Some remarks on complete theories modulo forcingbfe12ncu/arctic2.pdf · Some local notation: Given a theory ⌃ and a sentence , in the language of set theory, is a –consequence

Some remarks on complete theories moduloforcing

David Aspero

University of East Anglia

Arctic set theory workshop 2

Some local notation: Given a theory ⌃ and a sentence �, inthe language of set theory, � is a �–consequence from ⌃,denoted

⌃ `� �,

iff for every set–forcing P, if P forces every sentence in ⌃, thenP forces �.

� is for ‘forcing’.

This definition of course makes sense for choices of ⌃ for whichthis can be expressed. For choices of ⌃ where its membershave unbounded Levy complexity this might of course not bedefinable. Also, note that the definition makes sense also forchoices of ⌃ which are not even definable (as long as they arein V).

This gives a notion of logic |=�, possibly weaker than the logic|=GM of the generic multiverse.

We use �–true, �–satisfiable, �–complete and so on, in thenatural intended way. For example, a theory ⌃ is �–completefor a set � of sentences if andonly if for every � 2 � at least one of � |=� � and � |=� ¬� holds.

The usual (Woodin’s) definition of ⌦–logic can be phrased inthe above language, at least for (say) choices of ⌃ which aredefinable over !: Suppose ⌃ is definable over !. Then � is an⌦–consequence of ⌃ if and only if the sentence “for all ordinals↵, if V↵ |= for every 2 ⌃, then V↵ |= �” is a �–truth (whereof course the mention of ⌃ refers to the definition of ⌃).

We may also define relativized versions �� of �–logic fordefinable classes � of posets.

For example T is ��–complete for � iff for every � 2 � it holdsthat either

• for every P 2 �, if �P ' for every ' 2 T , then �P �, or• for every P 2 �, if �P ' for every ' 2 T , then �P ¬�.

⌃2 theories

For ⌃2 theories, i.e., theories of the form (9↵)(V↵ |= T )(equivalently, of the form (9)(H() |= T )) and ⌃2 sentences �,�–logic coincides with ⌦–logic:

T |=� � iff T |=⌦ �

Woodin: If there is a proper class of Woodin and the ⌦Conjecture is true, then:

1 the Pmax–axiom (⇤) is ⌦–satisfiable (equiv., it can alwaysbe obtained by set–forcing over any set–forcing extension).Hence, since (⇤) is �–complete for Th(H(!2)), if the ⌦Conjecture is true under every large cardinal hypothesis,then (⇤) is an axiom which is

• compatible with all large cardinals,• ⌦–complete for Th(H(!2)), and• which can always be set–forced after any set–forcing.

2 There is no ⌦–satisfiable theory which is ⌦–complete forTh(H(�+0 )), where �0 is the least Woodin cardinal.

Woodin: If there is a proper class of Woodin and the Strong ⌦Conjecture is true, then:

1 The ⌦ Conjecture is true.2 All theories which are ⌦–complete for Th((H(!2)) imply

¬CH.3 There is no ⌦–satisfiable theory which is ⌦–complete for

the ⌃23 theory.

In “Incompatible ⌦–complete theories”, JSL 2009, Koellner andWoodin contemplate the following very optimistic scenario:

Could it be, in a large cardinal context, that the following holds?(i) The ⌦ Conjecture is false.(ii) There is a sequence of ⌦–satisfiable ⌃2 theories which are

⌦–complete for the theory of larger and larger (all ?)reasonably specifiable initial segments of the universe.

(iii) All these theories give the same theory of the relevantinitial segments of the universe.

Koellner and Woodin show that if (i) and (ii) hold, then (iii) hasto fail (granting liberal use of large cardinals, as usual).

In “Incompatible ⌦–complete theories”, JSL 2009, Koellner andWoodin contemplate the following very optimistic scenario:

Could it be, in a large cardinal context, that the following holds?(i) The ⌦ Conjecture is false.(ii) There is a sequence of ⌦–satisfiable ⌃2 theories which are

⌦–complete for the theory of larger and larger (all ?)reasonably specifiable initial segments of the universe.

(iii) All these theories give the same theory of the relevantinitial segments of the universe.

Koellner and Woodin show that if (i) and (ii) hold, then (iii) hasto fail (granting liberal use of large cardinals, as usual).

They show that if there is a ⌃2 theory T which, modulo somelarge cardinal assumption LC, is ⌦–satisfiable and ⌦–completefor (say) Th(H()), for = (2@0)+, then there are ⌃2 theoriesT CH, T¬CH which, modulo slightly stronger large cardinalassumption LC0, are ⌦–satisfiable and ⌦–complete forTh(H(!2)) and such that

• T CH ` CH and• T¬CH ` ¬CH.

Proof proceeds by considering the theories that (essentially)say

� “I am a forcing extension of a model of T by Add(!1, 1)”(for T CH)

� “I am a forcing extension of a model of T by Add(!,!2)”(for T¬CH)

The main points are:• Add(!1, 1) and Add(!,!2) are definable over H() from no

parameters and homogeneous (given a sentence , allconditions force the same truth value for ).

• is large enough that all nice names for members ofH(!2) are in H().

CH, ¬CH is clearly not the only pair they can deal with. Asimilar result can be proved for any ⌃2 statement � such thatboth � and ¬� can be forced by some similarly nice forcing.

Proof proceeds by considering the theories that (essentially)say

� “I am a forcing extension of a model of T by Add(!1, 1)”(for T CH)

� “I am a forcing extension of a model of T by Add(!,!2)”(for T¬CH)

The main points are:• Add(!1, 1) and Add(!,!2) are definable over H() from no

parameters and homogeneous (given a sentence , allconditions force the same truth value for ).

• is large enough that all nice names for members ofH(!2) are in H().

CH, ¬CH is clearly not the only pair they can deal with. Asimilar result can be proved for any ⌃2 statement � such thatboth � and ¬� can be forced by some similarly nice forcing.

Down to H(!2)

Consider the question:

Question: Does the existence of an ⌦–satisfiable ⌃2–theory Twhich is ⌦–complete for Th(H(!2)) imply the existence ofanother such theory incompatible with T?

[Koellner–Woodin] does not address this question: their use ofAdd(!,!2) does address the problem of producing a theoryimplying ¬CH, but Add(!1, 1) is not suitable for building atheory implying CH (in our context): If CH fails, then there arenice Add(!1, 1)–names for members of H(!2) which are not inH(!2).

Addressing the question

Plan: Use [Koellner–Woodin]’s result in the following form:

Theorem [Koellner–Woodin] Suppose there is a proper class ofWoodin cardinals. Suppose ' is a ⌃2 large cardinal propertyand is a ⌃2 sentence such that T = ZFC + + “There is aproper class of Wodin cardinals” + “There is a proper class of'–cardinals” is ⌦–complete for Th(H(!2)). Let P ✓ H(!2) be aforcing such that T ⌦–implies that(1) P is definable over H(!2) (from no parameters).(2) P is homogeneous (for every x 2 V, �P '(x) or �P ¬'(x)).(3) P preserves !1 and has the @2–c.c. (in particular every

P–name for a member of H(!2) can be assumed to be inH(!2)).

Let TP be the sentence:

There is (,N,G) such that• is an inaccessible cardinal,• N |= T ,• G is PN–generic over H(!2)N , and• H(!2) = H(!2)N[G].

Then the sentence

ZFC+TP +“There is a proper class of Wodin cardinals” + “Thereis a proper class of '–cardinals”

is ⌦–complete for Th(H(!2)).

Let KA be:

There is a ladder system (C� : � 2 Lim(!1)) such that for everyclub D ✓ !1 there is some � such that

[C�(n),C�(n + 1)) \ D 6= ;

for a tail of n < !.

Here, (C�(n))n<! is the strictly increasing enumeration of C�.

KA follows from Club Guessing at !1 and is weaker than it.

This principle is something attributed to Kunen and calledKunen’s Axiom.

Consider the following forcing PKA.DefinitionConditions in PKA are pairs q = (~c, ~D) with the followingproperties.

• ~c = (c� : � 2 S) is a finite function with S ✓ Lim(!1) andsuch that for every � 2 S, c� ✓ ! ⇥ � is a finite strictlyincreasing function.

• ~D = (D� : � 2 T ) is such that T ✓ Lim(!1) is finite and forevery �, D is a finite set of cofinal subsets of � of order type!.

Given (~c0, ~D0), (~c1, ~D1) 2 PKA, (~c1, ~D1) (~c0, ~D0) iff:

(1) dom(~c0) ✓ dom(~c1) and c0� ✓ c1

� for every � 2 dom(~c0).

(2) For every dom(~D0) ✓ dom(~D1) and every � 2 dom(~D0),D0

� ✓ D1� .

(3) For every � 2 dom(~c0) and n, n + 1 2 dom(c1� ) \ dom(c0

� ), if� 2 dom(~D0), then [c1

� (n), c1� (n + 1)) \ D 6= ; for every

D 2 D0� .

FactPKA has the following properties.

1 It is definable over H(!2) from no parameters.2 It is homogeneous.3 It is proper and has the @2–c.c.4 It forces KA.

Let B denote Baumgartner’s forcing for adding a club of !1 withfinite conditions. Adding many Baumgartner clubs of !1:

Given a set X of ordinals, there is a forcing, which I will denoteby AddB(X ), with the following properties.

(1) For every AddB(X )–generic G and every ↵ 2 X one cannaturally extract a Baumgartner club CG

↵ from G. Moreover,CG↵ 6= CG

↵0 for ↵ 6= ↵0 in X .(2) AddB(X ) is proper and has the @2–c.c.(3) For every partition (X0,X1) of X into nonempty pieces,

AddB(X ) ⇠= AddB(X0)⇥ AddB(X1). In particular, if G isAddB(X )–generic and ↵ 6= ↵0 are in X , then CG

↵ isB–generic over V[CG

↵0 ].(4) AddB(X ) is homogeneous.

It follows from (1), (2) and (3) that if ot(X ) � !2, then AddB(X )forces ¬KA.

Let B denote Baumgartner’s forcing for adding a club of !1 withfinite conditions. Adding many Baumgartner clubs of !1:

Given a set X of ordinals, there is a forcing, which I will denoteby AddB(X ), with the following properties.

(1) For every AddB(X )–generic G and every ↵ 2 X one cannaturally extract a Baumgartner club CG

↵ from G. Moreover,CG↵ 6= CG

↵0 for ↵ 6= ↵0 in X .(2) AddB(X ) is proper and has the @2–c.c.(3) For every partition (X0,X1) of X into nonempty pieces,

AddB(X ) ⇠= AddB(X0)⇥ AddB(X1). In particular, if G isAddB(X )–generic and ↵ 6= ↵0 are in X , then CG

↵ isB–generic over V[CG

↵0 ].(4) AddB(X ) is homogeneous.

It follows from (1), (2) and (3) that if ot(X ) � !2, then AddB(X )forces ¬KA.

Defininition: Let X be a set of ordinals. AddB(X ) is the followingforcing: Conditions in AddB(X ) are pairs of the form p = (f ,F)with the following properties.(1) f is a finite function with dom(f ) ✓ X and such that

f (↵) 2 B for every ↵ 2 dom(f ).(2) F is a finite function with dom(F) ✓ !1 such that for every

� 2 dom(F),(a) � is a countable indecomposable ordinal,(b) F(�) is a countable subset of X ,(c) � 2 dom(f (↵)) and f (↵)(�) = � for all ↵ 2 dom(f ) \ F(�),

and(d) for every �0 2 dom(F � �) and every ↵ 2 F (�),

rank(F(�0),↵) < �.

Given (f0,F0), (f1,F1) 2 AddB(X ), (f1,F1) extends (f0,F0) iff• dom(f0) ✓ dom(f1) and f0(↵) ✓ f1(↵) for every ↵ 2 dom(f0),

and• dom(F0) ✓ dom(F1) and F0(�) ✓ F1(�) for every� 2 dom(F0).

We immediately get the following.

TheoremSuppose there is, under some sufficiently strong large cardinalassumption LC, a recursively enumerable ⌦–satisfiable⌃2–theory T such that T is ⌦–complete for Th(H(!2)).Then there are, under a slightly stronger large cardinalassumption LC0, ⌦–satisfiable recursively enumerable⌃2–theories T KA and T¬KA such that

1 T KA and T¬KA are both ⌦–complete for the theory ofH(!2).

2 T KA ` KA3 T¬KA ` ¬KA

Proof: By the above together with an application of theKoellner–Woodin argument with PKA (for T KA) and withAddB(!2) (for T¬KA). ⇤

Many natural questions spring from here. For example:

• Do all ⌦–satisfiable recursive ⌃2–theories which are⌦–complete for the theory of H(!2) imply the existence ofa Suslin tree?

(If yes, then of course the Pmax axiom (⇤) could not be⌦–satisfiable.)

• Do all ⌦–satisfiable recursive ⌃2–theories which are⌦–complete for the theory of H(!2) imply ¬CH?

Further advertising AddB(X ):Collapsing exactly @3

AddB(X ) has other interesting uses. Here is an example:

U. Abraham proves the following in On forcing without thecontinuum hypothesis, J. Symbolic Logic, vol. 48, 3 (1983),658–661:

Theorem(Abraham) (ZFC) There is a poset P collapsing !2 andpreserving all other cardinals.Abraham’s forcing is built as follows: Let A ✓ !2 such that!L[A]

2 = !V2 (and then of course !L[A]

1 = !V1 ). Then

P = Add(!,!1) ⇤ Coll(!1,!2)LA][G]

• P collapses !2 and has a dense subset of size @2.• Preservation of !1: If G is Add(!,!1)–generic,

Coll(!1,!2)L[A][G] is �–closed in L[A][G], but certainly not ingeneral in V[G]. However, Coll(!1,!2)L[A][G] is�–distribuitive in V[G]:

Given a Coll(!1,!2)L[A][G]–condition p and aColl(!1,!2)L[A][G]–name F in V[G] for a functionF : ! �! Ord, we may find a condition p0 p inColl(!1,!2)L[A][G] deciding all of F . We use the Cohen realsadded by G in order to guide this construction (in V[G]).

Question(in Abraham’s paper) Can this be extended to higher cardinals?In particular, is there, in ZFC, a forcing collapsing exactly @3?

Theorem(ZFC) There is a poset P collapsing !3 and preserving all othercardinals.

Construction of P: There is a poset P0 of size @2 preservingcardinals and adding a partial ⇤!1–sequence (C↵ : ↵ 2 S)such that {↵ 2 S : cf(↵) = !1} is stationary. In V1 = VP0 wemay fix A ✓ !3 such that !L[A]

3 = !3 and such that for everycardinal ✓ > !3, the set of N � H(✓) such that

• |N| = @1,• N \ H(!3)L[A] 2 L[A], and• N \ H(!3)L[A] is internally approachable in L[A]

is a stationary subset of [H(✓)]@1 . Still in V1, letP1 = AddB(!1)L[A] ⇤ Q, where Q is, in L[A]AddB(!1), a name forColl(!2,!3)L[A][G]. Our poset will be P = P0 ⇤ P1, where P1 is aP0–name for P1. ⇤

Question: Is there, in ZFC, a forcing notion collapsing @4 andpreserving all other cardinals? What about for any 6= !1, !2,!3?

General theories

A remarkable recent result of Matteo Viale concerning�–completeness of forcing axioms:

Viale defines a technical strengthening MM+++ of the forcingaxiom MM++ (= Martin’s Maximum++) fromForeman–Magidor–Shelah and shows that, under the existenceof a proper class of sufficiently strong large cardinals (LC), anytwo set–forcing extensions preserving stationary subsets of !1and satisfying MM+++ have the same theory of L(P(!1)). Inparticular,

MM+++ + LC

is �SSP–satisfiable (modulo LC) and it is �SSP–complete for thetheory of L(P(!1)) and hence for the theory of H(!2), whereSSP = {P : P preserves stationary subsets of !1}.

QuestionWhat is the foundational significance of this type of results?

Is it that they show that there is a recursive theory, namelyMM++++ LC, which is ��–satisfiable, granting LC of course,and which is ��–complete for the theory of H(!2), for areasonable class � of posets?

Well, V = L is of course also another such theory (for any � infact). On the other hand, V = L has a property which makes ita bad �–complete theory:

(1) V = L is not compatible with most large cardinals.









MM++++ LC of course does not have the undesirable property(1) (it is compatible with all standard large cardinal notions).

Another recursive theory �–complete for Th(V): The GroundAxiom, i.e., “V is not a set–generic extension of any groundmodel properly contained in V.”

The Ground Axiom is compatible with all (traditional) largecardinals and it is obviously �-complete for Th(V).But it has another feature that makes it a bad axiom for us:

(2) The Ground Axiom cannot be forced after any non–trivialforcing, so it is ��–complete for vacuous (uninteresting)reasons.









So it seems MM++++ LC is a good axiom (at least) in thesense that it satisfies the following:

(A) It is �SSP–complete for the theory of H(!2).(B) It is compatible with all large cardinal axioms.(C) It can always be forced by SSP set–forcing after any SSP

set–forcing extension in the presence of enough large cardinals.

My final observation will be that, under reasonable assumptions(for example the existence of a proper class of Woodincardinals and GCH holding on a tail of cardinals), satisfying(A)–(C) should not be a sufficient reason for a theory to begood (with � = “all posets” instead of � = SSP).

So it seems MM++++ LC is a good axiom (at least) in thesense that it satisfies the following:

(A) It is �SSP–complete for the theory of H(!2).(B) It is compatible with all large cardinal axioms.(C) It can always be forced by SSP set–forcing after any SSP

set–forcing extension in the presence of enough large cardinals.

My final observation will be that, under reasonable assumptions(for example the existence of a proper class of Woodincardinals and GCH holding on a tail of cardinals), satisfying(A)–(C) should not be a sufficient reason for a theory to begood (with � = “all posets” instead of � = SSP).

Observation: Suppose• there is a proper class of Woodin cardinals and• GCH holds on a tail of cardinals.

Let ✓ be any ⌃2–definable cardinal (!2, @!5 , the firstmeasurable cardinal, etc.). Let T = Th(H(✓)). There is aclass–forcing extension VP satisfying the following:

(1) H(✓) |= T(2) There is a proper class of Woodin cardinals.(3) GCH holds everywhere except on unboundedly many

intervals of the form [@↵,@↵+!), ↵ a limit ordinal, and oneach of these intervals [@↵,@↵+!),

{n < ! : 2@↵+n+1 = @↵+n+2}

codes the theory of H(✓).

Moreover, after any set–forcing extension of VP there is afurther set–forcing extension satisfying (1)–(3).

Now suppose• there is a proper class of Woodin cardinals and• GCH holds on a tail of cardinals.

Take your favourite recursive theory T0 such that H(✓) |= T0 canalways be forced by set–forcing. By going to a set–forcingextension we may assume H(✓) |= T0. By the observation wemay go to a class–forcing extension such that:

(1) H(✓) |= T(2) There is a proper class of Woodin cardinals.(3) GCH holds everywhere except on unboundedly many


{n < ! : 2@↵+n+1 = @↵+n+2}

codes the theory of H(✓),

and such that (1)–(3) can be recovered by set–forcing after anyset–forcing.

Consider the sentence �✓,T0 which is the conjunction of (↵) and(�):

(↵) On a cofinal segment of cardinals GCH holds everywhereexcept on unboundedly many intervals of the form[@↵,@↵+!), ↵ a limit ordinal and, on each of these intervals[@↵,@↵+!), the set

{n < ! : 2@↵+n+1 = @↵+n+2}

codes the theory of H(✓).(�) H(✓) |= T0

�✓,T0 is a ⌃4 sentence.

Now, in the current model,1 �✓,T0 is �–complete for Th(H(✓)), and2 �✓,T0 can always be forced by set–forcing after any

set–forcing.

So �✓,T0 is good in that it satisfies (A)–(C), i.e.,

(A) It is �–complete for the theory of H(✓).(B) It is compatible with all large cardinal axioms.(C) It can always be forced by set–forcing after any set–

forcing extension in the presence of enough large cardinals.

But �✓,T0 implies H(✓) |= T0 and T0 was arbitrary...

In conclusion: Every �–satisfiable theory H(✓) |= T0 can beextended to a ⌃4–theory �✓,T0 such that, in a class–forcingextension of V, �✓,T0 satisfies (A)–(C).

Proof of the Observation: Let r ✓ ! code Th(H(✓)). Build anEaston–support ‘progressively closed’ class–forcing iterationleaving H(✓) – and the fact that H(✓)V is H(✓) – untouched andforcing

{n < ! : 2@↵+n+1 = @↵+n+2} = r

for unboundedly many ↵. Make sure GCH is preserved outsideof these intervals (on a tail of cardinals).

By leaving long enough gaps in which we do nothing, makesure that your favourite large cardinals are preserved. Inparticular make sure in the end there is proper class of Woodincardinals.

Call this class–forcing extension W. In W:(1) H(T) |= T0,(2) There is a proper class of Woodin cardinals.(3) GCH holds everywhere except on unboundedly many


{n < ! : 2@↵+n+1 = @↵+n+2}

codes the theory of H(✓).Now let P 2 W a set–forcing. If � > |P| is a Woodin cardinal,then there is a condition p in PWP

<� (this is the full stationarytower at �) such that forcing with PWP

<� � p over WP resurrects“H(✓) |= T0” (H(✓) refers here to the new H(✓) in WP⇤(P<��p)).

(This is Woodin’s theorem on resurrection of ⌃2 truth.) ⇤

An advocate of (A)–(C) might object the following:

(i) We had to climb up to another generic multiverse (aproper–class forcing extension) for this argument.

(ii) We started from a ⌃2 theory (T0) and produced a ⌃4theory (�✓,T0). In particular, the argument does not apply toMM+++ (which is a ⇧3 theory), although it does apply toWoodin’s Pmax axiom (⇤) (in the presence of largecardinals, (⇤) is “L(P(!1)) is a Pmax extension of L(R)”, andis �–complete for Th(H(!2)); it is a very reasonableconjecture that something like MM++ implies (⇤)).

(iii) It works with � = “all posets” but not with � = SSP, forexample. Also in this sense it does not apply directly to theMM+++ result.

(iv) We need (something like) GCH holding on a tail ofcardinals. This is not a traditional large cardinal hypothesis.

We can address the objection realised in (ii) in a weak way, asfollows:

Well, we can still run basically the same argument as in the firstpart of the talk and argue that, if P ✓ H(!2) is definable,homogeneous, preserves !1, and has the @2–c.c., then

”I am a generic of a model of MM+++ by P” + LC

is �–complete for Th(H(!2)) and implies whatever P forcesover H(!2) (which could be b = !1, “There is a Suslin tree”, orKA).

Of course the argument does not apply to arbitrary theories. Itis not ruled out, for example, that all such complete theoriesimply ¬CH.

(iv) motivates various questions:

(1) Is there any reasonable axiom consistent with all largecardinals and blocking the possibility of a coding as in thisconstruction? This axiom would imply that the universe isvery “chaotic” or “inhomogeneous”.

(2) Woodin: Say that a ⌃2 statement ' is nice if is a �–truththat for every ↵ there is a set–forcing preserving V↵ andforcing '. So every finite collection of nice statements canbe forced to hold. Is “Every nice statement is true”consistent? This question goes in the way of addressing(1).

Some remarks on complete theories modulo forcingbfe12ncu/arctic2.pdf · Some local notation: Given a theory ⌃ and a sentence , in the language of set theory, is a –consequence

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