holds for hyperarithnieticity on w. In fact, one of the more elegant formulations of “hyperarithmetical” is based on this idea (see Kleene [5]). The A-scheme is the most natural candidate for formulating an abstract notion of hyperarithmeticalness, but compare with the following scheme.

/-scheme: the operator introducing the quantifer “for infinitely many” should be “recursive”. This certainly holds for hyperarithmeticity on w . In fact, the A-scheme and the I-scheme are equivalent on w (in the presence of bounded quantification):

C-scheme: constant functions should be recursive. This scheme is true for

IxP(x) - V y 3 x ( P ( x ) & y < x)

VXP(X) - fyVx(P(x) & x < y )

In some structures the two schemes are independent. Since the five schemes listed above are independent, we have 32(= 2 9

different formulations of recursion (of varying degrees of interest). To label thcm properly, we will use the terminology (C,E,S,A ,I)-recursive, (C,E,A)- recursive, etc. Thus t o say that F ( o n A ) is (C,E,S)-recursive in ( A ; G ) means that F c a n be generated combinatorially from the structure ( A ; G ) with the aid of schemes C,E,S. An exact formulation of what this means is given in the next section.

DISSECTING ABSTRACT RECURSION 407

An inquisitive reader will certainly question whether the five schemes listed above are exhaustive. In Section 5 we will show that they are (at least for countable structures) in the following sense: any scheme that holds for recursiveness (respectively, hyperarithmeticalness) on the natural numbers and can be formulated for arbitrary structures is derivable in (C,E,S)-recursion (respectively, (C,E,S, A ,I)-recursion).

Historical note. Perhaps the earliest formulation of abstract recursion is due to FrahC [ 11. However, he did not distinguish between the combinatorial and noncombinatorial (schemes C,E,S, A,I) aspects of recursion. Moschovakis [lo] seems to have been first to isolate the schemes C,E,S and a combined A/I scheme. The completeness of the C,E,S schemes was asserted by Lacombe [8] in a surreptitious way. The completeness of the C , E , S . A , / schemes was proved by Grilliot [2].

2. Syntactic description of recursion

Our choice of formulation of recursion is proof-oriented. Besides being fairly versatile, it has the advantage of bridging the computational and the model-theoretic aspects of recursion.

inductively as follows. 9 t- 6 i f

(a) I9 E \Ir or c p , l cp E 9 for some cp [initial] ; or (b) cp & $ E 9 and 9 U {p, $} (c) cp v $ E 9, 9 U {p} f 6 and 9 U {$} I- 6 [v-elimination] ; or (d) Vxcp(x) E 9 and 9 U {cp(t)} (e) 3xcp(x) E 9 and 9 U {cp(y)} f 0 where y is a constant not in *, 6

[weak 3-elimination]; or ( f ) 3xcp(x) E 9 and 9 U {p(b)} p I9 for all b E B [strong 3-elimination] : or (g) Zxcp(x)E9 a n d q U ( 3 x

(h) Cxcp(x)E9and\ I rU{3x l.. .x,Vy(p(y) v y = x l v . . . v y = x , ) } k0

Definition. We define 9 t 6 , where 0 is a sentence and 9 a set of sentences,

I9 [&-elimination] ; or

I9 for some term t [V-elimination] : or

x,(cp(x,)& ... & p ( x , ) & x l #x2 & x1 # x3 & ... & xnP1 # x,)}

for all IZ E w [C-elimination] .

6 for some n E w [I-elimination] ; or

Clearly Ix represents the quantifier, for infinitely many x, and Cx represents the quantifier, for cofinitely many x; thus Ix and 1 C x l are expected to represent the same thing. Because of our specialized purpose, there is no need

408 ThJ. GRILLIOT

to have a 1-elimination scheme (since 1 ’ s can be driven through the other connectives) nor introduction schemes (since B will always be atomic or negated atomic) nor equality schemes (since they can be incorporated into 9 as desired).

To say that Q is recursive in (B;P1, ..., P,) will mean that there is a formal description of Q in terms ofP , , ..., P, such that

(information about P,, ..., P,) U (description)

The description must be both complete (all true information about Q is derivable) and consistent (no false information about Q is derivable). The description may have intermediary predicate or function symbols in it. For example, the usual description of * (on a) in terms of 0, s = utilizes + as an intermediary. Let us formulate this definition of recursion precisely,

Definition. Let (B;P, , ..., P,) be a structure, where B is a set and PI, ..., P, are relations (possibly partial) on B. Let Q be another relation (possibly partial) on B. For a language that includes one or more names (constant symbols) for each element of B and one or more names (predicate symbols) for each of P,, ..., P,, Q, define AQ to be the set of formal sentences

(information about Q).

{Q(bl, ..., b,) : Q(bl , ..., b,) is true)

U { l Q ( b l , ..., b,) : Q(bl, ..., b,) is false}

and similarly for Apl, ..., Apn, where we use the same letter for an object and its formal name(s) so long as confusion is avoided. Q is (C,E,S,A,I)-recursive in the structure ( B ; P l , ..., P,) if there is a sentence cp such that

A= u Apl u _.. U Apn U (9) I- 8 for all 8 E AQ

and some expansion of ( B ; P l , ..., P,) is a model of cp. In this definition, some of the relations P,, ..., P,, Q may be replaced by (partial) functions where, if f i s a function, Af is defined to be

{ f ( b l , ...) b k ) = c : f ( b 1 , ..., b k ) “ C I

With regard to the formation of Af, if multiple names are used to denote one element of B , all names must appear as arguments of the function symbol but

DISSECTING ABSTRACT RECURSION 409’

only one name need appear as value. Also multivalued functions are acceptable, only in this case, the symbol = occuring in Af may have to be replaced by another symbol to avoid inconsistency with A=. An n-ary multivalued function amounts to the same as an (n+l)-ary relation in the presence of the E-scheme and S-scheme; but in the absence of the S-scheme the value-place of the function plays a role different from the argument-places as we shall see later.

we said that some expansion of ( B ; P , , ..., P,) must be a model of cp. This condition can be relaxed by saying some extension of (B;P , , ..., P,) must be a model of cp. In other words, it does not matter if the universe of the model is larger than B. This can be seen by the following little trick. Given cp with a model with universe larger than B , form a new description

As part of the definition of Q being (C,E,S,A,I)-recursive in ( B ; P l , ..., P, ),

& ... & vx l... Xk,(Pn(X1, ...) Xk,)+PA(fXl, ...) fXk,))

where P i , ..., PA, Q‘, f are new symbols and cp* is made from cp by replacing each P I , ..., P,, Q by P i , ..., PA, Q‘ and each variable or constant x by fx. ($ + 0 is regarded as an abbreviation for 1 $ v 0.)

Suppose Q is (C,E,S,A,/)-recursive in ( B ; P ) with description cp. Thus

We may think of cp as a program for computing Q from P as follows. Assume that all 1’s in cp are driven through the other connectives so that they occur only immediately preceding atomic formulas. To determine whether or not Q(a) is true, one begins with a computation node {p} /-. The connectives in cp are then systematically stripped in accord with clauses (b) through (h) so that one gets a tree of intermediate computation nodes of the form Jr where Jr is a finite set. If a t some node, P(b) E Jr or l P ( b ) E Jr or b = c E \I,

or b # c E Jr for some b , c E B , then one asks the oracle of P or = whether or not P(b) or b = c is true. If there is a conflict, clause (a) establishes that A- U Ap U \I, Q(a) and A= U Ap U Jr lQ(a) ; thus computation node

410 Th.J. GRILL107

\I,

Also if Q(a) E 9 (respectively, l Q ( a ) E q), clause (a) establishes 9 (respectively, \I, 1 l Q ( a ) ) ; thus node 9 is certified for Q(a) (respectively, lQ(a)) . As soon as all terminal nodes are certified for Q(a) (respectively, lQ(a) ) , the computation procedure ceases with the answer being Q(a) is true (respectively, false).

This computational procedure for (C,E,S,A,I)-recursion has five processes within it that are not purely combinatorial. We label these schemes by C , E , S , A , I . By removing one or more of these schemes as outlined below from the computational procedure, one gets 3 1 other forms of recursion ((C, E, S,A)-recursion, (C, E,S,Z)-recursion, ..., ( )-recursion).

Role and irredundance of the I-scheme. C-elimination is an infinitistic rule of proof and hence is not combinatorial. Its role is that of introducing the I x quantifier. Thus the I-scheme is retained or removed by retaining or removing the C-elimination rule from the inductive definition of k. This turns out to be equivalent to allowing or disallowing the quantifierszx, Cx in the description q. To verify the irredundance of the Z-scheme, we can apply a result of I1.C of [4] that there is a nonstandard model of first-order arithmetic in which every (C,E, S,A)-recursive set of standard numbers is finite. In particular, the set of all standard numbers (denoted by w ) is not (C,E,S,A)-recursive in this model. By contrast, w is (C,E,S,A,Z)-recursive in this model since x E o ++

1 1 y (y<x) . In other words, we have the following description of w :

is certified. Similarly if 8,18 E 9 for some 8, node 9 b is certified. Q(a)

& some equality axioms.

Role and irredundance of the A-scheme. The other infinitistic rule is strong 3-elimination. Its role is to introduce universal quantification over universe B. Thus the A-scheme is retained or removed by retaining or removing the strong 3-elimination rule from the definition of b. Consider the structure (w X a;<) where (a, b) < ( c , d ) iff a < c. The set ( 0 ) X w is (C,E,S,A,I)-recursive in (w X w ; < ) since x E ( 0 ) X w ++ 1 3 y ( y < x ) ; that is, the following is a description of { 0 ) X w :


However, the set cannot be (C,E,S,Z)-recursive in this structure. For suppose p is a description of {0} X w. Pick n E w so that no name for any ( m , n) occurs in y. Then y is still a description of {0} X w even when a new element (-1,n) is added to the structure and < is extended so that (-1,n) < (m,n) for all m E w . On the other hand, cp must be a description of the transplant of {0} X w under the canonical isomorphism from (w X w ; < ) to (w X w U

{(-l ,~)};<). Thus cp is a description of two different sets, a contradiction. Role and irredundance of the C-scheme. This scheme allows all constant

functions as automatically “recursive”. The C-scheme is retained or removed by allowing or not allowing names for elements of B (other than those listed in (B;Pl, ..., P, )) in descriptions. At first glance, allowing names for arbitrary elements of B in a description p i s a purely combinatorial feature. However, in the course of a /--tree such a name would have to be matched with its replicas in Apl U .._ U Apn. Such matching cannot be regarded as automatically combinatorial. For example, let w1 be the set of countable ordinals. The predicate P(x) ++ (x is finite ordinal) is (C,E)-recursive in the structure (wl;<) since it has the description:

Vx[(o < x -lPx)& (w = x + 1PX) & (x < w -Px)] .

The use of a name (0) for the first limit ordinal is crucial to this description. With a compactness-like argument one sees that P cannot be generated combinatorially from < and =; that is, P is not (E)-recursive in (wl ;<). With slightly more care, one can find a predicate P that is (C)-recursive in (a1 ;<) but not (E,S,A,Z)-recursive in (w,;<). It should be noted, however, that in those structures (B;P,, ..., P,) in which B is generated by P,, ..., P, (e.g., ( w ; 0, s, =)) the C-scheme is redundant.

relation is automatically “recursive”. At first glance, the scheme can be retained or removed by retaining or removing A= in the definition. However, the removal of A= may not be enough in some instances since subtle comparisons between objects of B may be made in the course of determining k. For example, {$ } t- 0 is established when $ is 8 . But to verify that $ is 0 , one must make a character-for-character comparison which often includes a comparison between two names for elements o f B which in turn insinuates a comparison of elements of B themselves. To avoid such surreptitious uses of equality, one must allow infinitely many names for each element of B. In this

Role and irredundance of the E-scheme. This scheme says that the equality

412 Th.J. GRILLIO’I

way, the combinatorial comparison of two names in no way establishes a noncombinatorial comparison of two elements of B. (Conversely, if one accepts the E-scheme, then one might as well use only one name for each element of B.) Consider the structure (a; 0, s). The predicate Z(a) t-, a = 0 is @)-recursive in ( w ; 0, s) (or, equivalently, is ( )-recursive in ( w ; 0, s, =)). However, Z is not ( )-recursive in (a; 0, s). For suppose there is a description cp of Z . Let a be a name for 0 different from any name occuring in cp. Then one cannot combinatorially determine that A, U {cp} f Z(a). It is of interest to note that Z and = are of equal strength in the presence of the predecessor function: p ( a ) =

a - 1 when a # 0 and undefined otherwise. That is, the structure ( w ; O,s,=)

and ( o ; s , p , Z ) are equivalent, and, in fact, the functions ( )-recursive in either structure are exactly the usual recursive functions, and the functions (A)- recursive (alternatively, (I)-recursive) in either structure are exactly the usual hyperarithmetical functions.

Role and irredundance of the S-scheme. This scheme allows one to search through the set B for some object that satisfies a “recursive” predicate. I t is the abstract analogue of the p-operator scheme. This search capability appears in the V-elimination rule in the definition of f . To determine whether \k U {Vxcp(x)} f 8 , one must check whether J( U {Vxcp(x), cp(t)} some term r . Thus one must search through all terms t , which is tantamount to searching through B. To remove the S-scheme, one must restrict V-elimination to instances where the terms involved can be generated combinatorially from information already given and the oracles of the functions in the structure. To exemplify this restriction, consider the structure (R;+ , .) where R is the set of real numbers. Suppose that in the course of some computation one encounters

B for

A+ U A. U {Vx(x * x # 2 v P ( d 3 ) ) ) k P ( d 3 ) . (1)

In the presence of the S-scheme, this can be established by replacingx by d2:

However, the new number 4 2 introduced cannot be found combinatorially from 2 , 4 3 , +, -; so one could not establish ( 1 ) in the absence of the S-scheme. The restricted V-elimination rule must be stated thus:


for some term t in which no name for an element of B occurs except those generated from the constant and function symbols occuring in 9 U {Vxcp(x), 0). (If 0 is of the form f ( b l , ..., b,) = c, the constant symbol c must be excluded from those “occuring” in 9 U {Vxq(x),O} unless it occurs elsewhere. This exception is made because the value(s) of a function being computed cannot be assumed to be given a priori. In this regard, n-ary multivalued functions are computationally different from (n+ 1)-ary relations.)

It should be noted that in those structures (B;P1 , ..., P,) in which B is generated by PI, ..., P, (e.g., (a; 0, s,=)), the S-scheme is redundant. To further exemplify the irredundance of the S-scheme, consider the structure (plane ; compass, straightedge). The functions “midpointing” and “angle- bisecting” are ( )-recursive in this structure; but “angle-trisecting’’ is not (C,E)-recursive in this structure, though it is (E,S)-recursive in this structure. More precisely, let A be the set of points in the plane, and let F , G,H be the following 0-, 1. or 2-valued functions on A :

F(a, b , c, d) = point of intersection of line ab and line cd

G(a, b, c ,d) = point(s) of intersection of line ab and circle cd

H(a, b ,c ,d) = point(s) of intersection of circle ab and circle cd

(Circle ab denotes the circle with center a passing through b.) Typical functions ( )-recursive in ( A ; F, G , H) are:

M(a, b) = midpoint of segment ab

B(a,b,c) = point d on line ab with the property that angle acd = angle dcb.

It can be shown that i f J is (E)-recursive in ( A ; F , G , H ) , thenJ(al, ..., a,)

takes on only finitely many values each of which is constructable from a l , ..., a, using only compass and straightedge. It follows that the following function is not (E)-recursive or even (C,E)-recursive in ( A ;F , G , H ) :

T(a, b,c) = point d on line ab with property angle acb = 3 angle acd.

However, T is (E,S)-recursive in ( A ; F , G , H ) with description

414 Th.J. GRILLIOT

Va bcde ( B (a , e , c) # d V B (d, b, c) # e V T(a, b , c) = d )

& (description of function B ) & (some equality axioms).

In the absence of the S-scheme but in the presence of the A-scheme, one may allow a V-elimination rule of another sort:

\Ir u {Vxcp(x)} k 0 if, for all terms t ,

and, for a t least one term r , \k U {Vxcp(x),cp(t)} k 8 .

Definition. Q is semi-(C, E,S,A,I)-recursive in (B;P , , ..., P,) if, for some cp,

Normally we will talk of only total predicates as being semi-recursive whereas we allow purtiul predicates that are recursive. The removal of the schemes C,E,S.A,I in semi-recursion is just as in recursion. I t should be fairly clear that a total predicate is (...)-recursive iff it and its negation are semi-(...)- recursive.

3. Semantic description of recursion: implicit definability

Each of the 32 variations of recuision that we discussed had something of

Q is recursive in ( B ; P l , ..., P,) if there is a description cp such that the following forin:

and an expansion of ( B ; P l , ..., P,) is a model of cp.

l.E of [4], we see that we can replace the syntactic k by the semantic I=. However,

These variations differed in that the meaning of k was varied. By 1.C and is complete at least in the case B is countable. Thus


simply means that every model of cp that interprets the symbols =, P,, ..., P, as the relations (functions) = ,P I , ..., P, must interpret the symbo! Q as the relation (function) Q. In other words, by replacing ing semantic formulation of recursion:

by we get the follow-

Q is recursive in (B;P,, ..., P,) if there is a description cp such that some model of cp interprets PI, ..., P, correctly and every such model interprets Q correctly. In effect, cp isan implicit definition of Q in terms ofP,, ..., P,.

Each of the 32 variations of recursion has such an implicit definability form. They differ only in the kind of description allowed and in the kind of models alloweq. These variations can be outlined very easily as follows.

Full (C,E,S,A,I)-recursion. cp may have V, 3, I , C quantifiers. Universes of models must be B.

Removal of the I-scheme. I and C quantifiers are not allowed in cp.

Removal of the A-scheme. Universes of models need not be B

Removal of the S-scheme. Models that expand substructures of (B;P, , ..., P,) are allowed. One must then say that every model with universe C that interprets PI 1 C, ..., P, 1 C correctly interprets Q 1 C correctly.

Removal of the C-scheme. cp may not have names of elements of B in it other than those in the list P,, ..., P,.

Removal of rhe E-scheme. Somehow one must allow models that interpret each element o f B as many objects.

Historical note: Fra’issC’s [ 13 definition of abstract recursion is essentially the implicit definability one. Kreisel [6] lays great emphasis on implicit definability. Indeed model-theoretic formulations of recursion have an air of completeness lacking in their combinatorial counterparts and thus add great weight to the validity of Church’s Thesis. The equivalence of implicit definability with Moschovakis’ scheme formulation of recursion is proved by Moschovakis [ 1 I ] , and also in the hyperarithmetical case by Grilliot [ 3 ] .

416’ Th.J. GRILL107

4. Semantic description of recursion: quantifier form

We wish to generalize the classical results that semi-recursive predicates have Zy quantifier form and semi-hyperarithmtical predicates have nl quantifier form. The latter result is virtually equivalent to the result that the hyperarithmetical functions are precisely the unique solutions of Z1 relationals. However, we must be more careful in the abstract case of describing quantifier forms than just by Ilr and Er, because we have the I and C quantifiers and also the contents of the matrix must be specified. For instance, if only O,s,= are allowed in the matrix (and hence not +, - , bounded quantifiers), then not every semi-recursive predicate is C1 but rather strict ~ nl . Let bold-faced quantifiers (3, U) denote a list of second-order quantifiers (quantifiers over relations of objects) of that kind. Thus every C 1 predicate has the form 3V3 over ( w ; O,s,=) and every strict - Z; predicate has the form ,3V over ( w ; O,s,=). The general results we would like are:

( 1 ) Every predicate V’3 over ( B ; P l , ..., P,,) is semi-(C,E,S)-recursive in

1

1

0 1

1

( B ; P , , ...,P,l):

( B ; P , , ..., P n ) ; (2) every predicateV’31 over ( B ; P l , ..., P,) is semi-(C,E,S,I)-recursive in

(3) every predicate V3V (also V3V3, V’3V3V, etc.) over ( B ; P l , ..., P,) is semi- (C,E, S , A )-recu rsive in (B;P , , ... , P,, );

(4) every predicate V 3 V (also V (any list of first-order quantifiers)) oveI ( B ; P 1 . ..., P,,) is semi-(C,E,S,A,I)-recursive in ( B ; P , , ..., P, ).

We must assume that B is countable. The converses of these assertions will be considered Fbortly. T o see ( 2 ) , suppose Q(a) +-+ V R ... VR,cp(a) where cp in prenex form is 31 2nd R ,. ..., R, are the relation symbols other than PI, .... P,,, Q in cp. I t follows that Q is (C,E,S,I)-recursive in (B;P , , ..., P, ) with description Vx(q(x) + Qcx)), which is a VC sentence. For let ‘u be a model of this description. Then % I B is also a model of it because it has quantifier form VC. I f Q ( u ) is true, then cp(u) is true in every model with universe H and so Q(u) is true in ’u I B and hence in “&.

The converses of (3) and (4) hold as is seen as follows. Suppose Q is semi- (C ,E ,S ,A ,I)-recursive in ( B ; P l . ..., P,,) with description q. First note that cp

can be reduced to another description with quantifier form VC by adding new

DISSECTING ABSTRACT RECURSION 41 7

intermediary relations. Indeed, any I-quantifier can be eliminated by noting that

Vx 3y (x < y & $ ( y ) ) & (< is irreflexive, transitive)

is a conservative extension o f l x $(x), where < is a new relation symbol. Next a description Vx 3yVzCu 3 u q has the following conservative extension:

VxVyVzVu(R4(x,y,z,u) + 3 u $ ) .

Drawing quantifiers forward judiciously we get VC3 (or V3C) quantifier form. In this manner, any description can be reduced to one in quantifier form VC3. Finally note that if cp is such a description of Q in terms of PI, ..., P,, then Q(a) ++ VR I ... VR,VQ(q+ Q(a)), where R predicate symbols in cp other than Q,P,, ..., P,. Thus Q is x 3 1 V over

..., R k are the

(B;P,, ..., Pn ). We do not know if the converses of (1) and (2) are generally true, but they

seem nearly to be true in the following sense. The converse of ( I ) is equivalent to the statement that every description cp for (C,E,S)-recursion can be made into one that is in V-form. We know from the discussion above that each description can be made into one that is in V3-form. However, 3-quantifiers are fairly inert in the absence of strong 3-elimination.

Historical note. Montague's [9] formulation of abstract recursion lays heavy emphasis on quantifier form. Moschovakis [ 121 first proved the abstract version of the Kleene-Suslin Theorem (HYP = A:).

5. Completeness of C, E , S , A , 1 schemes

We will show that (C, E, S)-recursion is the strongest abstract recursion that generalizes usual recursiveness on the natural numbers, and that (C, E , S ,A ,I)-recursion is the strongest abstract recursion that generalizes

418 ThJ. GRILLIOT

usual hyperarithmeticalness on the natural numbers. To see this, suppose Q is recursive, in some sense, in ( B ; PI, ..., P,,) where B is countable. Then Q is recursive in that sense in (B;Pl, ..., P,, 0, s, =) for any 0 E B and any successor function s on B (s is a successor function on B if s is injective and B = {O,sO,ssO,sssO, ...}). I t follows that, for any bijection from B t o w , the transplant of Q is recursive in the usual sense in the transplants ofPl, ..., P,. By the following assertion, Q is (C,E,S)-recursive in (B;P,, ..., P,). Let B be countable. Then

( 1 ) Q is (C,b’,S)-recursive in (B;Pl, ..., P,,) iff, for every bijection from B t o w , the transplant of Q is recursive (in the usual sense) in the transplants of PI, ..., P, (bijection may be changed to injection);

(2) Q is (C,E,S,/)-recursive in (B;P,, ..., P,l) iff, for every injection from B to w , the transplant of Q is hyperarithmetical in the transplants of P,, ..., P,:

(3) Q is(C,E,S,A,Z)-recursive in (B;P,, ..., P,) iff, for every bijection from B t o w . the transplant of Q is hyperarithmetical in the transplants of PI, ..., P,,.

The proofs of (2) and (3) are nearly identical, so let us consider (2). The necessity is straightforward: if Q is (C,E,S,Z)-recursive in ( B ; P 1 , ..., PJ, then Q is (C,E,S,I)-recursive in (C;P,, ..., P,) for all countable C 2 B, and so Q is (C, E, S,I)-recursive in (C; P,, ..., P,,, 0, s, =) for all countable C 2 B, 0 E C and successor functions s on C; this is equivalent t o saying that, for each injection from B t o w , the transplant of Q is hyperarithmetical in the transplants of P,, ..., P,. Conversely suppose that, for each injection from B t o w , the transplant of Q is hyperarithmetical in the transplants ofP,, ..., P,. Consider the theory A whose symbols include names for elements of B and Ors, < and whose axioms are A = U A p l U ... U A p plus the usual axioms of arithmetic concerning 0, s,< plus the axiom V x l I y ( y < x). A is clearly (C,E,S,I)- recursive in (B;P,, ..., P,). Also Q is (C,E,S,A,I)-recursive in every model of A . I t follows from 1I.B of [4] that Q is (C,E,S,Z)-recursive in (B;P,, ..., P,,).

Historical note. Lacombe’s [8] notion of V-recursiveness is that Q (on B ) is V-recursive in P (on B ) if, for every bijection from B to‘w, the transplant of Q is recursive in the transplant o f P . Lacombe asserted that V-recursiveness is equivalent to FrGsse’s invariant definability notion of recursiveness. Moscho- vakis [ 111 proved that V-recursiveness is equivalent to his schematic notion of recursiveness. Grilliot [ 21 proved the analogue for hyperarithmeticalness.


6. Extending the notion of finiteness

For simplicity, let us ignore the C,E,S,Z schemes. The difference between recursion with and recursion without the A-scheme may be regarded as a difference in the idea of finiteness. The A-scheme says that (for some super- being) the universe can be comprehended with ease and so in an extended sense of the word the universe is “finite”. There may be structures with sets other than the universe that are t o be regarded as “finite” in an extended sense, e.g., admissible sets. It is of interest to define recursion for such a structure. Let (B;P,, ..., P,) be a countable structure and let C be a countable collection of subsets of B that are t o be regarded as “finite”. The definition of Q being recursive in (B; P I , ..., P,) is just as usual except that the A-scheme is dropped and the inductive definition of allows for clauses that comprehend the elements of C ; namely, for C E C , \I, k 8 if \I, U {ip(c)} k 8 for all c E C and 3x E Cip(x) E \I,.

However, a problem of greater interest is the following. Given a structure (B; P,, ..., P,) and C a collection of “finite“ subsets of B , what other subsets of B must necessarily be regarded as “finite”? In other words, how does one effect closure under the idea of “finiteness”? For example, if w is regarded as “finite” for the structure (w;=) , then any finite or cofinite subset of w must be regarded as “finite”, but there is no reason to regard the set of even numbers as “finite”. On the other hand, if the structure is expanded t o (w; O,s, =),

then the set of even numbers - in fact, any hyperarithmetical set ~ is readily derivable from the universe and hence must be regarded as “finite” though there is n o reason t o regard a nonhyperarithmetical set as “finite”.

sets, characterize all the sets that are thereby “finite”. We cite three possible answers t o this question. (1) Any subset of B that is both El-definable and Il l -definable-on-B in every model of

To repeat the question: given (B;P, , ..., P,) and a collection C of “finite”

I 1

A Pl U . . . U A p n U {~,,,Vx(~~CvV,,~x=c)}.

The definitions of definable and definable-on-B are given on pp. 122-122 of [7]. The definitions are easily adapted to allow second-order quantifiers. (2) Any subset of B that belongs to every model of the set displayed above plus axioms that more or less state that two-element sets are “finite”, “finite” sets are closed under subset. and “finite” unions of “finite” sets are “finite”.

420 Th.J. GRILL107

These axioms are chosen because they are virtually complete for the notion of “countable” (see l.F of 141) and countability is completely degenerated finiteness. (3) Any subset o fB that can be built up from e using effectivized versions of the axioms: two-element sets are “finite”, “finite” sets are closed under subset, and “finite” unions of “finite” sets are “finite”. For arbitrary structures we d o not know how these three answers compare.

References

[ 1 ] R. F r a k t , Unc notion de r h r s i v i t k relative, Infinitistic methods (Pergamon Press,

[ 2 ] 7.5. Grilliot, Omitting types: application to recursion theory, J . Symbolic Logic,

[ 31 T.J. Grilliot, Implicit defjnability and hyperprojectivity, Scripta Mathematica, to

[ 4 ] T.J. Grilliot, Model theory for dissecting recursion theory, This volume. [ 5 ] S.C. Kleene, Recursive functionals and quantifiers of finite types, I , Trans. Amer.

[6 J G. Kreisel, Model theoretic invariants: application to recursive and hyperarithmetic

Oxford, 1961) 323-328.

vol. 37 (1972) 81-89.

appear.

Math. Soc., vol. 91 (1959) 1 -52.

operations, in: J . Addison et al. (eds.) The Theory of Models (North-Holland, Amsterdam, 1965) 190-205.

[ 7 ] G. Kreisel and J.L. Krivine, Elements of mathematical logic (North-Holland, A mst erdani , 19 67).

181 D. Lacombe, Deus g&n@ralizations de la notion de r&cursivitC relative, Comptes Rendus de I’Academie des Sciences de Paris, vol. 258 ( I 964) 341 0-34 13.

[ 9 ] R. Montague, Recursion theory as a branch of model theory, in: B. van Rootselaar and J.F. Staal (eds.) Logic, methodology and philosophy of science 111 (North- Holland, Amsterdam, 1968) 63-68.

vol. 138 (1969) 427-504.

Logic, vol. 34 (1969) 605-633.

Math. J . , vol. 37 (1970) 341-352.

[ 101 Y.N. Moschovakis, Abstract first order computability, Trans. Amer. Math. Soc.,

\ 1 I ] Y .N. Moschovakis, Abstract computability and invariant definability, J . Symbolic

[ 121 Y .N. Moschovakis, The Suslin-Kleene Theorem for countable structures, Duke

J.E.Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., I974

MODEL THEORY FOR DISSECTING RECURSION THEORY

Thomas J. GRILLIOT The Pennsylvania State University

I . Completeness theorems A. Usual predicate calculus with &, V, v, 3 , l B. Predicate calculus with equality C . w-logic D. lnfinitary &’s and V’s E . The infinity quantifier F. The uncountability quantifier

A. Compactness ( 3 A) B . Omitting types (VV) C . Omitting compactifiable types (VV 3 A )

111. Compactness after omitting types A. Application to uncountability quantifier B . Barwise compactness

11. Satisfying an infinite sentence

This paper summarizes some results of model theory that have an impact in examining recursion theory. No attempt has been made to document the results.

I. Completeness Theorems

A. Usual predicate calculus with &, V, V, 3 , l . First of all we make a blanket assumption that any set of axioms/rules is

strong enough to drive 1 ’ s through other connectives so that, if needed, we may assume that 1’s only appear before atoms. We show how a consistent set @ of sentences has a model. The key to this demonstration as well as most of those following is the formation of a set am of sentences with the following properties: (a) @ cpv $ €@=, then cpE@_ or I) E@-;(d) ifVxcp(x)E @= and f is a term formed from the function symbols of am, then cp(t) E @=; (e) if 3xcp(x) € am,

(b) if cp & $ € @-, then cp, $ € @=; (c) if

421

422 Th.J. GRILLIOT

then q(c ) E for some constant symbol c. Assuming that we have such a let be the set of all atomic or negated atomic sentences in am. We

can make the following observations: (1) since C am, any model of CP_ is also a model of both Q, and a:, and the consistency of am implies the consistency of both @ and atoms or negated atoms, its consistency implies that it has a model ; (3) by induction on the number of connectives in a sentence of am, one readily sees that a model \u of CP: with universe {t?,: f is a term formed from the function symbols of am} is a model for all of am. From these three observations it follows that the consistency of CP implies that it has a model provided we can form CPm in such a way that the consistency of am. We form assunie @ is countable. Let a0 be CP. Pick a sentence from CPo with a connective other than 1. If the sentence is y & $, then Q0 U {cp, $ } must be consistent, so let CPl be this set. If the sentence is y v $, then a0 U { y } or a0 U {$} is consistent, so let Vxcp(x), then a0 U { y ( t ) } must be consistent for any term t , so let be Q0 U {q(t)} where t is a variableless term. I f the sentence is 3xp(x), then a0 U { q ( c ) } must be consistent where c is a new constant symbol, so let a1 be this sct. In a similar manner, form a2, a3, Q4, ... in such a way that every connective (and every variableless term in connection with V) is acted upon at some stage. Then C P _ = U CPn is the desired set.

s am and

(2) since Cp: consists of only

implies the consistency of as a union of sets aO, a1,Q2, ... as follows. For simplicity

be :he one that is consistent. If the sentence is

B. Predicate calculus with e,quality. The following axioms when added to the usual predicate calculus axioms/rules are complete for =: Vx(x =x) ; x = y + O(x) tf O(y). This is proved just as in the preceding section except that the additional conditions are placed on am: (f) t = t € 9- for all t formed from function symbols of am; (g) if O(t) E @- and t = u E CP_ or u = t E CPrn, then O(u) E a_. The axioms mentioned above are just enough to permit the formation of with these two new conditions.

C. o-logic. Let A be a countable set whose elements have names in some countable

language such that a # b is an axiom when a, b € A and a # b. If one is interested in restricting the meaning of V and 3 t o quantification over A , then the following infinitary rules when added t o usual axioms/rules are complete: {@(a ) : a E A }/VxO(.x). The proof is as in the preceding section except that

MODEL THEORY FOR DISSECTING RECURSION THEORY 423

condition (e) concerning Q,= must be revised to read: (e) if 3xq(x) E aa, then q(a) E Q,= for some a E A . (One may have to add the axiom Vx3y (x = y ) to Q, to make the details work out readily.) Several adaptations can be made. For example, add A as a unary predicate symbol to the language. To make A(x) have the meaningx € A in any model, the following infinitary rules are complete: {O(a) : a €A}/Vx( lA(x) v O(x)). One can achieve the analogous result using countably many sets simultaneously.

D. Infinitary &’s and V’S.

One may wish to incorporate countably many infinitary conjunctions A and disjunctions V in a countable language. For this modification, the following axioms/rules are complete: rules allowing 1 ’ s to be driven through A’s and V’s according to deMorgan’s Laws; O n -+ VOi for any n ; { O o , 01, ...}I AOi. The proof is as in the case of the completeness of usual predicate calculus except that conditions (b) and (c) concerning Q,- are revised to read: (b) if Aqi E am, then qi E am for all i ; (c) if Vqi E Q,_, then pi E Q,- for some i.

E. The infinity quantifier. One may wish to add two new quantifierszx and Cx to a countable

language with intended meanings of “for infinitely many x” and “for cofinitely many x”. (Note: Zx and 1 C x 1 have same intended meaning.) For this modification, the following axioms/rules when added to the usual axioms/rules are complete: rules allowing 1 ’ s to be driven through Z’s and C’s according to deMorgan’s Laws; { 321xO(x), 322xO(x), ...}I ZxO(x); V2nxO(x) + CxO(x) for any n , where 32nxO(x) is an abbreviation for 3x, ... 3xn(O(x1) & ... & O(x,) & x1 #x2 & ...) and V2nxO(x) is an abbreviation for 132nx10(x). The proof is as in the case of completeness of usual predicate calculus except that the following two additional conditions must be added concerning am: (h) ifIxq(x) E am, then 32nxq(x) E Q,- for every n ; (i) if Cxq(x) € Q,-, then V2nxq(x) E Q,- for some n.

F. The uncountabaity quantifier. One may wish to add two new quantifiers UCx and CCx to a countable

language with the intended manings of “for uncountably many x” and “for cocountably many x”. Surprisingly, the additional rules needed for these quantifiers are finitary, and the completeness proof is quite different from those preceding. To simplify matters, use a two-sorted language with x , y , z

4 24 Th.J . GRILLIOT

denoting type-0 variables and X , Y ,Z type-1 variables and E a binary relation relation between type-0 and type-1. Let UCxO(x) be an abbreviation for 13XVy( B(y) c--, y E X ) and let CCxO(x) be an abbreviation for 1 UCx7B(x) . The following axioms are sufficient to make UC and CC have their intended meanings: 3 X ( y E X & z E X ) ; 3 Y V z ( z E Y - z E X & O ( z ) ) ; Vy EXCCzO(y , z ) -+ CCzVy E X O ( y , z ) . Informally, these axioms state that all two-element sets are countable, countable sets are closed under subsets and countable unions of countable sets are countable. The middle axioms (subset axioms) can be dispensed with if UCxB(x) is used as an abbreviation for 1 3 X V y ( O ( y ) + y EX) instead of for 7 3 X V y ( O ( y ) - y E X ) . Suppose @ is a set of sentences that includes the above axioms and the axiom of extensionality. We want to show that its consistency implies that it has a model in which UC and CC have their intended meanings. By the standard completeness theorem, @ has a countable model %,, say with type-0 universe A , and type-1 universe 2,. A, may be regarded as a subset of the power set o f A o . Let {q j (x) } be the collection of all formulas in the language of \uo in one free variable x such that '11, CCxqj(x) , and let $(x) be any formula such that '$1, UCx $(x). By the compactness theorem, ' $4 , clearly has an "extension" 211 in which $I1 + 3 x [ $ ( x ) & A q i ( x ) ] . The trouble is that the compactness theorem only insures that the type-0 universe A of 211 includes A 0 and not that the type-l universe A , of '$1, includesj,. In order for 2, CAI ,some types omitted in '$1, must remain omitted in %,. We are thus faced with the problem of superimposing compactness upon omitting-types. This is possible thanks mainly to the countable union axioms, but the details will be left to a more appropriate section (section 1II.A). Thus \uo has a bonefide elementary extension \ul with the property that l?ll one forms 'u2, '$I,, ... and indeed 'u, for any countable successor ordinal U.

If u is a limit ordinal, one lets 3, be u7<o%T. awl is the desired model.

3x[$(x) & A q i ( x ) ] . In like manner

I I. Satisfying an infinite sentence

A. Compactness (3A). Let CP be a set of sentences, for simplicity, countable. The problem of

compactness is to find a sufficient condition for when @ has a model of an infinite'conjunction or, what is virtually equivalent, an existentially quantified infinite conjunction. The answer is that @ U { 3 x A i E w q j ( x ) } has a model if, for each 12, @ U { 3 ~ A ~ . , ~ q ~ ( x ) } ~ has a model. (We have simplified the situa-

MODEL THEORY FOR DISSECTING RECURSION THEORY 4 25

tion by considering only countable conjuncts and one existential quantifier.) To see this, let C = @ U {cpi(c) : i E w ) where c is a new constant symbol. Clearly every finite subset of C has a model. Form Zm with the usual closure conditions(see section l.A) plus the condition that every finite subset of C _ has a model. Since C z consists only of atoms and negated atoms, it has a model because every finite subset of it has a model. Therefore, C_ and hence C and hence @ U { 3xAiG,cpi(x)} has a model.

B. Omitting types (VV).

find a sufficient condition for when Q, has a model that is also a model of a universally quantified infinite disjunction. The usual answer is that @ U {VxViEwcpi(x)} has a model if cb has a model and, for all $(x) such that cb U { 3x$(x)} has a model, Q, U { 3x($(x) & cpi(x))} has a model for some i. (We have simplified the situation by considering only one universal quantifier.) To see this, form @- from @ so that am has a model and satisfies the usual closure conditions plus the condition: if t is formed from the function symbols of aW, then qi(t) E am for some i. The hypothesis above is adequate to insure that this new condition on Qm can be achieved. As is noted in section 1.A, if Qm has a model Y f , then it has a submodel whose universe is {fa : t is formed from the function symbols of am}. This submodel is a model of @ U {VxViEwcpi(x)}. Variations can be made for languages with o-rules, infinitary conjuncts and disjuncts and quantifierszx, Cx. Also, adding an infinite conjunct before the universally quantified infinite disjunction is no problem. Thus for allj, for all $(x) such that @ U {3x $(x)) has a model, @ U { 3x($(x) & cpii(x))} has a model for some i. Another variation can be made for second-order quantifiers. Let the quantifiers 3 p , Vp vary through all relations of a specified number of arguments. Assume that p does not occur in @. @ U {Vx(ViEw 3pcpi(x) v ViE,Vp$i(x))} has a model if @ has a model and, for all O(x) in which p does not occur - though relation symbols other than those in @ may - with the property that @ U {3xO(x)) has a model, cb U { 3x(O(x) & cpi(x))} has a model for some i or @ U { 3x(O(x) & 1 $i(x))} has no model for some i. An important application is the following. Let @ include in its language names for the elements of a countable set A . Then any subset of A that is ll, -definable-on4 in every model of Q, is semi-represent- able in some finite extension of @.

Let @ be a countable set of sentences. The problem of omitting types is to

U {AjG,VxViE,cpq(x)} has a model if @ has a model and,

1

4 26 ' I3 .J . GRILLIOT

C. Omitting compactifiable types (VV3A).

a model if CP has a model and, for all G0(x), $l(x), $*(x), ... such that Q, U { ~ x & < ~ $ ~ ( x ) } has a model for eachj, Q, U {3~3vA,,~($,(x) & cpi,(x,y))} has a modelfor some i and all j . The proof is quite similar t o the one outlined in the preceding section except that the formation of Q,- is made so that the following condition holds: if t is formed from the function symbols of Q,-,

then cpij(t,c) E Q,- for some i and allj, where c is a constant symbol. Varia- tions of this result exist also. One important application is the following. Let Q, include in its language names for the elements of a countable set A . Then any subset ofA that is Xi-definable in every model of Q, is finite. In fact, there exists one model of Q, in which every Xl-definable subset o fA is finite.

Let Q, be a countable set of sentences. Q, U { V ~ V ~ ~ ~ 3 y A ~ ~ ~ c p ~ ~ ( x , y ) } has

1

I I I. Compactness after omitting types

A. Application to uncountability quantifier.

second-order model % in which 91 + CCxcpi(x) for each i and % UCx$(x), find an elementary extension Since 3x($(x) & Ai<,qi(x)) is true in % for all n, this would follow trivially from the compactness theorem if it were not that we insist that the type-1 universe of % include the type-1 universe of a. In other words, if W is a type-1 object of 'I( and ao, a l , ... are the type-0 objects of a for which 'u ai E W , then the universally quantified infinite disjunction Vx(x 4 W v ViEwx = ai) must hold in % . Since we are making infinitary requirements on % of the sort VV, the usual compactness theorem is of no value by itself. Compactness must be achieved by using the axioms satisfied by 91, especially the countable union axioms. Let Q, be the set of 0 for which 2! k 6 together with $(c), cpi(c) for i E o, where c is a new constant symbol. We assume that cpo(x), cpI(x), ... is a complete list of those one-place formulas such that CCxcpi(x) is true in %. We need to show that Q, U (Aw,%Vx(x 4 W v VaEWx = a)} has a model. If i t does not, then by section 1I.B there exists B(x,c) such that Q, U {3xB(x,c)} has a model, but that, for some W E %, Q,

Q,

C C y [ $ ( y ) ~ V x ( B ( x , y ) ~ x E W ) ] and Vz€ WCCy[$(y)-+Vx(O(x,y)-+x#z)] are true in 91. Since 8 satisfies the countable union axioms, the bounded quantifier Vz E W can be drawn through the CCy quantifier so that

In section 1.F we were confronted with the following situation. Given a

in which 3x($(x) & AiE,cpi(x)) is true.

Vx(O(x,c)+x E W ) and Vx(B(x,c) + x # a ) for alla E W. Therefore, we have that

MODEL THEORY FOR DISSECTING RECURSION THEORY 421

CCy [$ (y ) + Vx(O(x,y) -+ x 4 W)] is true in ‘91. Combining this with the one above we get that CCy[$(y) + VxlO(x,y)] is true in %. This means that one of the cpi(c) is $(c)+ VxlB(x,c), contradicting the fact that @ U {3xB(x,c)} has a model.

B. Barwise compactness.

infinite conjunctions and disjunctions. Assume that Q, is closed under components, the finite propositional connectives, and the distribution of 1’s. In other words, cp E @ iff l c p E @, cp & $ E @ iff cp, $ E @, cp v $ E @ iff cp, $ E Q,, Vxcp(x) E @ implies cp(t) E Q, for all variableless terms t formed from a set of function symbols that includes an infinite number of constants, 3xcpfx) E @ implies cp(t) E @ for those same terms, Aiqj E @ implies cpi E @ for all i, Vicpi E Q, implies cpj E @ for all i, lAicpi E Q, iff V i 1 q E @, lVicpi E Q, iff A i l p i E @, lVxcp(x) E @ iff 3xlcp(x) E Q,, 13xcp(x) E @ iff Vxlcp(x) f 41. Barwise compactness gives a sufficient condition for when a subset of @ has a model. Let the notation cp E 9/ denote that $ is a conjunction and that cp is one of its components, and for a subset z of Q, let $ C C denote that every cp E $ is an element of 2. Let cp, $, B vary through elements of CP. A subset C of Q, has a model i f

(1) every cp 5 C has a model;

and, for any $, $’, $ a conjunction,

(2)

Let Q, be a countable collection of sentences including, possibly, some with

if, for every cp E $, there is a O c z such that B t= $’ v cp,

then there is B 2 C such that 8 k $’ v $.

Note that if there are no infinite connectives in Q,, then condition (2) is automatic and so Barwise compactness reduces to classical compactness. Con- dition (2) has the form of a union axiom or a replacement axiom. Indeed condition (2) is automatically satisfied when Q, can be represented by an admissible set in such a way that z becomes a set. (In an admissible set, k is a C, relation.) To prove Barwise compactness, suppose that C satisfies the two conditions above. We form Zm in the usual manner as outlined in section 1.D except that each stage C, must satisfy condition (1). This is obvious for 2, because Zo is z. To see that it is true for C l , recall the ways that Z1 may be constructed from C,. The difficult one is when Viqj E Co and C, must be zo U {pi} for some i. Such an i can be chosen; for otherwise, for each i, some

428 Th.J . GRILLIOT

$i c Zo U {pi} has no model which means that $ j - {qi} I= l q j ; by condition (2), for some $ C Zo, $ contradiction. One proceeds in this manner to show that every C, satisfies condition (1). (One must form the En’s so that each is only a finite extension of C.) I t follows that 22: has a model and hence that C has a model.

A i l p i ; thus $ A Viqi C Zo has no model, a

J. E. Fenstad, P. G. Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., 1974

AXIOMATIC THEORY OF ENUMERATION

Andrzej GRZEGORCZYK University of Warsaw

Sets may be considered to be simpler than functions. Hence I propose to study first an axiomatic theory with a fundamental epsilon-like notion E . ( x E y means: x belongs to the set with number parametery.) The stronger theory of enumeration of functions may be developed later. Besides the relation E we need some other individual constants or functions (primitive recursive in the standard model) as primitive notions. First the pairing function and its inverses:

then the shifting function S and its axiom:

Al . x E S ( y , z ) - ( x , z ) E y .

The next axiom A2 is a collection of six comprehension schemas (ClLC6). I shall use them in two parallel forms as existential formulas and as definitions of new individual constants (whch is the same in the model):

c1 V u A x ( x E u - p= 8) C'1 xEc-p= \1 /

c 2 V u A x f x E u -p#$) C'2 x E c - p # $

c 3 V u A x ( x E u - p E $ ) C'3 x E c - p E $

c 4 V u A x ( x E u - V y ( x , y ) E a ) C'4 x E c - V y ( x , y ) E a

C5 Vul \x(xEu - (xEa A xEb))C'S x E c - (xEa A x E b )

C6 V u A x ( x E u - ( x E a v x E b ) C ' 6 x E c - ( x E a v x E b ) .

429

430 A. GRZEGORCZYK

In C 1 -C6 cp, $ , a and b are terms containing no occurrences of the variable u , but possibly containing some variables as parameters. In C'l-C'6, cp and $ may contain only x as variable, a and b must be constant terms, and c is a new individual constant.

In order to pass to the enumeration of functions we must postulate the existence of a universal function 21 and the Lachlan function &:

To close the theory we postulate that:

A4 A x V w , u ( A z ( z E x V y ( z , y ) E w ) Az(zEu - 1 z E w ) ) .

(Every element is the projection of a dual element.)

with respect to the equivalence. I f P ( x l , ..., x,) is a polynomial built o f S and x l , ..., x,, then the combinator associated to P is an element ap such that the following formula is a theorem:

The shifting function S allows us to consider some elements as combinators

(1) xES( ... S(ap,xl), ... ) X , ) - XEP(Xl, ...) X n ) .

Combinatory Property. For evety P(x l , ..., x,) there is a combinator associated to P.

Proof. By C3 we can define ap to satisfy the formula:

(... (x,x,) , ..., x I ) E a p -xEP(x l , ..., x,,)

Then applying n times A 1 . we get (1).

I shall mention some other properties,

Fixed point theorem. There is a function 71 which produces fixed points:

AXIOMATIC THEORY OF ENUMERATION 431

(2 ) zES(x ,n (x ) ) - zEn(x) .

Proof. Accordingly to A2 there is an element Q such that:

h t t i n g y = S ( Q , x ) we get:

Hence the function: n(x) = S(S(Q,x) , S (Q ,x ) ) satisfies the formula (2).

Definiti0n.x is dual - V y A z ( z E y - 1 z E x ) .

There are dual elements. There are also elements which are not dual, e.g. the element c satisfying the equivalence:

The supposition that c is dual leads to a contradiction by Russell's argument: if for somey,Az(zEy - l z E z ) , then:yEy - 1 y E y .

Definition. x isfinite - A y ( A z ( z E y + z E x ) + y is dual).

The intuition is that every infinite set contains a non dual set. Every finite element is of course dual. Every element defined by alternation of identities with constants is finite:

x E c - ( x = a l v ... v x = a , ) .

Considering the Boolean operations U, n, and / as defined by means of E as epsilon, we get that the dual elements constitute a Boolean algebra. On the other hand, having non dual elements we can prove that the complement of c defined above (or defined by: xEc - x # x ) is infinite because it contains a non dual element.

432 A. GRZEGORCZYK

There is a sequence of infinitely many different infinite elements:

x E a O - x = x

xEa,+l -x#aO A ... A x f a , ( 3 )

Proof. aoEao. Suppose that aiEai for every i < n . Accordingly to the definition laiEa,+l. Hence ai # a,+l, and by the definition Ea,+l.

Definition. x is closed -Ay, u(Az(zEy -zEu) + EX e y E x ) ) .

Rice’s The0rem.x is closed h aOEx A l ( b o E x ) + x is not dual.

Proof. Suppose x to be dual. Hence for some x‘:

(4) zEx‘ - l ( z E x )

By A2 there is an h such that:

( 5) (n,y)Eh-((rzEx ~ y = b , ) v(nEx‘ ~ y = a ~ ) ) .

The element h considered as a set of pairs is a function which maps {n : nEx} to bo and { H : 1 n E x ) to ao. By A2 for h there is an rn such that:

Using the futed point theorem we get no = ~ ( m ) such that:

(7) zES(nz,nO) - zEnO ,

The element h as a function is total (by (4) and (5)). Hence for no there is a y o such that:


By (6)-(9) and A l we get that:

When x is closed, (10) implies that:

On the other hand (4) and (5) imply that:

(12) (n,y)Eh -+ (nEx -1 yEx) ,

and (8) and (12) imply that:

(1 1) and (13) give a contradiction.

Non-extensionality theorem. For every element x and for every n there exist more than n elements which are extensional with x.

Proof. Suppose that there are only n elementsxl, ..., x, which are extensional withx. This means that:

(14) Az(zEu e z E x ) - ( u = x l v ... V U = X , ) .

According to C’1 and C’6 there is an elementy such that:

uEy - (u=x1 v ... V U = X , ) .

By (14) y is closed. I t is not empty because xEy , and by ( 3 ) there is some u such that 7uEy. According to C’2 and C‘5, y is dual. But this contradicts Riceis the orem.

Notice that for the above argument we need Rice’s theorem in a uniform formulation. Instead of h take: S(S(S(H,x), ao),,bO) for suitable H , and instead of m take S(M, h) for suitableM.

434 A. GRZEGORCZYK

The axiom A3 enables us to consider the enumeration of functions. Ac- cordingly to A3 the element U may be considered as a partial function and we can write:

U(z ,x ) = y instead o f ((x,y),z)E?d.

We shall use the abbreviations and 4.

Kleene’s Sr-theorem. There is a shifting function S’ such that:

U(S ’(a, b), x ) = U(a, (b , x ) ) .

Proof. By A2 there is an element Uo such that:

According to A 1:

By A3a, the elementy is unique. Hence, applying A3b, we get that:

PuttingS’(a,b) = g(S(S(ao,b) ,a)) and applying (15)-(17) we get our theorem.


Proof. By A2 there are elements al, a2, a3 such that:

Putting (Y = &(a3), by A3 and Kleene’s Sr-theorem we can deduce:

Similarly by A2 there are elements e l - e4 such that:

Putting J / = &(e4) we easily verify conditions: c , d, and e .

There is of course a standard arithmetical model for AO-A4 in which the universe consists of natural numbers and “xEy” means: the number x belongs to recursively enumerable set having the numbery. The other model consists of the recursive ordinals and metarecursive enumeration.

Is it a natural theory, or is it perhaps too weak to give more involved interesting theorems? One can trye to develop the hierarchies in it, but perhaps it may be more appropriate to add the weak second order logic.

Another question which seems to be interesting is: taking AO, A 1, how

436 A. GRZEGORCZYK

many instances of A2 can one take and still get a theory compatible with extensionality? If it were possible to define two combinators of the X-calculus, we would have a model for the X-calculus, if the definitions could be compatible with extensionality.

J.E. Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., 1974

POST’S PROBLEM FOR ADMISSIBLE SETS

S.G. SIMPSON The University of California, Berkeley

In 1944 Post proved that there exists a recursively enumerable subset of w having intermediate many-one degree. Post then asked whether there exists a recursively enumerable subset of w having intermediate degree of unsolvability. In 1956 Friedberg and Muchnik solved Post’s problem affirmatively by proving that there exist two recursively enumerable subsets of w having incomparable degrees of unsolvability.

Recently, Sacks and Simpson [ 51 generalized the Friedberg-Muchnik theorem to the context of recursion theory on admissible sets of the form La. Admissible sets of this special form retain the following basic property of w :

the universe is well-ordered by

a recursive relation.

Kreisel [ 2 : p. 1731 asked whether property (W) is in any way essential or “significant” for generalizations of the Friedberg-Muchnik theorem.

In the present paper we partially answer Kreisel’s question. Namely we prove: there exists an admissible set M for which both (W) and the Friedberg- Muchnik theorem fail. However, our proof has one serious defect: it uses AD, the so-called axiom of determinacy, which is actually not an axiom but rather an unsupported (though pragmatically interesting) hypothesis. We conjecture that this defect can be eliminated. In the meantime, for background material on AD, the reader may consult [3].

Research partially supported by NSF Contract GP-24352. See also Kreisel’s 1973 Zentralblatt review of [4] in which Kreisel suggests that

this question must be answered before it is reasonable to start thinking about axiomatics for post-Friedberg recursion theory.

437

438 S.G. SIMPSON

Our main theorem, Theorem 1 below, is stronger than what was stated above, in two ways. First, while our admissible set M will not have property (W), it will have the following property:

the universe is prewellordered by an M-recursive

relation whose initial segments are uniformly

M-finite.

Second, not only will theM-analog of the Friedberg-Muchnik theorem fail, but so will the M-analog of Post’s weaker theorem mentioned in the first sentence of this paper.

Definition. LetM be an admissible set. B S M is complete- E(M) if (i) B is E(M); and (ii) for each E(M) set A E M there is a E(M) relation C C M XM such that

(a) VX3Y C(X,Y> (b) VxVy (C(x,y) + (x € A -y EB) ) .

Remark. I f M is E-uniformizable then (ii) is equivalent to every E(M).set being many-one reducible to B. In any case, (ii) implies that every X(M) set is A ( ( M , B)) .

Theorem 1. Assume AD. Let M = R+, the next admissible set after the continuum. Then every X (M> set is either A (M> or complete.

In particular, the Friedberg-Muchnik theorem fails for R’. Note that R + is a “Friedberg theory” in the sense of Moschovakis [4].

Before proving Theorem 1, we establish some notation. Let R = w w , the real continuum. Let M = R+, the smallest admissible set such that R EM. Put K = On n M . Clearly M = L,(R) where the constructible hierarchy over R is defined by

LO(R) L,+I(R) = {X C L,(R) I X is first-order definable over (L,(R), €)

= transitive closure of R ;

alowing parameters from L,(R)};

POST’S PROBLEM FOR ADMISSIBLE SETS 439

L,(R) = u { LJR) I a! < A} for limit A;

L(R) = U {LJR) I a! an ordinal}.

Some Berkeley set theorists have conjectured that AD ‘‘holds” in L(R) in the sense that plausible large cardinal axioms may be found which imply this. Note that our theorem and proof take place entirely within L(R).

Let H be the Moschovakis system of notations for the ordinals less than K .

Let 11 be the corresponding norm. ThusH is a subset of R and 11 maps H onto K .

For eacha<rc putMa = LJR) andH, = {x €HI 1x1 <a!}.

Facts. 1. H i s complete E(M). 2. Foreacha!<K,M,EMandH,EM. 3. The sequences (Ma I a! < K ) and (Hal a! < K ) are Z(M). Hence II is E(M). 4. For each a! < K there is i E M such that i maps R onto Ma.

The proofs of the above facts do not use AD and are buried in the writings of Moschovakis (see for example [ 11).

Let I and J be subsets of R . We write I I J if there exists a continuous function f : R + R such that x E Z * f ( x ) E J for aU x E R . We shall not actually use AD but only the following consequence of it due to Wadge [7].

Lemma 1. For evety I , J C R either12 J o r J S R - I.

Lemma 2. Let S be a subset of K such that Va! < K (S fl a! EM). m e n S is A (M).

Proof. Assume hypothesis. We shall show that S is Z(M). Put K = {x €HI Ix I E S}. It suffices to show that K isE (M). We shall do this by showing that K suppose H I R - K via f. Then for all x E R we have

H. By Lemma 1 it suffices to show that H $ R - K . So

I personally do not subscribe to this conjecture. However, I am impressed by the fact that a number of people have tried and failed to deduce a contradiction from ZF + AD.

44 0 S.G. SIMPSON

whence H is n ( M ) a contradiction.

Proof of Theorem 1. Let A C M be C(M). In this Case we shall show that A is A (M). Let A be defined over M by

Case I : A " M a EM for all a < K .

x E A - 3yD(x , y )

where D is A (M). For each x EM let h ( x ) be the least 77 such that D(x, y ) holds for some y E Mv. Thus dom (h ) = A and h is C(M). By the Case hypothesis and the admissibility of M , h [M,] is bounded below K for each a < K .

Let g(a) be the least upper bound of h [M,]. Thusg : K -+ K and for each x E M we have

By Lemma 2 g is A (M) hence A is H ( M ) q.e.d. Case IZ: negation of Case I. Let a < K be such that A O M , 4 M . Let

i E M map R ontoM,. Put I = {r E H 1 i(r) E A} . T h u s Z S R and I is L(M) but not A(M). In particular Z $ R - H hence by Lemma 1 H <Z hence Z is complete L(M). From this it is immediate that A is complete Z(M).

The proof of Theorem 1 is complete.

The following theorem is an immediate consequence of Wadge's lemma [7].

Theorem 2. Assume PD, projective determinacy. Let M = H , 1, the herditarily countable sets. Then every L(M) set is either A (M) or complete.

Addendum.

are indispensable for priority arguments. Consider the following slight strengthening of property (PW):

The reader should not conclude from Theorems 1 and 2 that wellorderings

Hence there exists a countable admissible set Mo for which the same conclusion holds. The proof that M o exists does not require the assumption of PD outright but only the assumption that PD has an admissible model.

POST'S PROBLEM FOR ADMISSIBLE SETS 4 4 1

M has A(M) prewellorderings < and < such that the initial segments of < are

uniformly M-finite, and the initial segments

of < are M-small.

where Y C M is said to be M-finite if Y E M, and M-small if Y (l A E M whenever A is E (M). We tentatively propose that an admissible set be called thin if it has property (T). Admissible sets of the form L, are thin via the (pre)wellorderings x <y and f ( x ) <f(y) where f : a + a* is a-recursive and one-one. Not every thin admissible set has property (W). For thin admissible sets one can imitate the proof of Theorem 4.l(i) in [6] yielding a version of the Friedberg-Muchnik theorem.

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PART V

BIBLIOGRAPHY OF GENERALIZED RECURSION THEORY

J.E.Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Cornp., 1974

SOME PAPERS ON GENERALIZED RECURSION THEORY ARRANGED ACCORDING TO SUBJECT MATTER

A number after an author’s name singles out an item in the Uncritical Bibliography. Thus Grilliot (32) refers to: [32] T. Grilliot, Selection functions for recursive functionals, Notre Dame Jour. Formal Log. X (1969) 225-234. TV after an author’s name refers to his paper in this volume.

Recursion in objects of finite type: Aczel and Hinman (TV). Gandy (26, 27). Grilliot (3 1,32,33). Harrington (TV). Kleene (48). MacQueen (75). Moschovakis (80, TV). Platek (92). Sacks (103, TV). Shoenfield (109).

Recursion on ordinals: Jensen and Karp (42). Kin0 and Takeuti (43). Kreisel and Sacks (61). Kripke (62). Lerman (68). Lerman and Sacks (69). Owings(88). Platek(92). Sacks(lOl,lO2). Sacks and Simpson(l05). Shore (1 10,111). Simpson (TV). Takeuti (123,124). TuguC (127).

Admissible sets: Barwise (8, TV). Barwise, Gandy and Moschovakis ( 1 1). Platek (92).

Inductive definability and hyperprojective sets: Aanderaa (TV). Aczel and Richter (TV). Cenzer (TV). Gandy (TV). Grilliot (34). Harrington (TV). Hinman and Moschovakis (40). Moschovakis (81,82). Richter (99). Spector (1 17).

Model theoretic, axiomatic and other views of generalized recursion theory: Fenstad (TV). FraissC (21). Friedman (22,23). Gordon (30). Grilliot (TV). Kreisel(58,59). Kunen (63). Lacombe (65,66). Lambert (67). Montague (76,77). Moschovakis (83). Strong (1 18,119). Wagner (128,129).

445

44 6

AN UNCRITICAL BIBLIOGRAPHY OF PAPERS ON GENERALIZED RECURSION THEORY

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[ 121 S. Bloom, The hyperprojective hierarchy, Zeit. Math. Log. Grund. Math. 16 (1970)

[ 131 G. Boolos and H. Putnam, Degrees of unsolvability of constructible sets of integers,

[ 141 R. Boyd, G. Hensel and H. Putnam, A recursion-theoretic characterisation of the

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[ 161 D.A. Clarke, Hierarchies of predicates of finite types, Memoir Amer. Math. SOC.

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INDEX *

Aanderaa, S., 207, 221,245,263,

Aczel, P., 3, 19,41,4345,49,50, 305,381,445

52, 110, 121,212,220,221- 223,238,263, 266,270-271, 278,287,291,296-297,301, 306,333,340, 381,387,397, 40 1,404,445446

26 1,263,446 Addison, J.W., 218, 220, 250,

Amstislavskii, V., 446

Bachmann, H., 296-297 Barwise, K.J., 97, 107, 110, 112,

114, 120, 122, 196, 198,200, 204, 263,265-266,270-272, 278, 281,287,295,297, 334- 335,354,381,401404,427, 439,441,445-446

Beth, E., 275,297 Blass, A., 260 Bloom, S., 446 Boolos, G., 446 Boyd, R., 446 Bridge, J., 296-297

Cenzer, D., 221-223,225,229, 233,240,247,256,263, 304,381,445

Chang, C.C., 446

Chong, C.T., 190 Clarke, D .A ., 446 Cohen, P., 172

Devlin, K., 123,313-314,316,

Driscoll, G. C., 446 38 1

Enderton, H.B., 446

Feferman, S., 89,93,271,294,

Fenstad, J.E., 220,385,437,441,

Fisher, E., 446 FraissC, R., 407,4 15,418,420,

445,447 Friedberg, R., 172,437 Friedman, H., 112,121,385,404,

296-297,447

445

445,447

Candy, R.O., 4,33,39,40-41,43, 49,50-52,58,78,84,92-93, 107, 110, 121, 200,204,221, 263,265,270-272,297,303, 334-335,339,354,373,381, 402404,439,441,445-447

Gentzen, G., 296 Godel, K., 86,93, 144, 149,

192,258,263

* Boldface numbers refer to title pages of authors’ chapters in this volume.

453

454 INDEX

Gordon, C., 107,270,297,445,

Costanian, R., 286, 298 Grant, P., 281 Grilliot, T.J., 49, 50-52, 58, 68,

447

81-82,93,219, 238, 270-271, 278, 289, 290, 293-295, 298, 340,400,404,405,407,410, 414-415,418,420,421,445, 447

447 Grzegorczyk, A., 271, 298,429,

Hanf, W., 306,381 Harrington, L., 19,25,43, 220.

445.447 Harrison, J., 447 Hensel. G.. 446 Ilinman, P.G.,3,41, 43-45. 49,

50, 52,92, 238,293, 295, 297, 387, 397,401,404,445, 447

Hirano. M., 447

Isles, D., 296-297

Jensen, K., 123-125, 140-141, 144, 163, 166, 191-193, 260, 263, 273, 298.313-314,317. 320,322,324,445,447

Jockusc1i.C.. 191

Karp,C., 273,298, 313-314, 317.320, 322, 324,376, 381,445,447

Kechris, AS., 53. 78,419 Keisler, J.H., 113, 115, 121 Kino, A . , 445,448 Kleene, S.C., 3 , 4 , 7 , 2 9 , 33-34,

36,41,43,52, 54-55,57,67,

71-72,78-79,81-83, 86,88, 93, 167, 221,263, 266,268- 269,286,288-289,298, 302,344,364-367,373,381, 386,400,434435,445-446, 448

Kreider, D. L., 448 Kreisel, G., 86, 93, 166, 193,

198,201,204,232,263, 271,289,298,415,420,

Kripke, S., 98, 152, 257, 263,

Krivine, J.L., 198, 201, 204,

Kunen,K., 110, 121, 196,271,

437,445,447-448

279,302,312,381,445,448

419,420

298.445,448

Lachlan, A . , 190, 193,430 Lacoinbe, D., 407,418, 420,

Lambert , W ., 445,449 Lerman, M., 166, 190, 192-193,

Levy. A. , 84 ,93 ,99 , 105, 115,

445,448-449

445,449

121,258,263,269,306-308, 330-33 1,38 1.449

Linden, T., 318,381 Liu, S.C., 449 Lopez-Escobar, E., 284. 298 Lorenzen, P . , 449 Lyndon, R., 294

Machtey, M., 166, 193,449 MacIntyre, J.M., 190, 193 MacQueen, D.B., 52, 82, 292, 298,

Martin, D., 208, 220 Martin-Lof, P., 296-298 Moldestad, J., 387,398,400, 404

445,449

INDEX 455

Montague, R., 270,298,397,400, 445,449

Morley, M., 115 Moschovakis,Y.N., 7,41,49, 52,

53, 57-58,60,64,68-69,73-74, 79,84,93, 107, 110, 121,200,

266,269,270-271,275278, 204,208,218,220,263,265-

288,292,295,297-299,302, 304,334-335,354,381,385- 392,394-396,398404,407, 415,417-418,420,421-423, 425 ,44544,449

449 Mostowski, A., 271,298-299,447,

Muchnik, A.A., 437 Myhill, J . , 449

Normann, D., 387,398-399 Novikoff, P.S., 250

Ohashi, K., 449 Owings, J . , 445,450

Parikh, R., 450 Paris, J . , 191 Peano, G., 405 Pfeiffer, H., 295, 299 Platek, R., 4-6, 9, 19, 21, 26, 33-35,

41,43-45,50, 52,79,84-85, 93,98,103,152,257,263,279, 286,288,292-293,299,302, 445,450

Post, E., 268,302,437 Putnam, H., 222,264,302-303,

356,381,446,450

Rice, H., 432-433 Richter, W., 22,29,41,4344,48,

52,212,220,221-222,242-

243,263-264,270-271,297, 299,301,302-303,306,333, 340,380-381,445446,450

Robinson, I., 450 Rogers Jr., H., 68, 170, 172, 177,

185, 191, 193,208-210,220, 264,330,381,448

Ryll-Nardzewski, C., 271, 298,447

Sacks, G., 52,53,79,81,92-93, 165-167, 169-170, 182, 189- 190,193,232,262,292,299, 304,381,399,404,437,441, 445,447-450

Scarpellini, B., 450 Schmidt, D., 291,299 Schutte, K., 296,299 Scott, D., 306,381 Shoenfield, J. , 4,9,22,24-28,41,

44-45,52,82,88,93,97, 12 1, 189-190, 193,233,249-250, 258,264,358,381,445,448, 450

193,445,450 Shore, R., 166, 182, 189, 191,

Simpson,S., 165, 166, 170, 182, 190, 192-193,404,437,441, 445,450-45 1

Smullyan, R.M., 268,299 Solovay, R., 260, 263 Spector, C., 67,79,86,93, 178,

190,220,221,233,264,268, 299,303,381,445,451

Strong,H.R., 385,404,445,451 Suzuki, Y ., 45 1

Takahashi, M., 45 1 Takeuti, G., 165,445,448,451 Tait, W.W., 204,447 Tanaka, H., 305,381,451

456 INDEX

Thomason, S.K., 451 Tugue, T., 445,45 I

Vegem, M . , 398,404 Ville, F., 195, 271

Wadge, W., 439,440-441 Wagner, E.G., 385,404,434,

Wang, H., 448,45 1 445,451

[SLFM 079] Generalized Recursion Theory - Fens Tad, P.G.hinman [Studies in Logic and the Foundations of Mathematics] (NH 1974)(T)

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