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Interpreting Frege's Grundgesetze in an Adaptation of Quine's New Foundations
Ryan Beaton
Department of Mathematics and Statistics,
McGill University, Montréal
Québec, Canada
August, 2004
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements of the degree of
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Abstract
We first give a modern presentation of the formai language of Frege's Grundgesetze.
There follows a comparison of the motivations for Frege's "Cumulative Type The
ory" and for Russell's Type Theory and of the basic arithmetical definitions in each.
Quine's New Foundations and, in particular, extensions of Jensen's modification,
NFU, are introduced and consistency results are discussed. Finally, an interpretation
is given in an NFU framework of a modified form of the Grundgesetze theory. It
is shown that an "Axiom of Counting" necessary for arithmetic in NFU is needed
in an analogous way for arithmetic in our interpretation; it is further demonstrated
that from the statement of this axiom in NFU, the appropriate analogue is provable
for our interpretation. The development of arithmetic in an NFU framework is seen
essentially to be that intended by Frege in the Grundgesetze.
11
Résumé
En premier lieu, nous donnons une présentation moderne du langage du texte Grundge
setze de Frege. Nous procédons à une comparaison entre la "théorie cumulative des
types" de Frege et la "théorie des types" de Russell et des définitions arithmétiques
de chacune. Nous discutons, par la suite, des "Nouvelles Fondations" de Quine
et de certaines extensions de la modification, NFU, de Jensen. Finalement, nous
interprétons, dans le cadre de NFU, une forme modifiée de la théorie des Grundge
setze. Nous démontrons qu'un "axiome de dénombrement," essentiel à l'arithmétique
dans NFU, est d'une manière analogue nécessaire à l'arithmétique dans notre in
terprétation. Nous prouvons qu'à partir de l'énoncé de cet axiome dans NFU, l'énoncé
analogue est démontrable pour notre interprétation. Il est clair que le développement
de l'arithmétique dans le cadre de NFU est essentiellement celui visé par Frege dans
les Grundgesetze.
III
lV
Acknowledgments
l thank Professor Michael Makkai for his many hours of help and am deeply appre
ciative of his enthusiastic support for this project. l am grateful for the financial
support l have received from the National Sciences and Engineering Research Coun
cil (NSERC) of Canada and from Dr. Richard H. Tomlinson, benefactor of McGill
University. The generous contributions of NSERC and Dr. Tomlinson have funded
my two years as a Master's student at McGill. Thank you finally to aU the professors
of the department from whom l've had the opportunity to learn and to the friendly
staff of the McGill Mathematics Department.
v
VI
Table of Contents
Abstract
Résumé
Acknowiedgments
Introduction
1 The FormaI and Meta-Language of the Grundgesetze
1.1 The "Cumulative Type Theory" Behind the Grundgeseize
1.2 The Theory GG .
1.2.1 Types ..
1.2.2
1.2.3
Basic definitions: primitives, terms and sentences
Axioms and Rules of Inference .
1.3 Interpretation of Primitives ....
2 Type Theory and the Grundgesetze: Basic Arithmetic and the Para-
i
iii
v
1
5
6
11
11
12
17
20
doxes 25
2.1 Frege's Levels and Russell's Types. 26
2.1.1 The Theory of Types . . . . 26
2.1.2 Frege's Theory of Levels and The Definition of Number
3.1.1 A Concise Statement of Quine's New Foundations
3.2 The Consistency of NF and Variants ...
3.2.1 Specker Models and Models of NF
3.2.2 Jensen's New Foundations with Urelements .
3.3 Constructing Models of NFU .......... .
3.3.1 Weak Zermelo Set Theory and Extensions
3.4 Arithmetic in NFU . . . . . . .
3.4.1 The Axiom of Counting
4 Interpreting GGree in NFUT
4.1 The Language of GGree . ..
4.2 NF UT, Interpreting GGree Primitives, and F-stratijication
4.2.1 Extending the Language of NFU: NFUT .
4.2.2 Translating GGree into NFUT
4.2.3 F-Stratification ....
4.3 Recovering Frege's Arithmetic
4.3.1 Basic Laws of GGree
4.3.2 Arithmetic in GGree
4.3.3 The Axiom of Counting
Conclusion
Vlll
30
35
38
41
43
43
46
47
47
49
51
52
55
56
59
59
61
62
67
71
77
77
80
84
93
A Frege's Cumulative Type Theory 95
IX
x
Introduction
Largely ignored in his own lifetime, recognition came posthumously to Gottlob Frege.
It is now generally acknowledged that if anyone can rightfully be called the father
of modern logic, then surely it is Frege. In 1879, his first publication, Begriffss
chrift ("Concept Writing"), introduced, among other things, the modern notions of
quantification, formaI language and formaI derivation, and provided precise logical
definitions of "sequence," "following in a sequence," etc., paving the way for his log
ical reconstruction of arithmetic.
Begriffsschrift was followed by several publications of a more philosophical tone
supporting Frege's "logicist program" of reducing basic mathematics to a branch
of pure logic. This line of thinking culminated in the Grundgesetze der Arithmetik
("The Basic Laws of Arithmetic"), the first volume of which appeared in 1893 and was
written by Frege at the height of his intellectual powers. Satisfied with his lengthy
philosophical reflection, he wrote the Grundgesetze as an ultimate vindication for
logicism, a mathematical book which develops arithmetic from purely logical founda
tions. As the second volume, treating rational and real numbers, was being published
in 1902, Frege received a letter from the young Bertrand Russell, pointing out the
inconsistency in the formaI system of the Grundgesetze.
The Grundgesetze came to be seen as a brilliant failure, including by Frege himself,
who published litt le after 1902, certainly nothing of the same scope or ambition. The
extent to which his work does indeed accomplish the founding of mathematics in logic
1
and sets the stage for modern mathematicallogic has slowly come to be appreciated.
There are now countless books discussing Frege's influence on a variety of fields,
including, of course, logic.
However, it remains an interesting question to what extent Frege's actual devel
opment of arithmetic in the Grundgesetze is consistent and viable. SpecificaUy, is it
possible to interpret a sufficient part of the Grundgesetze theory, hereafter denoted
by "GG," in a consistent set theoryl in such a way that Frege's basic definitions and
proofs regarding cardinals, natural numbers, arithmetic, etc., can be reproduced?2
Frege can hardly be faulted for having implicitly assumed a naive set theory in 1893;
(un)fortunately, he had the brilliance to capture in a precise, formaI system the in
consistency that had been lurking. There was nowhere to hide the inconsistency once
spotted and Frege's system coUapsed; however, this alone does not mean the model
of arithmetic he describes and attempts unsuccessfuUy to axiomatize is incoherent.
After aU, Frege's definitions of cardinal, successor, natural number, etc., are very
natural, more so perhaps than the usual ZF definitions.
The main purpose of this essay is twofold. For one, we aim to give a set-theoretic
interpretation of GGree , a modified form of the theory GG, that will aUow, as much
as possible, the development of arithmetic as it is carried out in the Grundgesetze.
A ZF -style set theory is completely unsuited to this. For instance, Frege takes the
natural number n to be the class of aU classes having n elements.3 In particular,
IThat is, one we are reasonably confident is consistent.
2Boolos [4], Wright [33] and others have written extensively on set theories and theories of second
order logic equivalent to fragments of GG; however we are interested here especially in interpreting
Frege's own language and giving definitions of natural number, etc., in this language, so that we may,
for one thing, bring out conceptual similarities between the Grundgesetze, Russell's Type Theory and
Quine's New Foundations.
3Roughly speaking. This is indeed the conception of natural number Frege arrived at in Grundla-
gen der Arithmetik ("Foundations of Arithmetic") (1884) and still held at the time of the Grundge
setze. However, technical features of the for mal language he presents in the Grundgesetze force Frege
2
the cardinal "One" is "the class of all unit classes." No such thing exists in ZFC,
and while there is such a class in, e.g., Morse-Kelley set theory, we still cannot form
the class of all natural numbers, so defined, in MK. The iterative conception of set
motivating these set theories is completely incompatible with GG.
On the other hand, there are deep conceptual similarities between Frege's system
and Type Theory, as first introduced by Whitehead and Russell. This is especially
clear from Frege's discussion of "concept levels," from which 1 extract a "Cumulative
Type Theory," denoted "CTT" (see Appendix). Russell's theory of types, even as
simplified by Ramsey, can be rather awkward in mathematical practice, as it main
tains a strict separation of types and strong restrictions on expressions considered
grammatical or meaningful. There is also a proliferation of "copies of sets," e.g.,
there exists an empty set and a universal set for every type 2': 1. A further sim
plification was proposed by Quine in his New Foundations article. The system of
New Foundations, NF, rejects the separation of types to the extent that it formally
contains a single type; however, the key notion of "stratification" is drawn from Type
Theory. Roughly speaking, a formula is stratified if it contains no free variable in
stantiating different types, i.e., within this single formula. For instance, x E x is an
unstratified formula. The details are given in Chapter 3. The main feature of NF
is that it restricts comprehension to stratified formulas. Unlike Type Theory, NF
recognizes x E x as a meaningful formula; however, no comprehension principle is
available from which to form a set of objects satisfying it (even if these objects are
restricted to the elements of sorne set).
This stratified comprehension principle creates a set theory very much different
from ZFC, and ideal for interpreting GGree . The consistency of NF is still very much
in question, but modifications have yielded interesting consistency results. It is in
an extension, our NF UT, of the best known such modification, Jensen's NFU (New
to modify the definition slightly; this is discussed fully in Chapter 2.
3
Foundations with Urelements) , that we give our interpretation of GGree . It should
be mentioned that, as the modern day doyen of NF studies, Randall Holmes, puts it,
NFU is not being promoted as a replacement for ZFC, but "those who are interested
in studying the foundations [of mathematics] should be familiar with alternatives." [16]
ZFC itself was developed in the aftermath of Frege's great "failure," which led to the
first deep studies of the role of the paradoxes in mathematicallogic. An understanding
of alternative set theories and logical foundations of arithmetic, su ch as Frege's, can
only deepen our understanding of set theory in general. The second of the principle
aims of this essay is therefore to draw the conceptual similarities between GG, TT,
NF and NFU4, in particular, to show that arithmetic in NFU is arithmetic as Frege
intended.
The main text of this essay is divided roughly as follows. Chapter 1 presents the
formaI language GG. The presentation is preceded by an informaI discussion of CTT,
which is itself given a formaI presentation in the Appendix. Chapter 2 discusses the
relation between CTT and simple Type Theory, TT, and basic arithmetical definitions
therein. Russell's paradox is discussed and derived for GG. Chapter 3 is a discussion
of NF and its modifications and extensions. Chapter 4 presents our main results. The
primitives of GGree are defined in NFUT. The fragment of the GG axioms recovered in
GGree is discussed. The development of arithmetic in the reconstruction is presented,
with basic results relating this development to the usual development in NFU proper.
Finally, we give some concluding remarks.
4Cocchiarella [5] is a fairly in-depth philosophical and logical analysis of these similarities and
a nice complement to some parts of the present essay. In some instances, we develop and make
mathematically precise certain statements in [5]; see 2.2.1 and the Appendix. Indeed, this present
essay may be seen as exploring a middle ground between the two paths of research in logicjFrege's
studies as found in Boolos [4] and as found in Cocchiarella [5].
4
Chapter 1
The FormaI and Meta-Language of
the Grundgesetze
The main purpose of this chapter is to introduce the formaI language of the Grundge
setze and to explain some of its more prominent features. First, however, some
discussion is needed of the background logical theory to which Frege's study has
brought him at the time of the Grundgesetze. This theory, which Frege outlines in
the opening sections of the Grundgesetze is extremely unwieldy for the purposes of
even the most basic arithmetic and Frege greatly simplifies things before presenting
the formaI language of his system. However, an understanding of Frege's "Cumula
tive Type Theory" will be important for discussions in Chapter 2 and subsequent.
To this, the first section of this chapter is therefore devoted.
5
1.1 The "Cumulative Type Theory" Behind the
Grundgesetze
Much of the brilliance and originality in his work, Frege derives ultimately from a
rigourous analysis of language. In logic, Frege takes an analysis of statements as his
starting point. A major advance in logic came with his separation of statement into
argument and function rather than subject and predicate. While it may be hard
to appreciate the depth of such a change now, this break with thousands of years
of tradition paved the way for the first satisfactory presentation of quantification in
logic, in Frege's 1879 publication, Begriffsschrift. By the time of the Grundgesetze,
this linguistic analysis led Frege to a theory of "concept levels," which he discusses at
the beginning of the Grundgesetze (cf. §§1-5, 21-33), and which l will call "Cumulative
Type Theory," or "CTT."! For instance, let us consider the statement, "Hydrogen is
lighter than oxygen." If we remove the object name "Hydrogen" from this statement,
replacing it with the variable symbol "x" , we obtain the name of a function, Frege says,
namely "x is lighter than oxygen" , whose value is true or false for every appropriate
argument. As only an object name in the place of "x" is appropriate (to obtain
a meaningful statement), Frege calls the function named above a unary first-order
function. And clearly then "x is lighter than y" is the name of a binary first-order
function. The types of these two functions are to be sharply distinguished from each
other as well as from the types of second-order functions. One example of the latter is
"There exists an x such that f(x)", where x is an object variable and f is a (variable)
function of x. Now, only the name of a unary first-order function is appropriate
1 l have not come across the term "Cumulative Type Theory" anywhere, but it seems very natural
and may weIl have been coined previously. In the Appendix, l give a presentation of the formaI
language CTT, based on my interpretation of Frege's informaI discussion. Terminology and notation
in the Appendix rest to some extent on definitions and explanations in the main text and it is assumed
the reader has read the main chapters before the Appendix.
6
to take the place of "f" in the above. Clearly we can continue on with third-Ievel
functions, etc., (though examples may become less natural). Note that Frege also
allows "mixed level" functions: e.g., "x is such that f(x)" is a name for a function
requiring two arguments, the first of object type and the second of unary first-order
function type. Again, the type of such a function is to be sharply distinguished from
the types of the functions discussed above.
Three points should be mentioned. First, in relation to statements, Frege uses
the word "function" in a literaI sense. That is, his analysis of statements led him
to the conclusion that concepts, such as "is lighter than oxygen", really are special
cases of functions, of the same nature as those familiar in mathematics. Just as "x
+ 2" is a name for a function taking arguments of object type to values of object
type2 , so Frege views the function named by "x is lighter than oxygen". The latter
is a special case of first-order unary function. For such a first-order function, that
is, for one whose value for every argument is a truth value, Frege reserves the name
concept. A first-order binary function whose value for every pair of arguments is a
truth value Frege calls a relation. In particular, then, Frege is led to the conclusion
that truth values, the Ttue and the False, or T and -.l, are objects of the same nature
as, e.g., numbers. This has, for one thing, the effect that logical connectives (officially:
material implication, negation and univers al quantification in the Grundgesetze) are
functions, and each statement is a name for T or -.l. In fact, Frege includes a further
truth-functional connective, the truth function --x, a first-order concept such that,
for each object a, --a = T if a = T and --a = -.l if a i= T 3. Since the logical
2Frege insists that every such function must be defined for all objects "that there really are,"
including, e.g., Julius Caesar. Though this causes Frege philosophical spasms, it is never an issue in
mathematical practice, and we may keep in mind a universe of purely mathematical entities when
working with GG, as Frege himself inevitably does. 3 Aczel [1] gives an interesting analysis of --x as an "internaI truth definition" that is the
ultimate source of inconsistency in GG.
7
connectives are functions, and GG will contain only functions as primitives4, there
are no formulas in the language, only terms. As we will see, any term, cp, can occur
in the argument position of --x. Therefore, for any term cp, --cp equals T or
1- (for aIl values of the free variables in CP); we say that --cp is a GG statement.
An axiom is interpreted as saying that a certain statement equals T and rules of
inference as saying that, given a (finite) set of statements, one of which is identified
as the conclusion and the remaining as premises, if, for a given set of values of aIl free
variables in the set of statements, we have that each of the premises equals T, then
so does the conclusion. Derivations are defined as usual; see 1.2.2.
The second point is that the number of fun ct ion types will multiply very rapidly,
as any two functions are of the same type iff they have the same arity, n, and, for
each i ~ n, the ith variable of each is of the same type. By the definition of function
in GG, the value of any function for each appropriate set of arguments is an object.
So, for example, the differential operator, d~~) Iy is of mixed level, needing a unary
first-order function, F(x) and an object, a, as arguments (for the variables f(x) and
y, respectively) in order to return an object. This framework is very cluttered and
completely impractical for "actually doing" mathematics, even simple arithmetic.
The situation is resolved in large part through the introduction of an abstraction
operator. Frege takes it as a law of logic that to every first-order concept there
corresponds a collection, or class, itself of object type, who se members are exactly
those objects which fall under (satisfy) the given concept. Since his analysis has
brought Frege to the conclusion that concepts are a special case of unary first-order
functions, he extends this basic law of logic to aIl unary first-order functions, with
4Including equality. We will denote Frege's "functional" equality symbol by "~,, and keep "=" for
the usual equality predicate. Note that the above discussion of --x concerns the meta-theoretical
interpretation of --x and the symbol "=" has its usual meaning. Note also that there are no
constant names for T and ..L in GG; they are used here, as by Frege, to discuss the meta-theoretical
interpretation of the GG primitive symbols.
8
the slight modification that to each of these functions there corresponds a graph,
or "course-of-values" ("Werthverlauf"), as Frege puts it. In the special case of a
concept, we therefore have the graph of its characteristic function rather than sim ply
the collection of objects falling under it. The abstraction operator is a second-order
unary function, taking first-order unary functions to their corresponding graphs, the
graphs themselves being objects. But in this way, any second-order unary function has
a corresponding first-order function, in which the variable ranges over the graphs of
first-order functions, and this first-order function has a corresponding graph (object).
Thus we can "reduce" all higher order functions to their corresponding graphs5 . This
is explained in detail in Chapter 2, where an explicit example of such a reduction
is also given in discussing Frege's definition of "natural number". Iteration of the
abstraction operator, as explained in 1.3, is used for functions of higher arity. (Note
the difference between higher arity and higher order). Frege also adopts a strict law
of extensionality: two courses-of-values are identical iff their corresponding functions
have the same value for every argument. The picture of the logical universe as given
by CTT is thus nicely tidied up (or at least contained), and Frege formally retains
only three types: object, first-order unary function and first-order binary function.
However, with an unlimited comprehension principle and a strict law of extensionality
in hand, Frege is clearly on the edge of inconsistency. The final push cornes in the
form of a unit class operator, from which Frege can define an application operator;
the resulting inconsistency is demonstrated formally at the end of Chapter 26.
5Cocchiarella [5] also mentions this "collapse" of aU function types to object type, or "level 0";
in Chapter 2 we prove a theorem making precise this idea of collapse, or reduction.
6In the appendix to the Grundgesetze, Vol. II [10] that deals with Russell's paradox, Frege
returns to the initial framework of CTT and asks whether it ultimately is necessary to keep ail
levels separate, with a different "type" of equality for every level. Of course, he quickly despairs
at placing mathematics on such a hopelessly convoluted foundation. To his credit, even before the
discovery of any paradox, he called attention the his Basic Law V, dealing with abstraction and
9
The third and final point l wish to mention is Frege's concern with "securing
a reference" for aIl the terms of his language. There are two reasons a reader of
the Grundgesetze der Arithmetik might come to believe that Frege do es not want to
commit fully to a rigourously formaI language. One reason is the scathing attack Frege
launches on "formalists."7 Another is Frege's insistence that each term of his language
pick out a single, specifie mathematical entity, its reference, existing in a logical realm
independent of human thought. Frege clearly intends that these references are fixed
and provide the only correct interpretation of his language. In point of fact, however,
Frege is entirely committed to formaI rigour and seems to have been the first to
present a formaI language, with a fully explained grammar and rules of inference
based solely on syntactical features. Yet, as a diehard realist, he is not satisfied that
some interpretation of his language can be given; rather, he requires that the correct
interpretation of his language can be given, e.g. of the term in GG for the cardinal one
as the cardinal one, of the functional equality symbol as the equality function, and so
on. Frege is opposed to trends he sees in certain formalists of ignoring the question of
mathematical meaning and, ultimately, truth, and instead simply looking for clever
rules for shifting symbols around. For Frege, the focus has to be on logical and
mathematical truth, from which consistency would follow8 . While he made the point
against "pure formalists", Frege's own mathematical realism seems rather far-fetched
from the modern point of view and l have no intention of discussing it further. l simply
wish to point out that Frege's insistence on a specifie semantical interpretation in no
way compromises his formaI rigour, and while l have "modernised" the presentation of
the language GG below, this is basically a notational modification. The essentials of
extensionality, saying that if there \vere any weakness in or point of attack against his project of
reducing mathematics to logic, it must surely be this law. ([11] Introduction, p.VII) 7See especially [10] §§86-l37.
8See [8], pp.4-24, for an exchange between Frege and Hilbert, in which Frege questions Hilbert's
emphasis on consistency over truth.
10
the following definitions are all found in Frege's own presentation in the Grundgesetze,
and his notions of formaI language, formaI deduction, etc. are the modern ones. (Or,
rather, the modern notions are Frege's.)
1.2 The Theory GG
1.2.1 Types
The language of the Grundgesetze is a multi-sorted language and we introduce the
following variable types:
• object variables: xo, Xl, '" , X n , .... We say these variables are of object
type or level O.
• unary function variables: fl, il, ... , f~,
• binary function variables: f5, if , '" , f~, Both unary and binary
function variables are said to be of (first-order) function type or level 1.
We will follow Frege in distinguishing bound variables from free variables.9 This
distinction isn't necessary, but it allows a clearer statement of substitution rules
below. The variable symbolslo listed above will therefore stand for free variables and
we introduce the following bound variable types:
9Frege in fact further distinguishes between variables bound by quantifiers and those bound by
the abstraction operator (see 1.2.2 for definition of the abstraction operator). Our free variable
symbols correspond to Frege's Latin letters, our bound variable symbols correspond to both his
lower-case German letters (bound-by-quantifier variable symbols) and his lower-case Greek letters
(bound-by-abstraction variable symbols).
lOWe are going to adopt the simplifying convention that all symbols of the formaI language,
and expressions built up from these symbols, name themselves. E.g., --(x ~ x) both equals
T (since indeed --(x ~ x) = T) and names the expression --(x ~ x). This allows us to
avoid complications such as Quine's "quasi-quotes" for expressions like 1/J ---> cjJ or VxcjJ, which mix
11
• bound object variables:
object type or level O.
... ,
• bound unary function variables: fo 1 , il,
• bound binary function variables: R, li,
. . .. We say these variables are of
o •• ,
o •• , j;, .... Both unary and binary
function variables are said to be of (first-order) function type or level 1.
In practice, we will generally drop the arity superscript on function symbols when
writing terms, as the arity of any function will be clear from the context.
1.2.2 Basic definitions: primitives, terms and sentences
We now introduce nine primitive operation symbols, which Frege calls the "pri
mary function names" ("ursprüngliche N amen von Functionen") of the Grundgesetze.
These are the primitives of GG, and include the functional analogues of the usuallog
ical connectives. The interpretation which Frege intends for these function symbols
is discussed in 1. 3 below.
primitive unary operations:
• truth function: --x,
• negation: -.:., x, 11
• unit class function: \x,
• universalization over objects: \jif(i) ,
meta-theoretical names of expressions ('Ij;, qy) and parts of expressions themselves (-->, V, x). 1 thank
Professor Makkai for pointing this simplification out to me.
n A dot over a logical connective indicates that Frege's "functional analogue" to the usual con-
nective is meant, and serves to distinguish the two.
12
• universalization over unary functions: VP'iJ!(p) ,
• universalization over binary functions: V j2'iJ! (j2),
• abstraction: 5:f(x).
primitive binary operations:
• equality: x ~ y,
• implication: x -----+ y.
In the Grundgesetze formalism, a second-order function, such as 'iJ!(fl) or 'iJ!(j2)
ab ove , is given by a term <p and specified free function variable( s) (e.g. JI or j2),
as explained below. Frege does not state things quite like this, as he does not define
"term" in the way we will. His discussion is somewhat informaI, but it is clear that, for
him, the type (or linguistic role) of a meaningful expression (which we caU a term) <p
in his language is determined by specified Latin letters (i.e., free variables) occurring
in <p (these correspond to the variables in our Xq,; see Definition 1.1 below)12. In -1
connection with this, we need to explain notations such as <p[j1i~~l which occur below.
In what foUows, given expressions "(, Œi, f3i built from the primitive symbols of
GG (i.e., GG expressions), "(['K"'" tl is the expression obtained from "( as foUows:
For each i ::; n:
(i) If f3i is an ob ject variable (free or bound), then for each occurrence of f3i in "(
we substitute Œi.
12We take the slightly more liberal view that a function may also be determined by a term, cP,
and specified variable(s) not necessarily occurring in cP.
13
(Note for use in the following two clauses that, for expressions a, 'ljJ, X and object
variable z, w, the expressions a[~l and a[~, ~l are defined by (i). E.g., for a unary
(resp., binary) function variable f, we have that f(z)[~l (resp., f(z, w)[~,~]) is the
expression f('ljJ) (resp., f('ljJ,x))·)
(ii) If (Ji is f(w), with f a unary function variable (free or bound) and w a free
object variable, and 'ljJ is any GG expression13, then for each occurrence of the form
f('ljJ) in '1 we substitute ad~]' this last expression being defined by (i). For example,
(f(a) ~ f(a))[(;(~~)l is the expression (a ~ b) ~ (a ~ b);
(iii) If (Ji is f(u, v) with f a binary function variable (free or bound) and u, v
free object variables, and 'ljJ, X are GG expressions14 , then for each occurrence of the
form f('ljJ, X) in '1 we substitute ai[~, ~], this last expression being defined by (i). For
example, (f(a, b) ~ f(a, b))[j(~~~l stands for (a ~ b) ~ (a ~ b).
In every use we make of expressions of the form '1 [~: ' ... , ~ l, the (Ji fall under one
of the above three cases, such that 'I[~, ... ,~l is al ways well-defined by the above
three clauses. Note that if '1 and each ai are assumed to be terms in the ab ove , then
one can prove by an entirely straightforward and tedious induction, that 'I[~, ... , ~l
is also a term, according to the following definition of "term."
We give an inductive definition of (well-formed) term, with a simultaneous induc
tive definition of the set of free variables (sfv)) X qn of a term cP·
Definition 1.1. A GG term and the set of free variables of a GG term are defined
as follows.
13 As we have yet to define "term," 'l/J, like "(, here stands for any GG expression; however, in
practice, 'l/J and "( will always be terms, except in cases of binding variables, such as \ixr,b[~], where
we have the bound variable x in the role of 'l/J above.
14Same note for 'l/J and X as for 'l/J above.
14
1. Each free object variable, Xi, is a term with sfv {Xi}'
2. if cP is a term with sfv XrjJ, then each of -- cP, ~ cP, and \cP is a term with sfv
XrjJ;
3. if cP and 'ljJ are terms with sfv XrjJ and X'Ij;, respectively, then each of cP ~ 'ljJ
and cP ~ 'ljJ is term with sfv XrjJ U X'Ij;;
4. if cP and 'ljJ are terms with sfv XrjJ and X'Ij;, respectively, then each of P (cP) and
r (cP, 'ljJ) is a term, with sfv XrjJ U {fI} and sfv XrjJ U X'Ij; U {f2}, respectively,
where p is a free unary function symbol and r a free binary function symbol;
5. if cP is a term with sfv XrjJ, X is a free object variable and i; is a bound object
variable not occurring in cP, then each of Vi;cP[~l and icP[~l is a term with sfv
X4>-{x};
6. if cP is a term with sfv XrjJ, f is a free unary (resp., binary) function variable, f is a bound unary (resp., binary) function variable not occurring in cP, and x, y
are free object variables, then vjlcP[{~~~}l (resp., vj2cP[{~~~:~m is a term with
sfv XrjJ - {f};
7. nothing else is a term.
Note that these rules do not allow a term to have a quantifier that falls within the
scope of another quantifier binding the same variable. Frege himself does allow such
"nesting" of quantifiers; we have simply found it easier to state substitution rules
below with such nesting of quantifiers ruled out. Frege adopts a standard set of rules
for scope and binding, as do we, without repeating them here. We will call a term cP
closed if it has no free variables, i.e., if XrjJ is empty.
Every term cP is an object name in the sense that cP may be meaningfully substi
tuted for any free object variable in a meaningful GG expression. Specifically, cP may
15
be meaningfully substituted for any free object variable occurring in a GG term. Sub
stitution rules are given formally below. A unary first-order function is given in GG
by a term cp together with a specified free object variable x (which may or may not be
in X</». A binary first-order function is given in GG by a term cp together with two
specified free object variables x, y (which may or may not be in X</». A unary second
order function is given in GG by a term cp together with a specified free function
variable f (which may or may not be in X</». EtC. 15 To this extent, we find a trace of
Cumulative Type Theory in GG and this way in which terms represent objects and
functions in GG determines the exact statement of the rules of substitution; see 1.2.3.
In meta-theoretical discussions, we will generally use the lower case letters a, b, c, ...
to indicate objects, capitals F(x), F(x, y), G(x), G(x, y), H(x), H(x, y), ... to indicate
first-order functions, and capital Greek letters 1>(f), 1>(f, g), 'IJ(f), 'IJ(f, g), ... to in
dicate second-order functions.
Now, on the intended interpretation of --x, for any term, cp, -- cp is equal
(for aIl values of the free variables in cp) to T or -L and, as mentioned, we call -- cp a
G G statement. Frege reserves the term sentence ("Begriffsschriftsatz") for a sequence
of symbols of the form f- cp, with cp a term. The symbol f- consists of two parts for
Frege: the "judgment stroke" ("Urtheilstrich"), l, and the "horizontal stroke," --.
The latter is a function symbol as already described, while the former indicates the
assertion that the term to the right is (a name for) T. l believe this is how the
now ubiquitous turnstile was born. Frege calls an expression of the form -- cp a
"thought" ("Gedanke"), rather than a sentence, in the sense that we may contemplate
a "thought" without asserting it. In Frege's notation, an expression of the form f- cp
15Frege does not state or see things quite this way. Using our own terminology, we may say that
he cons id ers closed terms to be object names, terms <p such that Xq, contains a single object variable
to be unary first-order function names, terms <p such that Xq, contains exactly two object variables
to be binary first-order function names, terms <p such that Xq, contains a single function variable to
be unary second-order function names, etc ..
16
appears only if cp can be formally deduced in GG. This judgment stroke is a convenient
notation for marking theorems of GG, and we use it as such.
Officially, we adopt the following definitions. Any expression of the form --cp, where cp is a GG-term, is called a GG-statement or simply a statement. A formal
deduction, L:, is a finite sequence, CPl, CP2, .. , CPn, of statements, such that each CPi is
an axiom of GG, or is the conclusion of an instance of a rule of inference of GG all of
whose premises precede CPi in L:. (The axioms and rules are given in 1.2.3 immediately
below.) L: is a formal deduction of length n if L: is a formaI deduction and L: is a
sequence of length n. L: is a formal deduction of cp if L: = (CP1, CP2, ... ,CPn) is a formaI
deduction of length n and cp is CPn. cp is a GG-theorem, or simply theorem, if there
exists a formaI deduction of cp. From the following axioms and rules, it is easy to
see that if a theorem, cp, contains free variables, it is equivalent to its (functional)
universal closure, in the usual sense that the two statements are inter-derivable16 . We
will sometimes write f- cp to indicate that cp is a theorem.
1.2.3 Axioms and Rules of Inference
We come now to the axioms and rules of inference.
Axioms
Frege states six "Basic Laws" ("Grundgesetze") for his system. They are:
16This rais es a rather inelegant point. Consider the statement --(x ~ x). Technically, its
universal closure is Vx(--(x ~ x)) and the statement of its universal closure is
--Vx(--(x ~ x)). But the latter is equal to Vx(x ~ x), provably in GG (of course, technically
everything is provable in GG, but we mean "provably" in a more straightforward sense here), and . .
by definition in the meta-theoretical interpretation ofV. There is thus no harm in calling Vx(x ~ x)
the universal clos ure of --(x ~ x), and referring to it as a statement. Exactly the same point
that is made here for V can be made for ~, similarly for ~.
17
• Basic Law 1.
a) f- x ~ (y ~ x)
b)f-x~x.
Frege shows in Begriffsschrift that b) is in fact derivable from a) in his logic;
however he states b) here for economy of derivations.
• Basic Law II.
a) f- vyf(y) ~ f(x)
b) f- VgM{3(g((3)) ~ M{3(J((3)).
b) is actually an axiom schema: f and gare (first-order) unary function variables
and M{3 is a second-order unary function (of a first-order unary function), which
is equivalent formally to saying it is a GG term, </>, together with a specified
unary function variable, f. Given such </> and f, an instance of Basic Law II
b) is obtained by substituting Vg</>[~~~~l for VgM{3(g((3)) and </> for M(3(J((3)).
Since Frege doesn't allow quantification over second-order functions, b) must
be stated as a schema 17.
• Basic Law III. Substitutability of equal terms.
f- g(x ~ y) ~ g(vj(](y) ~ ](x))).
Note that f- (x ~ y) ~ vj(](y) ~ ](x)) is a special case.
• Basic Law IV.
f- (~ (-x ~~ y)) ~ (-x ~ -y).
This law is intended to formalize the notion that there are two truth values,
i.e., two possible values of the function --x. It certainly captures the notion
that there are ai mosi two; indeed a revision of GG proposed by Frege in light
17The f3 subscript in expressions of the form M(3(g(f3)) indicates simply that M(3(g(f3)) is a function
of the (first-order) unary function 9 only, and that if the argument in any occurrence of g(x) in
M(3(g(f3)) is changed, then the resulting second-order function M(3*(g(f3*)) is different.
18
of the paradoxes fails because it cannot prove that f--':" (--x ==-.:., x), i.e., that
T -=/: ~18.
• Basic Law V. Extensionality.
f- [if (x) == ig(x)] == [vx(J(x) == g(x))].
The interpretation is that the LHS (of the main equality symbol) is equal to T
iff the RHS is also, which is to say the two courses-of-values are equal iff the
corresponding functions are equal for every possible argument.
• Basic Law VI.
f- x == \ Y (x == y).
This law governs the unit class operator, \x, explained below in 1.3.
Rules of Inference
As Frege showed in Begriffsschrift, he requires as rules only modus ponens, univers al
generalization and "substitution rules," together with obvious allowance for renam
ing of bound variables in a term to obtain an alphabetic variant. However, for the
sake of economy of derivations, he officially sanctions several derivable rules in the
Grundgesetze. We won't present the latter here as we don 't intend on carrying out
many derivations in GG, and there is nothing exceptional about any of the derived
rules Frege uses. Here are the non-derived rules:
• Modus Ponens. If <p and 'ljJ are any two terms of GG such that <p and <p ~ 'ljJ
are theorems of GG, then 'ljJ is also a theorem of GG.
• Universal Generalization. If <p ~ 'ljJ is a theorem and the free object variable
x do es not occur in <p, then <p ~ (vx'ljJ[~]) is a theorem. Similarly, if <p ~ 'ljJ
18See Boolos [4] for a detailed study of the consistency of fragments of GG.
19
is a theorem and the free (unary or binary) function variable f does not occur
in <p, then <p ~ (',fj'lj;[;~~jD is a theorem19 .
• Universal Specification (s). a) If <p is theorem of the form vx'lj;[~J2° for sorne term
'lj;, then, for any term X, 'lj;[~l is a theorem.
b) If <p is theorem of the form V jl'lj; [~~ ~~~ 1 for sorne term 'lj;, then for any term
X21 and object variable y, 'lj;[~l is a theorem.
c) If <p is theorem of the form V j2'lj; [~~~~:~~ 1 for sorne term 'lj; , then for any term
X and object variables z, w, 'lj;[j2(;,w)l is a theorem.
1.3 Interpretation of Primitives
The intended interpretation of sorne of Frege's primitives has already been touched
upon. As mentioned, the truth function, --x, takes every object other than T to..i,
while taking T to T. By definition, each functional analogue (negation, implication,
universalization and equality here) of a usuallogical connective returns a truth value
for each (pair of) argument(s); keeping this in mind, we may complete their definitions
as follows:
• negation: For any object, a, -.:, a = T {::}def --a = ..i.
19We must add the clause that each ofvx7,b[~l and vj7,b[~l is a term, which, given our definition
of term (and in particular our disallowance of nesting quantifiers binding the same variable), is
equivalent to saying that x do es not occur in 7,b, and that j does not occur in 7,b, respectively.
20Technically --vx7,b[~]' see the final foot note in 1.2.2.
21X may or may not contain y. Note the different substitution rules for 7,b[~1 and 7,b[~]; see
1.2.2. Similar remarks apply for clause c).
20
• implication: For objects a, b, (a ~ b) = ~ <;::}def (--a = T and --b = ~).
• quantification: (i) For any first-order unary function F(x),
\jxF(x) = T <;::}def for aIl objects a, --F(a) = T (equivalently, F(a) = T);
(ii) for any second-order unary function WU) of a unary (resp. binary) first
order function variable j, vjW(]) = T <;::}def for aIl unary (resp. binary) first
order functions F, --W(F) = T (equivalently, W(F) = T).
• equality: For any objects a, b, (a ~ b) = T <;::}def a = b.
The remaining primitives are the abstraction and unit class operators. We describe
abstraction first. Frege's notion of abstraction applies to junctions, specificaIly, unary
first-order functions. In the formaI system GG such a function is given by a term <fJ
and a specified object variable x. Abstraction is the Grundgesetze analogue of set
formation; however because <fJ is a term and not a formula, we cannot take i<fJ[~l to
be the set of x such that <fJ(x). Rather, i<fJ[~l is to be thought of as the graph or
"course-of-values" of y = <fJ(x) (where y is a variable not free in <fJ), as mentioned
earlier. Frege takes i<fJ[~l to be primitive in the sense that he does not specify any
structure for courses-of-values (e.g., as sets of ordered pairs), saying only that we find
a visual representation ([11], §1) of the course-of-values of a (unary) function as a plot
in the xy-plane. In any case,. Frege takes it as a logical truth that for every unary first
order function, F(x), there exists a corresponding course-of-values, iF(x), satisfying
Basic Law V. The notion of a "double course-of-values" ("Doppelwerthverlauf') is
already contained in the above notion of course-of-values. For if we consider a binary
first-order function, F(x, y), then for any fixed object a, iF(x, a) is a (simple) course
of-values corresponding to the unary function F(x, a). Therefore, iF(x, y) is itself a
unary function of the object variable y, who se value for any argument is a course
of-values. The meaning of the term y(iF(x, if)) is thus completely determined from
the original definition of course-of-values. Frege himself discusses and makes use of
21
only simple and double courses-of-values (and thus technically only simple courses
of-values - the point here being that he never iterates abstraction more than once);
however the further generalization to courses-of-values corresponding to functions of
n object variables is straightforward from the above22.
Now, just as modern set the ory reduces all mathematical entities to sets, Frege
would like to construe all mathematical objects as courses-of-values. (Note, however,
that functions for Frege remain fundamentally different from objects and cannot be
construed as courses-of-values.) In his interpretation of GG, the only objects to which
Frege is committed are T, ..i and those which can be named by terms of GG. Since
it is clear by inspection of the primitives used in term-formation that all objects
which can be named by GG-terms must be courses-of-values or names of Tor ..i, this
amounts to finding a way to construe T and ..i as courses-of-values. It is not clear to
me why this is an appealing path for Frege to take, and his "permutation argument,"
given in §10 of [11], supporting such a move is perhaps less than convincing. We will
only mention here for the sake of completeness that Frege cornes to the conclusion
that we should take T to be the course-of-values corresponding to the unary function
taking T to T and every other object to ..i, while we should take ..i to be the course
of-values corresponding to the unary function taking ..i to T and every other object
to ..i23 . Strange objects to say the least, and it is all the more strange that Frege
says the reason these identifications are permissible is that he has shown they do
22 Interestingly, when we later reconstruct our modified form of the Grundgesetze in NF UT, this
restriction to only simple and double courses-of-values plays a more significant role.
23Formally, we could give the definitions as T =def i(--x) and ~ =def i(x ~ (~\;y(ij ~ fj))). There is nothing especially peculiar about these definitions as they stand. What is peculiar is rather
Frege's interpretation of T and ~ bath as the objects used to define (meta-theoretically) functions
such as --x and x ~ y and as the two (formally definable) courses-of-values given above. If
we were to interpret courses-of-values as sets of ordered pairs in the usual way, we would have, in
particular, that (T, T) E T and (~, T) E ~.
22
not contradict (which can only mean "are consistent with") Basic Law V, governing
courses-of-values. This is certainly peculiar for someone who bitterly and sarcastically
argues against mathematicians who think they can "magically bestow" properties
on objects through definitions, without contemplating the (independently existing)
objects themselves to see if they possess these properties (cf. [10], §139 and §143).
It is interesting that the identification within GG of T and --L with the courses-of
values described ab ove , is entirely analogous to the suggestion made by Quine in
Mathematical Logic that, in his set theory, we should identify concrete objects, i.e.
non-sets, with their unit sets. E.g., if we admit the existence of T as a non-set in
ML, then we have T = {T}. This seemingly trivial point we will return to in our
discussion of NF, NFU and variants, as it is at the centre of work on the consistency
of these systems.
Given an abstraction operator on functions as ab ove , it is natural to expect an
application operator. Such an application operator is more naturally the complement
of abstraction than the unit class operator and the main use Frege makes of the unit
class operator is indeed to define an application operator. The function \x, formally
a primitive of GG, is to be interpreted as the function which Frege defines as follows.
Definition 1.2. \x = y {:}def (there is a unary first-order function G(z) such that
x = iG(z) and G(y) = T and, for all u, G(u) = T =} u = y) OR (there is no such
function and y = x).
The idea is that if there is some concept24 , G(z), such that x = iG(z) and such
that exactly one object falls under the concept G(z), then the unit class operator
returns this object; otherwise it simply returns the argument. (As can be seen by
24The function G(z) need not, strictly speaking, be a concept; it need only be a function whose
value is T for exactly one argument; in practice, however, we will generally not make use of any
function whose range includes both truth values and other objects (e.g. numbers).
23
the fact that Basic Law VI governs only the case where there is such a function, the
actual value returned by the "garbage clause" is irrelevant.)
The application operator, x ~ y, is then formaUy defined as:
Definition 1.3. x ~ y =def \t(-.:, \;g(y == ig(i) ---.:.....-':' (il, == g(x)))).
By "formally defined," we mean that the left-hand sicle of the definition is to be
understood simply as a meta-theoretical short-hand notation for the right-hand side.
Given the obvious meta-theoretical definitions of 3 and Â25, this definition can be
expressed more readably as:
x ~ y =def \t(3g(y == ig(i) Â il, == g(x))).
80 if Y is a course-of-values, iG(i), for sorne G(z), then x ~ y = G(x); otherwise,
for every u, 3g(y == ig(i) Â u == g(x)) = .1, and
x ~ y = t(3g(y == ig(i) Â il, == g(x))), that IS, x ~ y is the course-of-values
corresponding to a (every) function whose value is uniformly .1:
x ~ y = i(-':' (x == x)). However, since Frege intends for aU objects to be courses-of
values, the latter case is not supposed to arise.
As a final note in this section, we point out that ((.), (.) ~ (.)) constitutes a
À-system for any model of GG. (Let us ignore for a moment the fact that there are
no such models and the last statement is vacuously true.) Peter Aczel develops an
analysis of the language of the Grundgesetze as a À-structure in his article [1].
25For definiteness, ~x4> abbreviates ~ ('-ix(~ 4») and 4> A 'lj! abbreviates ~ (4) .-:..... (~ 'lj!)), while,
for future reference, 4> V 'lj! abbreviates (~ 4» .-:..... 'lj!.
24
Chapter 2
Type Theory and the
Grundgesetze: Basic Arithmetic
and the Paradoxes
In the decade after uncovering the paradoxes plaguing Frege's Grundgesetze, Russell,
together with Alfred North Whitehead, developed his own logical analysis of mathe
matics. The monumental Principia Mathematica presented the Theory of Types as a
logical framework for mathematics which steered clear of the paradoxes. This chap
ter focuses on the relation of simple Type Theoryl, denoted "TT" below, to Frege's
Grundgesetze, especially the similarities of TT to CTT and the careful separation of
levels in both. This will help to clarify the basic GG arithmetical definitions which
we present and to draw a conceptual link between the Grundgesetze and Quine's
New Foundations, which is in a sense derived from TT. Of course, the system of the
IThis is Russell's type theory as simplified by Ramsey: simple or "unramified" type theory. "In
spirit," simple type theory is still Russell's type theory, to state things vaguely. More specifically,
the handling of the paradoxes in simple type theory remains essentially that which motivated Russell
to develop his original type theory; see 2.1.1.
25
Grundgesetze is inconsistent. We discuss how the addition of the abstraction and ap
plication operators creates the inconsistency and, finally, we derive this inconsistency
in GG.
2.1 Frege's Levels and Russell's Types
2.1.1 The Theory of Types
As Russell notes in reflecting on the Principia, "It will be found that in aIl the logical
paradoxes there is a kind of reflexive self-reference [ ... ]" ([24], p.83). In the most
common statement of Russell's paradox, we ask whether the (set-theoretic) property
of not being a member of oneself is applicable to the collection of sets/objects which
are not members of themselves. In deriving this paradox, we assume that, given the
property "not being a member of oneself," we may inquire whether it is applicable to
individual elements of sorne domain, which may be left indefinite in every way, except
that it must "already" contain the collection ofthose objects/sets/things which satisfy
"not being a member of oneself." Put this way, perhaps the intuitive, unrestricted
comprehension of naive set theory is not so obvious, in the sense that we must at
once, or "on the same level," be able to state a property and ask whether it applies
to the collection of aIl things satisfying it.
In any case, Russell says that it was by reflecting on this self-referential nature of
the paradoxes that he was led to formulate his Theory of Types. Here we will formally
present and discuss TT, the version of type theory that is now generally considered,
that is, Russell's theory as simplified by Ramsey, a.k.a. "simple Type Theory."
TT is a theory in first-order predicate logic with distinct variable types (or sorts)
for each natural number i,2 that is, we have a countable list of variables, xb, xl, ... ,
2We could of course interpret TT in a "pure" (unsorted) first-order set theory by adding, for
each i, a unary predicate symbol, Si, to the signature of TT and using these to "sort" the variables,
26
x~, ... ,for each i E N. We will not introduce separate symbols for free and bound
variables in TT. The signature of TT contains the single primitive symbol E, a binary
predicate symbol.
The atomic formulas of TT are all the expressions of the form Xi = yi and Xi E yi+1,
for i E N. Finally, the axioms of TT consist of two schemas:
Axiom Schema of Extensionality:
VXi+1V y i+1(VZi(Zi E XH1 f--+ Zi E yHl) -+ XH1 = yHl)
and
Axiom Schema of Comprehension: 3xH 1Vyi(yi E XH1 f--+ CP),
where the variable i ranges over the natural numbers in both schemas, and cp is a TT
formula not containing XH1 free.
This is the complete statement of TT as it is usually given. Note that for the
purposes of arithmetic, one must add a further axiom of infinity, ensuring an infinite
number of "urelements," i.e., entities of type O.
For any variable, X, of any type, the expression X E X is ungrammatical in TT;
we cannot even make statements of self-membership, let alone form the collection of
all non-self-membered entities, and Russell's paradox never gets off the ground.
2.1.2 Frege's Theory of Levels and The Definition of Number
Frege's concept levels were discussed in some detail in Chapter 1. The parallels
with TT should be clear. In particular, CTT, which can fairly be called the logical
system of the Grundgesetze prior to the introduction of the abstraction and unit
class operators, avoids the "self-referencing" that Russell points to as the source of
the paradoxes.
Certainly, it is Frege's definition of natural number that Russell reproduces in
while also appropriately restricting quantification. E.g., \lxi would be interpreted as \Ix E Si (or,
more precisely, \lX(SiX ---+ •.. )).
27
TT. Frege arrived at his notion of cardinal number after much reflection and critical
analysis of previously suggested definitions of number. It is a testament to Frege's
insight and clarity of thought that his definition of cardinal number strikes us now
as very natural and, to a certain extent, obvious3 . We sketch very roughly the line of
thinking leading to his definition.
Frege points out that we run into difficulty when we try to assign a number
directly to objects. For instance, if we spot a soccer match, sayon field A, we may
say that there are two teams playing or that there are 22 players playing, and if these
numbers are properties of objects, i.e. of the people on the field, then we have that
they are both two and 22. Frege concludes that the property we assign in assigning a
cardinality must be a property assigned to a concept. That is, we assign the property
of "being two" to the concept "team on field A," and of "being 22" to "player on
field A," because there are two and 22 objects, respectively, faUing under these two
concepts. Thus, we have the conclusion that, for n a natural number, "being n" is a
second-order concept, i.e., a property that can be meaningfuUy predicated only of a
first-order concept.
Up to this point, RusseU's analysis of "natural number" is in essential agreement
with Frege's and the natural numbers (though not the set of natural numbers!) "exist
at level two" in a model of TT: the natural number n is the set of aU sets of type 1
that contain n elements (of type 0). (Note, however, that we will also have a "copy"
of the natural numbers for each type k greater than two: the natural number n "of
type k" will be the set of aU sets of type k - 1 that contain n elements of type k - 2.)
However, Frege does not hold the view that the natural number n is a second-order
3 Although it need not concern us here, we may note that the von Neumann definition of cardinal
- i.e. a special case of the von Neumann ordinal - would also have been rejected by Frege. For,
though these might serve nicely for the purposes of arithmetic, the idea that the number one is an
element of the number two would certainly have seemed preposterous to Frege.
28
concept; it is "being n," or, more clearly put, "being true of n objects" which is a
second-order concept. The natural number n is, for Frege, an object, as indicated by
the definite article. This same line of reasoning Frege follows in paradoxically and
(somewhat) famously stating "the concept 'horse' is not a concept" ([12], pp.67-69).
The point Frege wishes to make is naturally that the concept "horse" is the extension
of "is a horse," and, while "is a horse" can be (either truly or falsely) predicated
of any object, it is not grammatical to ask whether "the concept 'horse'" is true
of an object. We may ask if "fans under the concept 'horse'" is true of an object,
but now we have again named a concept, to be sharply distinguished, Frege says,
from the concept "horse" (and from the concept "falls under the concept 'horse'"
!). In 2.2.2, we show how Frege defines the natural number n as the course-of-values
corresponding (via a first-order concept) to the second-order concept "being n". Now,
we might at first think that "being n" could also be a third-order concept for Frege,
so that, in analogy with the situation in TT, "being n" would apply to second-order
concepts that are satisfied by n first-order concepts. Frege disagrees on the grounds
that counting is always do ne of abjects falling under some specified first-order concept.
So that even if we "count concepts," e.g. we count the number of "distinct species of
four-Iegged animaIs," we are really counting classes (which are, for Frege, objects),
i.e. the extensions of the concepts, "tiger," "zebra," "German shepherd," etc., and
not the concepts themselves, which are of a fundamentally different nature than
their extensions. For Frege, these extensions, or courses-of-values, are of fundamental
importance in logic. He believes that (first-order) concepts have extensions, i.e.,
collections or classes consisting of those objects falling under the given concept, and
that these classes genuinely exist independently of human thought. Russell on the
other hand, believes that "classes are merely a convenience in discourse," that a class
."is only an expression" ([24], p.81-82)4. Accordingly, in a model of TT, there is no
4And numbers are thus also "merely linguistic conveniences" ([24], p.71). It's somewhat amusing
29
object (entity of type 0) corresponding to a property - or formula - qy(x) of one free
object variable; the set of objects x such that qy(x) is an entity of type 1 ("merely a
linguistic convenience" for Russell) and its existence resides essentially in the formula
qy(x).
We have here a fundamental difference between Frege's and Russell's philosophical
outlooks on logic. Ultimately, his more liberal view on the existence of classes is what
brings inconsistency into Frege's system, for it translates formally into the abstraction
operator discussed in Chapter 1. Before demonstrating this inconsistency, however,
let us give the basic arithmetical definitions of the Grundgesetze.
2.2 Formally Defining Cardinality
We have earlier alluded to the fact that the abstraction operator, though strictly
speaking it is applicable only to first-order functions, can be used to "reduce the
level" of higher-order functions as weIl. We will now see this for the second-order
concept "being true of n objects" discussed above, as the natural number n will be
defined as the course-of-values corresponding to this second-order concept. We first
need sorne preliminary definitions, however.
2.2.1 The Application Operator and "Reduction"
In §34 of the Grundgesetze, Vol.! ([11]), Frege introduces the application operator,
as in Definition 1.3:
x ,-., y =def \fi(~ Vg(y = ig(i) ~~ (fi = g(x)))).
to note Russel! claims that with the definition of the natural number n as the class of al! classes with
n members (which he credits to Frege and, independently, to himself ([24], p.70)), we "get rid of
numbers as metaphysical entities." ([24], p.71) It's hard to imagine that Frege would have accepted
this conclusion.
30
As mentioned in 1.3, the meaning of the unit class operator is such that we then
have, for any unary fun ct ion G(x) and any object a, that a ~ iG(i) = G(a). For
any binary function F(x, y) and any objects a, b, we similarly have that
a ~ (b ~ y(iF(i, fj))) = a ~ iF(i, b) = F(a, b). (Recall that iF(i, y) is a unary
function of y, and y(iF(i, fj)) is the corresponding course-of-values.)
In §35, Frege gives sorne examples of second-order functions and their correspond
ing ("entsprechende") first-order functions. For instance, con si der the second-order
80 for any unary function, F, E2 (F) = T iff there is an object a such that
F(a) = T, otherwise, E2(F) =~. Corresponding to this second-order function, E2'
we have the first-order function El:
Definition 2.2. "Existence operator, reduced":
El(Y) =def':' Vi(':' (i ~ y)), or El(Y) =def 3i (i ~ y).
Now we have that for any unary function, F, E1(iF(i)) = T iff there is an object
a such that a ~ iF(i) = F(a) = T, otherwise E1(iF(i)) =~. And so, for any
unary first-order function F(x), E2(F) = E1(iF(i)).
Clearly, in this correspondence between second- and first-order functions, there is
nothing particular about the Existence operator; in any unary second-order function,
\If (f), of a unary first-order function, we rnay replace each occurrencé of f (x) with
x ~ z to obtain a first-order function G1lt (z) such that, for any function F,
\If(F) = G1lt (iF(i)). Frege says this much in the Grundgesetze ([11], §§35-37) without
giving a precise general staternent of this "reduction of second-order functions". It
will prove quite useful for us to do so and we state the following (meta-)lernrna.
5Recall that, in GG, functions are given by a term and specified variable(s).
31
Lemma 2.1. Let cp be any GG-term and 9 any unary function variable. Then there
is a well-defined GG-term CPg-red and an abject variable Yg su ch that: (i) Xq, - {g} =
Xq,g_red - {Yg} and (ii) for the functions cp(g) and CPg-red(Yg) , we have, provably in
GG,6 that cp(G) = CPg-red(iG(x)) for every unary first-order function G(x) (and set
of values for variables in Xq, - {g} ).
Pro of. Let f2 be a free binary function variable and let f be a free unary function
variable different frorn g. Given cp, we define CPg-red inductively as follows.
1. If cp is x for sorne object variable x, then CPg-red is x;
2. if, for sorne terrns 'l/J and X, cp is (-'l/J) or (.:. 'l/J) or (\'l/J) or ('l/J ----t X)
or ('l/J ~ X) or f ('l/J) or P ('l/J, X) then CPg-red is, respectively, (--'l/Jg-red) or
(.:. 'l/Jg-red) or (\ 'l/Jg-red) or ('l/Jg-red ~ Xg-red) or ('l/Jg-red ~ Xg-red) or f( 'l/Jg-red)
or p ('l/Jg-red, Xg-red);
3. if, for sorne terrn 'l/J and free object variables x, y, cp is i'l/J[~] or vx'l/J[~] or
Vf'l/J[;~~j] or Vf2'l/J[t~~~:~j], then CPg-red is, respectively, i'l/Jg-red[~] or VX~g-red[~l ~f-nl, [Ï(x)] ~f-2nl, [i2 (x,y)]. or v If'g-red j(x) or v If'g-red j2(x,y) ,
4. if, for sorne terrn 'l/J, cp is g('l/J) , then CPg-red is 'l/Jg-red r-., Yg, where Yg is a free
object variable not occurring in 'l/J;
5. finally, if, for sorne terrn 'l/J, cp is Vg'l/J[;i~i], then CPg-red is Vyg'l/J~-red[~:], where ni/ . ni, [i(z~yg)] If' g-red lS If' g-red yg .
6Note that the proof does not contain any "trick" relating to the inconsistency of GG. Specifically,
if we placed restrictions on the existence of courses-of-values, for instance, to obtain a consistent
fragment of GG, then the following proof would still be valid, with clause (ii) modified to read, " ...
for every unary first-order function G (x) such that i:G (x) exists ... ."
32
It's obvious from this definition that X'" - {g} = X"'g-red - {Yg}, and we assume
this without proof. The proof of (ii) is by induction on the structure of cp, specifically,
on the number of GG primitives in cp. Note that the variables 9 and Yg, as in the
statement of the lemma, are fixed in the proof of the inductive step. We need to
consider each of the above cases:
1. In case 1, we have that both cp and CPg-red are x and, so we have triviaUy that,
for every value a of x and every unary first-order function G (z), cp (G) = a =
CPg-red(iG(z)).
2. For each instance of case 2, it foHows directly from the induction hypothesis that,
for any G(z), cp(G) = cp(iG(z)) (for all values of the variables in X", - {g}).
3. Suppose cp is i?jJ[~l. By the induction hypothesis, the fun ct ions ?jJ(g) and
?jJg-red(Yg) are such that ?jJ(G) = ?jJg-red(iG(z)) for every unary first-order G(z)
and aU values of the variables in X'Ij; - {g}. In particular, then, for all values of
the variables in X'Ij;-{g, x} = X",-{g}, and every G(z), we have that, for all val
ues a of the variable x, ?jJ(G) = ?jJg-red(iG(z)). That is, for the functions ?jJ(x, g)
and ?jJg-red(X, Yg), we have \;x(?jJ(x, G) = ?jJg-red(X, iG(z))) for aH G(z) and aH
values of variables in X", - {g}, which is to say \;x(?jJ[~](G) = ?jJg-red[~](iG(z))).
FinaUy, by Extensionality (Basic Law V), this is equivalent to saying that
i?jJ[~](G) = i?jJg-red[~](iG(z)) for aU G(z) and aU values of variables in X",-{g}.
I.e., cp(G) = cpg-red(iG(z)) for aU G(z) and all values of variables in X", - {g},
as required. The proofs of the inductive steps of the remaining possibilities of
case 3 are similar. (Note, in particular, that the proof of the inductive step for
the case where cp is \;x?jJ[~l is essentially contained in the pro of just given.)
4. Suppose cp is g(?jJ) , so cjJg-red is ?jJg-red '" Yg' Note that (*) CPg-red(Yg)
?jJg-red(Yg) r--- Yg and cp(g) = g(?jJ(g)),7, and by the IH, the functions ?jJ(g) and
70f course, it is certainly possible that g not appear in 'ljJ (and therefore also that Yg not appear
33
'ljJg-red(Yg) are such that 'ljJ(G) = 'ljJg-red(iG(i)) for every unary first-order G(z).
Now, for any G(z) (and values of the variables in Xrj; - {g}), we have <p(G) =
G('ljJ), and <pg-red(iG(i)) = 'ljJg-red(iG(i)) ~ iG(i) = G('ljJg-red(iG(i))), by
definition of the application operator. Now, G( 'ljJg-red(iG(i))) = G( 'ljJ( G)) by
the IR, and G('ljJ(G)) = <p(G) from (*). So agàin, <p(G) = <Pg-red(iG(i)) for aU
G(z) and aU values of variables in Xrj; - {g}, as required.
5. FinaUy, suppose <p is Vg'ljJ[~], so <Pg-red is Vyg'ljJ~-red[~l. Now, for aU values
of the variables in Xrj; (note that 9 is not an element of Xrj;), we have that
each of <p and <Pg-red is equal to a truth value. Therefore, it suffices to show
that <p(G) = --L iff <pg-red(iG(i)) = --L8, for any G(z). Since 9 is not free in cp
and Yg is not free in <Pg-red, <p(G) has the same value for every G(z), as does
<Pg-red(iG(i)). As the values of these functions do not vary with G(z), we will
suppress the argument of <p(g) and that of <Pg-red(Yg) in what foUows. Suppose
<p = --L. Then for sorne H(z), 'ljJ(H) =1= T, so by the IR, 'ljJg-red(iH(i)) =1= T.
Now note that by the definition of 'ljJg-red, every occurrence of Yg in 'ljJg-red is
within an expression of the form X ~ Yg, where X is a term (or a term with
bound variables substituted for free variables). Since a ~ b = a ~ i(i ~ b) for
aIl a, b, we have that 'ljJ~-red(b) = 'ljJg-red(i(i ~ b)) = 'ljJg-red(b) for aIl values b
of the variable Yg in 'ljJ'(Yg). So from the above, 'ljJ~_reAiH(i)) =1= T also, which
is to say <Pg-red = --L.
Conversely, suppose <Pg-red = --L, i.e. there is an object b such that 'ljJ~-red(b) =1=
T. Therefore, 'ljJg-red(i(i ~ b)) =1= T, and by the IR, 'ljJg-red(i(i ~ b)) = 'ljJ(z ~
b).9 So 'ljJ(z ~ b) =1= T, and therefore <p = --L, as required.
in 1jJg-red). 8Note that since T and -1 are not symbols of GG, an expression such as cP = -1 should be taken,
within GG, as an abbreviation for, e.g., cP ~..:., (x ~ x):
9It's to obtain this equality that we replace 1jJg-red by 1jJg_red[i(z;:yg)] in the definition of cPg-red
34
So, once again expressing the arguments, we have cjJ( G) = cjJg-red(iG(i)) for
every G(z) and aU values of the variables in X</>( -{g}), completing the proof.
o
ActuaUy, our main interest in this result is something that Frege does not mention
at aU, namely that quantification over function variables can, in the sense given by
above the above lemma, be replaced by quantification over object variables. This
is important for our interpretation of GG in NF UT, because the latter, as a theory
of sets recognizes only one type of first-class existent, that is, there exists a single
variable type (corresponding to Frege's object type) over which we can quantify.
For now, however, we turn to Frege's definition of cardinal number.
2.2.2 Bijection and Equinumerosity
The basic notion at the heart of Frege's definition of cardinal number is that of
"equinumerosity" ("gleichzahligkeit"). Two sets are said to be equinumerous if there
is a bijection between them. We therefore need a definition of bijection; to this end,
Frege gives first a definition of injection. Note that in the context of GG this apphes
to a binary function.
Definition 2.3. A binary function F(x, y) is said to be injective ("eindeutig") if
\;x\;i)(F(x, i)) -.:..... (\;i(F(x, i) -.:..... i) ~ i))) = T, i.e. if, for each x, there is at
most one y such that F(x, y) = T. "Being injective" is most naturaUy a second
order concept, as an injection is a first-order function. However, in practice Frege
always makes use of the corresponding first-order concept, for which he introduces
4Expressions in quotation marks are defined below, and y = 0 abbreviates
:3u(Set(u) Il 'tfv(v ~ u) Il y = u).
68
8. </J is Vx1P[~]:
. x (y = VX1P[-]) <=?def [(Vx(1P
X T) !\ Y T) V (3x( 1P =1= T) !\ Y .l)];
~ x (y = x1P[ -]) <=?def
X
[(3w("wisaunaryF - function"!\ VxVz((x,z) E w <--t Z = 1P)!\y = w))
V (-,3w( "w is a unary F - function" !\ VxVz((x, z) E w <--t Z = 1P))!\ y = 0)];
10. </J is fi1P[~, ~]:
-- x z (y = xz1P[-, -]) <=?def
X Z
[(3w( "w is a binary F - function" !\ VxVzVu((x, z, u) E w <--t Z = 1P) !\ y = w))
V(-'3w("wisabinaryF-function"!\VxVzVu((x,z,u) E w <--t Z = 1P))!\y = 0)].
AIl occurrences of GGree terms 1P (or X or p) in the RHS of the definitions above
are within expressions of the form x = 1P,5 for sorne variable x, such that the number
of occurrences of GGree primitives in 1P is strictly less than in the term </J for which
y = </J is being defined. So it is clear that iterative use of the definitions above will
translate y = </J into a formula, </J:FUT, of NFUT for any GGree term </J. It should also
be clear that the free variables in the final NFUT formula </J:FUT are exactly those
in the original expression y = </J, i.e., x is a free variable of </J:FUT iff x E Xq, U {y}.
We explain now the expressions in quotation marks above. As mentioned, once
ordered pairs, and n-tuples generally, are defined (see 4.2.1), we define relation and
function as usual. That is,
50r T = 1/; / ..l = 1/;, which could of course be replaced by =:ix(x = T Ax = 1/;) / =:ix(x = ..lAx = 1/;).
69
Definition 4.1.
"x is an n - ary relation" {:}dej
Vy(y E x ---+ :3zo, Zl,···, Zn-l(y = (zo, Zl,···, Zn-l))).
"x is an n - ary function" {:}dej
"x is an n + 1 - ary relation" A VyVz[Vwo, Wl, ... ,Wn-l(((WO, Wl,· .. ,Wn-l, y) Ex
A (wo, Wl,···, Wn-l, z) EX) ---+ y = z)].
An n-ary F-function is simply a n-ary function whose domain is vn, where V =dej
{xix = x}, i.e., V is the universe (which exists as a set in NFU(T)).
Definition 4.2.
"x is an n - ary F - function" {:}dej
"x is an n - ary function" A Vwo, Wl, ... ,Wn-l (:3y(( wo, Wl, ... ,Wn-l, y) Ex)).
Actually, the notion of F-function is hardly more restrictive than that of function,
in an NFU context. For, taking the case of unary functions for simplicity, given any
function f in NFU (i.e., a set of ordered pairs satisfying the conditions of being a
function, as defined above) , the domain, Dj, of f, is a set, as is its complement,
DCj. The identi ty function restricted to Dy, Id Dy' is then also a set and, finally, so
is the union fUI d DG, which is an F -function that agrees with f on D j, and is the f
identity function for aIl other arguments. Thus, any function is easily extended to an
F -function in this way.
In the arithmetical definitions we use below, it will be necessary that certain
formulas be stratified and, in sorne cases, this will require that variables in the different
argument places of binary functions be assigned the same type. For this reason,
70
we cannot do with only simple (unary) abstraction and application operators. For,
given an F -stratified (defined below) GGree term cp and free variables x, y, the term
y(x1>[~,~]) indeed defines a function in NFUT6. However, it does not define a binary
function, in the sense of the above definitions. Rather, y(xcp[~, Q]) de fines a unary x y .
function F(y) such for any a, F(a) = Fa(x), where Fa(x) is itself a unary function,
namely the function defined by (xcp[~, ;:])1. This means that the arguments of the
function F(y) are one type higher than the arguments of the functions Fa(x), so that
a stratification assignment for a term of the form z ,,-... (w ,,-... y( xcp[~, ;]) will need to
assign a type one higher to w than to z. Of course to state this clearly, we need to
formulate a precise notion of stratification for GGree terms, and this should be done
without resorting in each case to a full translation into NFUT primitives. Therefore
we introduce now the notion of F-stratification.
4.2.3 F -Stratification
Note that defining the interpretations of GGree terms in NFUT as above amounts
to defining a definite description term corresponding to each GGree term, e.g., the
definition for y = (x == z) is equivalent to defining
(x == z) =def ~y((x = z 1\ y = T) V (x =1 z 1\ y = ~)).
Similarly for the other definitions; to complex terms will generally correspond nested
definite descriptions. E.g., we may translate the GGree term (x == y) ~ z as, in the
first instance, ~w[((x == y) = T 1\ z =1 T 1\ w = ~) V (((x == y) =1 TV z = T) 1\ w = T)],
60f course, in NFUT we are adopting the usual set-theoretic identification of a function with its
graph.
7Note the different meanings for [~l and [~l: the former indicates a substitution of symbols and
corresponds to the meaning we have given to expressions such as &:<p[~l throughout this essay; the
latter indicates that a is a parameter replacing the variable y. x<p[~, ~l does not in fact stand for
any term of GGrec , and perhaps the notation &:<p[~l(a) is more appropriate, as long as we keep in
mind that a is here an argument of &:<p[~l(y), a function of the variable y.
71
and finally as
LW[(LU[(X = y 1\ U = T) V (x =1- y 1\ U = l..)] = T 1\ z =1- T 1\ W = l..)
V ((LU[(X = y 1\ U = T) V (x =1- y 1\ U = l..)] =1- T V z = T) 1\ W = T)].
From what was said above concerning stratification and definite descriptions, it's
essentially automatic to read off the requirements that the notion of F-straiijication
must place on a GGree term 1> to ensure that the translation 1>:FUT of y = 1> is
stratified. Below, we state these requirements precisely and verify that, for any GGree
term 1> satisfying our definition of F-stratification, 1>:FUT, the translation into NFUT
of y = 1>, is indeed stratified. (Stratified in the original sense (as modified for NFUT) ,
so that, as mentioned in 4.2.1, we do not ultimately rely on any notion of stratification
extended to formulas with definite descriptions.)
So we place the following conditions on a stratification assignment for a GGree
term. We may assign an arbitrary type to the term as a whole. Now, assume that 1>
is a subterm of the term in question and that 1> has been assigned type n. Then
1. if 1> is --1/;: We may assign an arbitrary type to 1/;;
2. if 1> is ~ 1/;: We may assign an arbitrary type to 1/;;
3. if 1> is 1/; ~ x: We may assign arbitrary types to 1/; and X;8
4. if 1> is 1/; ~ x: We may assign an arbitrary type to 1/;, but must assign this same
type to x;
5. if 1> is 1/; ,-.., x: We must assign type n to 1/; and type n + 1 to x;
6. if 1> is 1/;X;::::'P: We must assign type n to 1/; and to X, and type n + 1 to p;
8Note that this freedom in assigning types to the immediate subterm(s) of a term whose main
connective is --, ..:" -:.... or \; is analogous to the fact that an NFU formula ,cp or cp ---> 7jJ or \:/xcp is stratified iff cp (and 7jJ) is (are).
72
7. if qy is vx7jJ[~l: We may assign an arbitrary type to 7jJ[~1;9
8. if qy is i7jJ[~l: We must assign type n - 1 to x and to 7jJ[~1;
9. if qy is ii7jJ[~, ~l: We must assign type n - 1 to x, to fi and to 7jJ[~, ~l.
Let us call a GGrec term qy F-stratijied if there is an assignment of types (natural
numbers) to all subterms lO of qy such that the above conditions are satisfied. Such
an assignment will be called an F-stratijication assignment for <p. We now prove a
lemma to the effect that "F -stratification implies stratification." The proof will make
use of the following obvious fact: If 7jJ is any subterm of a GGrec term <p, and T is
an F-stratification assignment for <p, then T restricted to the subterms of 7jJ is an
F-stratification assignment for 7jJ. In the statement and pro of of the lemma, "NFUT
formula" is a formula written purely in the primitives of NFUT (in particular, not
containing definite description terms), so that "stratification assignment" for such a
formula me ans an assignment of types to variables only.
Lemma 4.1. Let <p be an F-stratijied GGrec term and T be any F-stratijication as
signment for qy. Let <p:FUT be the translation dejined above (see 4.2.2) of y = qy into
the primitives of NF UT. Then there exists a stratijication assignment, Œ, for qy:FUT
such that (i) Œ(Y) = T(<p), and (ii) Œ(U) = T(U) for each free variable u of <p,u
Proof. The proof is by induction on the structure of GGrec terms, specifically, on
the number of occurrences of GGrec primitives in a term <p. If the number is zero,
then qy is x for some free variable x and T(X) = n trivially defines an F-stratification
9Note that, technically, 1j;[~l is not a GGree term; however, for stratification assignments, expres
sions such as 1j;[~], which would form GGree terms on substitution of free variables for the bound
variables that are "free" in them (e.g. i; in 1j;[~]), will be considered terms and assigned types in the
same way as other subterms. lOSee previous footnote. llRecall that the free variables of cp:FUT are exactly those of cp together with y.
73
assignment for </J for any natural number n. </J:FUT is then y = x and a-(y) = a(x) = n
is clearly a stratification for </J:FUT such that a(y) = T( </J).
Induction Hypothesis: For all GGree terms 'ljJ containing n or fewer occurrences of
GGree primitives, if T is any F-stratification assignment for 'ljJ, then there is a strati
fication assignment, a, for the NFUT formula 'ljJf:FUT such that (i) a(x) = T('ljJ) and
(ii) for every free variable, u, of 'ljJ, a(u) = T(U).
Assume the GGree term </J contains n + 1 occurrences of primitives. From the
definition of "GGree term" (see 4.1), we have the following cases:
1. </J is --'ljJ for sorne term 'ljJ. Suppose T is an F-stratification assignment for
</J, with T(</J) = m. Now, </J:FUT is the translation of y = </J, equivalently, of
3x( 'ljJf:FUT 1\ y = --x), which is to say </J:FUT is
(*) 3X('ljJf:FUT 1\ ((x = T 1\ Y = T) V (x # T 1\ Y = .-L))),
where 'ljJf:FUT is the translation of x = 'ljJ. 'ljJ contains n occurrences of GGree
primitives and T restricted to the subterms of'ljJ is an F -stratification assignment
for 'ljJ, so by the IH, there is a stratification assignment, a*, for 'ljJf:FUT such that
a*(x) = T('ljJ) and a*(u) = T(U) for every free variable U in 'ljJ. It's easy to verify,
from (*), that a =def a* U {(y, m)} is a stratification assignment for </J:FUT.
By definition, a(y) = m = T(</J) and from the IH, a(u) = T(U) for every free
variable U in </J.
2. The proof of the inductive step is similar for the cases where </J is ~ 'ljJ, 'ljJ ~ X
or \;fx'ljJ[~l.
3. </J is 'ljJ ~ X for sorne terms 'ljJ and X. Suppose T is an F-stratification assignment
for </J and T( </J) = m. By the definition of F -stratification, there is sorne k such
that T('ljJ) = T(X) = k. Now, </J:FUT is
(*) 3x3z('ljJf:FUT 1\ XljFUT 1\ ((x = z 1\ Y = T) V (x # z 1\ Y = .-L))).
The restrictions of T to subterms of'ljJ and to subterms of X are F-stratification
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assignments for 1jJ and X, respectively. Thus, by the IH, there are stratifi
cation assignments, 0"1 and 0"2, for 1jJ!:FUT and X1jFUT, respectively, such that
O"l(X) = r(1jJ) = k = r(x) = 0"2(Z) and O"l(U) = r(u) for every free variable u in
1jJ!:FUT and 0"2(U) = r(u) for every free variable u in X1jFUT. Note in particular,
that 0"1 (u) = 0"2 (u) for every free variable common to both 1jJ!:FUT and X1jFUT.
Alphabetic variants of 1jJ!:FUT and X1jFUT may be chosen so that no bound vari
ables are common to both these formulas, and therefore the intersection of the
domains of 0"1 and 0"2 contains only free variables common to both formulas.
Then it is easy to verify from (*) that 0" =def 0"1 U 0"2 U {(y, m)} is well-defined
and is a stratification assignment for </J:FUT such that the two conditions of the
lemma are satisfied.
4. </J is 1jJ ~ X for sorne terms 1jJ and X. Suppose r is an F-stratification assignment
such that r(</J) = m. Then, by definition of F-stratification, r(x) = m + 1 and
r(1jJ) = m. Now, </J:FUT is
(* ):3x:3z(1jJ:FUT /\ X:FUT /\ (( "z is a unary F - function" /\ (x, y) E z)
V ("zis NOT a unary F - function" /\ y = 0))).
The restrictions of r to subterms of 1jJ and to subterms of X are F-stratification
assignments for 1jJ and X, respectively. By the IH, there are stratification as
signments, 0"1 and 0"2, for 1jJ!:FUT and X1jFUT, respectively, such that O"l(X) =
r(1/J) = m, 0"2(Z) = r(x) = m + 1 and O"l(U) = r(u) for every free variable u in
1/J!:FUT and 0"2(U) = r(u) for every free variable u in X1jFUT. Note, in particular,
that O"l(U) = 0"2(U) for every free variable common to both 1jJ!:FUT and X1jFUT.
Again, we may choose alphabetic variants of 1/J!:FUT and X1jFUT such that they
have no baund variables in common. So 0" =def 0"1 U 0"2 U {(y, m)} is well-defined
and, from (*), we can verify that 0" is a stratification assignment for </J:FUT such
that the two conditions of the lemma are satisfied. (Remember that, on our
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definition of ordered pair, a stratification assignment for y = (x, z) will assign
the same type to each of x, y and z.)
5. Similarly for the case where </> is 1/Jx~p.
6. </> is i1/J[~l for some term 1/J, free variable x and bound variable x. Suppose r
is an F -stratification assignment such that r( </» = m. Then, by definition of
F-stratification, r(x) = r(1/J[~]) = m - 1. Now, </>~FUT is
(* K:Jw( "w is a unary F - function" 1\ VxVz( (x, z) E w f--+ 1/J:FUT) 1\ y = w))
V (-,:::Jw( "w is a unary F - function" 1\ VxV z( (x, z) E w f--+ 1/J:FUT)) 1\ Y = 0).
The restriction of r to subterms of 1/J[~l is an F-stratification assignment for 1/J[~l.
(Recall that, by convention for the purposes of F-stratification assignments, x in 1/J[~l is considered a free variable symbol and therefore 1/J[~l is considered
to be a GGree term.) By the IH, there is a stratification assignment, cr*, for
1/J1jFUT, such that cr*(z) = r(1/J[~]) = r(5:) = m - 1, and cr*(u) = r(u), for u a
free variable of 1/J[~l. It's easy to verify, from (*), that cr =def cr* U {(y, m)} is a
stratification assignment for </>~FUT. By definition, cr(y) = m = r( </» and from
the IH, cr( u) = r( u) for every free variable u in </>.
7. Similarly for the case where </> is :§1/J[~, ~l.
D
The importance of this lemma is that it establishes a precise sufficient condition on
the structure of a GGree term </> which guarantees the existence of i</>[~l and iY</>[~,;l
as courses-of-values. We state this as a theorem.
Theorem 4.3. Let </> be an F-stmtified GGree term. Then for any free variables
x, y, the courses-of-values i</>[~l and iY</>[~,;l exist as unary and binary F-funciions,
respectively.
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Pro of. Note that, by definition, our interpretation of GGree terms guarantees the
existence of sets corresponding to the terms i4>[~l and iy4>[~,;l regardless of strat
ification restrictions. However, if the NFUT formula translating z = 4>, namely
4>~FUT, is unstratified, then both i4>[~l and iy4>[~,;l may simply equal 0. On the
other hand, if 4>~FUT is stratified, then i4>[~l is the unary F-function F such that
VxVz((x, z) E F +--> 4>~FUT), and iy4>[~,;l is the binary F-function G su ch that
VxVyVz((x, y, z) E G +--> 4>~FUT). But sin ce 4>~FUT is stratified if 4> is F-stratified,
F-stratification of 4> is a sufficient condition for the existence of i4>[~l and iy4>[~,;l
as the desired functions, or courses-of-values. D
We may now examine the extent to which we recover Frege's arithmetical devel
opment in GGree-
4.3 Recovering Frege's Arithmetic
Theorem 4.3 above relating F-stratification and the existence of courses-of-values is
needed to state in a reasonable and self-contained way the axioms of GG as recovered
in our interpretation of GGree . In this section, we first state these modified basic laws
and proceed to develop Fregean arithmetic in GGree , drawing parallels to arithmetic
in NFU(T).
4.3.1 Basic Laws of GGree
We state six "Basic Laws" of GGree , statements that equal T on the interpretation
given and that refiect Frege's six laws, stated in 1.2.3. OfficiaIly, we define the theory
GGree as the set of aIl GGree statements 4> such that 4> = T on our interpretation, so
the following is not meant as an axiomatization of GGree .
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Axioms
• Basic Law 1.
a) f- x ~ (y ~ x)
b)f-x~x.
These are clearly true on our interpretation.
• Basic Law II.
f- Vi:(i: r--- y) ~ X r--- y.
This is again clearly true on our interpretation for any term cp. This law corre
sponds to BL II a) in 1.2.3. As we no longer have quantification over function
variables, we have no axiom here corresponding to BL II b). Recall that quan
tification over function variables is "replaceable" by quantification over object
variables, in a sense made precise in 2.2.1.
Note, however, we may also state a schema, true on our interpretation of GGree ,
corresponding to BL II a) in 1.2.3. This schema would follow from the above
in the presence of Unrestricted Comprehension (Abstraction); however, given
the restriction to F-stratified Comprehension (see Basic Law VI below), this
schema is more general:
f- Vi: ( cp[~]) ~ cp, for any GGree term cp.
• Basic Law III. Substitutability of equal terms.
L_ rI-.[(y~z)l . rI-.[('iw(z~W~y~w»l J: GG t ri-. ,- 'f' x ----+ 'f' x ' lor any ree erm 'f"
Note that f- (y ~ z) ~ Vw(z r--- W ~ y r--- w) is the special case of the above
taking cp to be x. As for Basic Law II ab ove , we could have stated a single
axiom 12, from which the above would follow in the presence of U nrestricted
Comprehension, but which is weaker than the above given only F-stratified
Comprehension. The axiom would be stated as:
12We stated the schema first for Basic Law III because we think the meaning is (slightly) clearer.
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\- (y ~ z) ~ x ~ [(\>'w(z ~ W ~ y ~ w)) ~ x], from which the special
case above could still be recovered by substituting for x the term i(x), i.e. the
course-of-values for the identity function (which exists as a unary F-function by
Basic Law VI below).
• Basic Law IV
\- (~ (-x ~~ y)) ~ (-x ~ -y).
This law is clearly true on our interpretation.
• Basic Law V Extensionality.
a) \- (icp[~l ~ i~[~]) ~ vx(cp[~l ~ ~[~]), for any F-stratified GGree terms cp and
~ and variable x.
b) \- (fYcp[~, ~l ~ fY~[~,~]) ~ VxVy(cp[~, ~l - 'l/J[~, ~]), for any F-stratified
GGree terms cp and ~ and variables x, y.
Extensionality applies to courses-of-values corresponding to F-stratified GGree
terms. This law is true (in this form) on our interpretation as we showed that
courses-of-values corresponding to F-stratified GGree terms exist as graphs of
(unary or binary) F-functions and two such graphs are clearly equal iff the
functions to which they correspond are equal for all arguments.
• Basic Law VI. F-stratified Comprehension.
\- :Jy\>'x(x ~ y ~ cp[~]), for each F-stratified GGree term cp and variable x.
If cp is F-stratified, then i(cp[~]) exists as the intended unary F-function and
and satisfies Vx(x ~ i( cp[~]) ~ cp[~]) = T, and Basic Law VI, as stated here,
therefore holds in GGree .
The most significant difference between the Basic Laws as stated here and as
stated in 1.2.3 is the restriction to F-stratified terms in the axioms of Extensionality
and Comprehension. Clearly, this is (at least roughly) equivalent to the restrictions, in
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NFU(T), of Extensionality to sets (as opposed to urelements) and of Comprehension
to stratified formulas. We must examine to what extent this affects the development
of Arithmetic in GGree .
4.3.2 Arithmetic in GGree
In this section, we give basic arithmetical definitions in GGree , an appropriately mod
ified version of the presentation given in 2.2 for GG.
Bijection and Equinumerosity
Definition 2.3, of "p is (the graph of) an injective function", is changed slightly as
the "iterated application" of GG is replaced by the "binary application" of GGree .