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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
Master of Science
Copyright © Ryan Beaton, 2004
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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
2.2 Formally Defining Cardinality .............. .
VIl
27
30
2.2.1 The Application Operator and "Reduction"
2.2.2 Bijection and Equinumerosity . . . . . . . .
2.2.3 Successor, Ancestral and the Natural Numbers .
2.3 Russell's Paradox . . . . . . . . . . . . ........ .
3 New Foundations, Modifications and Consistency Results
3.1 Quine's Original Theory . . . . . . . . . . . . . . . . . .
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
concept of "existence":
Definition 2.1. "Existence operator":
E2(f) =def':' Vi(':' f(i)), abbreviated E2(f) =def 3if(i).
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
the foUowing short-hand notation:
I(p) =def VxVi)[(x ,---., (i)'---" p)) -.:..... (Vi(x ,---., (i ,---., p) -.:..... i) ~ i))].
above.
35
Thus, if p is the extension y(&:F(i, fj)) of sorne binary function, F(x, y), then
I(p) = T iff F is injective as defined above. Frege then gives the following two
definitionslO .
Definition 2.4. )p =def y(&:(I(p) À Vû(û ~ i ~ :3v(v ~ fj À û ~ (v ~ p))))).
Thus, for each object p, )p is a relation such that, given objects a, b, a ~ (b ~
)p) = T iff [I(p) = T and for each u such that u ~ a = T, there is a v such that
v ~ b = T and u ~ (v ~ p) = T]. In such a case, Frege says "the p-relation maps
the a-concept into the b-concept." 11 This is the GG equivalent of an injection from
the set {ulu ~ a = T} into the set {vlv ~ b = T}.
Definition 2.5. Inverse relation. U(p) =def y(&:(fj ~ (i ,--.., p))).
This gives, for any objects a, b,p, that (a ~ (b ~ U(p))) = (b ~ (a ~ p)), so
that, in particular, --(a ~ (b ~ U(p))) = --(b ~ (a ~ p)). It follows, for any
a, b,p, that a ~ (b ~)U(p)) = T iff b ~ (a '--"')p) = T. We can now capture the
notion of a bijective correspondence, or equinumerosity, as follows.
Definition 2.6. Given objects a, b, the concepts --(x ~ a) and --(x ,--.., b) are
said to be equinumerous if there exists an object p such that
[a,--.., (b '--"')p) 1\ b ~ (a ~)U(p))] = T, i.e., p injects the a-concept into the b-concept
and U(p) injects the b-concept into the a-concept.
With the above definitions in hand, Frege can now define, for any object a, the
cardinal of the a-concept ("Anzahl des a-Begriffes"). Recall that "being n in number"
or "having cardinality n"12 is for Frege a second-order concept. However, to this
lOFrege himself never uses ~, Â or V in his definitions, but for the sake of readability we will use
them freely.
Il "Die p-Beziehung bildet den a-Begriff in den b-Begriff ab." ([11], §38) Note that Frege is implicitly
defining, for an object a, the a-concept as --(x r-- a).
12n is not intended to indicate a restriction to finite numbers here.
36
there corresponds a first-order concept by the association of first-order functions with
their courses-of-value and the use of application, as discussed above. This first-order
function may be stated as "The x-concept has cardinality n," where x is an object
variable. Now if we apply the abstraction operator to this function, we obtain the
cardinal n. That is, n is the course-of-values corresponding to the function that takes
an object a to T iff the a-concept has cardinality n, and takes a to 1.. otherwise. If we
were to replace concepts with the sets for which they are characteristic functions, then
we would have that n is the set of all sets with n elements. The number associated
with an object a is the extension (course-of-values) of the concept "equinumerous
with the a-concept." Formally:
Definition 2.7. The cardinal, CardF(u), of the u-concept.
CardF(u) =def i[:3Y(u r--- (x r---)U(Y)) A x r--- (u r---)Y))].
We can then define the cardinals zero and one as follows.
Definition 2.8. Zero. OF =def CardF(i(':' (x ~ x))). (OF for "Frege 0".)
That is, OF is the cardinality of the concept "not equal to itself." Note that to
every function F(x) is associated a cardinality CardF(iF(x)) and many extensionally
different 13 functions have cardinality OF, e.g. any function whose range is the natural
numbers, assuming T is not equal to any natural number. But the cardinality operator
defined in 2.7 is intended primarily as an operator on extensions of concepts and any
function F(x) can be converted into a concept --F(x) by an application of the
truth function. (For F(x) a concept, --F(x) = F(x) for all x.) And modulo such
an application, all functions with cardinality OF are extensionally equivalent, i.e. take
the same value (uniformly 1.. in this case) for every argument.
13Two functions being "extensionally different" meaning that there is at least one (set of) argu
ment(s) for which the functions return different values, which, given Basic Law V, is equivalent to
the courses-of-values corresponding to the two functions being unequal.
37
Definition 2.9. One. IF =def CardF(i(i ~ OF )).
Any concept, e.g. "equal to OF," , under which a unique object falls has cardinality
IF. Note that, having defined OF and IF as such, a proof (though an easy one) is
needed in GG to show that IF is the successor of OF, with successor as defined below.
2.2.3 Successor, Ancestral and the Natural Numbers
We come now to Frege's definitions of successor, ancestral and, finaIly, natural number.
The first two definitions are of relations, and the third of a concept.
Definition 2.10. Successor.
SF =def y(i(~il~v(i ~ CardF(i(z ~ il  -:., (z ~ v)))  v ~ il  fJ ~ CardF(ù)))).
We'Il write XSFY for x ~ (y ~ SF).
80 bis a successor of a, aSFb = T, iff there are objects c,d suchthat (i) a is the
cardinality of the concept "falling under the c-concept but not equal to d," (ii) d falls
under the c-concept and (iii) b is the cardinality of the c-concept. Roughly, b is a
successor of a if b is the cardinal of a concept under which exactly one more object
falls than under a concept for which a is the cardinal.
The following definition of the "ancestral of a relation" is intended mainly to be
applied to the extension (double course-of-values) q of sorne given relation, but can of
course be rneaningfully applied to any object. Given a relation F(x, y), the ancestral
of F, denoted <~(iF(i,:iï))14 is intended to capture the relation of "precedes by a finite
(and non-zero) nurnber of places in the F-relation."
Definition 2.11. Ancestral. Given an object q, we define the ancestral of the q-
14Essentially, Frege's own notation; see [11] §45.
38
relation as
<q=dej y(i(Vg[(\ffi\fîi(g(fi) A fi ~ (îi ~ q)) ~ g(îi)) ~
\fiiJ(i ~ (iiJ ~ q) ~ g(iiJ)) ~ g(Y)])).
Let us calI a concept, G(x), hereditary in the q-relation if . . .
[\ffi\fîi(G(fi) 1\ fi ~ (îi ~ q)) ~ G(îi)] = T. Then the above definition states that
a <q b = T ifIfor every concept, G(x), which is hereditary in the q-relation, it follows
from the fact that G(c) = T for every "q-successor" c of a (i.e. \fiiJ(a ~ (iiJ ~ q) ~
G(iiJ)) = T), that G(b) = T.
The following definition is straightforward.
Definition 2.12. Equal ta or finitely preceding in the q-relation.
~q=dej y(i(i <q fJ V i == fJ))·
We come finally to Frege's definition of natural number as equal to zero or finitely
preceded by zero in the SF-relation.
Definition 2.13. An object nF is a (Frege) natural number if OF ~SF nF = T.
lmmediately after laying down this definition in §46 of the Grundgesetze ([11]),
Frege says that he has indicated in §82 of his Grundlagen ([18]) that, for any natural
number nF, the cardinality of the concept "belonging to the series of natural numbers
ending with nF" ("der mit n endenden Anzahlenreihe angehorend" 15) is the successor
of nF. As a theorem of GG, this may be stated as:
f- (OF ~SF nF) ~ (nFSFCardF(nF ~~SF)).
Note that a ~ (nF ~~SF) = T ifI a = nF or a precedes nF in the SF-sequence, i.e.
15Frege, of course, did not subscript his natural numbers with "F."
39
nF ~::;SF is the extension of the concept "belonging to the series of natural numbers
ending with nF." (In other words, nF ~::;SF= i(i ::;SF nF).)
It's interesting that Frege calls attention to this theorem as central to his devel
opment of arithmetic, as we must discuss this exact theorem in sorne detail when
reconstructing a modified form of GG in NFUT. What is clear is that an inductive
proof will be required. That the principle of induction holds for natural numbers
in GG is clear from the definition of Ancestral, 2.11 above. For suppose G(x) is a
function such that G(OF) = T and G(x) is hereditary in the SF-relation, i.e., that
G(a) = T implies G(b) for ban SF-successor of a, formally:
(*) f- \lu\lv[(G(u) À USFV) ~ G(v)].
If nF is a natural number, then by definition nF = OF or OF <SF nF, i.e. nF = OF or
f- yg(\lu\lv((g(u) À USFV) ~ g(v)) ~ [YW(OFSFW ~ g(w)) ~ g(nF)])
from which it follows by an instance of Universal Specification, that . .
f- [\lu\lv((G(u) 1\ USFV) ~ G(v))] ~ [\lW(OFSFW ~ G(w)) ~ G(nF)].
Then by (*) above and an instance of Modus Panens, we have
(t) f- \lW(OFSFW ~ G(w)) ~ G(nF).
It's quite easy to demonstrate that cn f- OFSFa ~ a ::::: IF. (In §87, Frege proves
sentence 71: f- I(SF), i.e. that there is at most one SF-successor to any object16,
and it's easy to show that f- OFSFIF, so (t) follows.) From the assumption f- G(OF)
together with (*) and (t), we may deduce f- G(IF). From f- G(IF) and (t), it now
follows that
f- \lW(OFSFW ~ G(w)).
Finally, then (t) and Modus Panens give us that
f- G(nF).
16This does not require induction, and it is not limited to objects, a, whose corresponding concepts,
-(x ".--, a), are finite.
40
2.3 Russell's Paradox
The inconsistency in Frege's painstakingly constructed logical theory, the project
to which he had dedicated his career, was brought to light in his 1902 exchange
with Russell ([8], pp.59-85). Frege did not immediately recognize the extent of the
disaster, quickly appending an analysis of Russell's paradox to the second volume
of the Grundgesetze, an analysis that, although quite detailed and insightful, ends
optimistically with the statement that should the problem17 not be so fully resolved
as he had believed, he "does not doubt that the way to a solution has been found"
([10] Nachwort, p.265). He soon came to realize that no simple solution presented
itself and, in print at least, abandoned the project of reducing of mathematics to pure
logic.
Once it is pointed out, the paradox is easy to derive. Frege gives the following
derivation in his Appendix. In Volume 1 ([11]), Frege has proved
f- f(x) ~ x ~ yf(fj) (special case of sentence 77, §91). We may substitute
-:. (x ~ x) for f(x) and i( -:. (x ~ x)) for x to obtain
f- [-:' (i(-:' (x ~ x)) ~ i(-:' (x ~ x)))] ~ i(-:' (x ~ x)) ~ i(-:' (x ~ x)).
It then follows immediately (reductio ad absurdum: sentence la, §49) that
(t) f- i(-:' (x ~ x)) ~ i(-:' (x ~ x)).
Similarly, starting from f--:' f (x) ~-:. (x ~ y f (fj)) (also a special case of sentence
77, §91), we arrive at
(:n f--:' (i(-:' (x ~ x)) ~ i(-:' (x ~ x))).
(t) and (:n give the (un)desired contradiction. Sentence la, mentioned ab ove , is
f- x ~ (-:' x ~ y), from which, using (t) and (:j:), we can now deduce any sentence
we wish.
Different interpretations have been given of the exact cause of the contradiction in
17In particular, Frege says, the problem of how we can grasp and recognize numbers as logical
objects.
41
Frege's Grundgesetze. The most conspicuous source of inconsistency is the unlimited
abstraction principle, which correlat es one-to-one unary first-order functions (and, ul
timately m-ary nth-order functions for any fixed m, n) with courses-of-values18 , which
are themselves arguments of these first-order functions. This correlation, together
with the application operator which gives us the means to apply a course-of-values
to itself, would seem, at least at first thought, to be the source of problems, and this
is indeed the "majority opinion"; see, for instance, Boolos [4]. However, there have
been other interpretations, as weIl. For instance, Dummett [7] seems to primarily
blame the second-order quantification in Frege's logic; Boolos offers a rebuttal of this
opinion in [4]. Aczel [1] offers yet another perspective in his analysis of the underly
ing structure of the formaI language of GG as a À-structure, and in particular as a
construction Aczel caUs a "Frege structure." Aczel shows that, in a Frege structure,
the introduction of any function playing the role of the truth function --x results
in contradiction, and so fingers --x as the real culprit. Unfortunately, any serious
discussion of these various conclusions would take us too far astray from the stated
goal of our essay. We turn our attention now to NF and NFU, the set-theoretic
framework we will use in Chapter 4 to interpret GGree .
18The correlation is one-to-one if we take an extensional view of functions, Le., that two given
functions are identical exactly if they take the same value for each (set of) argument(s).
42
Chapter 3
New Foundations, Modifications
and Consistency Results
This chapter introduces Quine's original theory, as well as Jensen's modification,
New Foundations with Urelements. A celebrated consistency result of Specker's is
presented for NF, and its application to NFU is shown. The consistency of NFU
relative to a weakened version of Zermelo set theory then follows by a theorem of
Jensen's. Finally, extensions of NFU, including the addition of an "Axiom of Count
ing" important for Chapter 4, are discussed and recent results are mentioned. As the
results in this chapter are not original and are presented mainly for use in Chapter 4,
we state lesser results without proof and sketch proofs for more important theorems,
giving references to the original sources.
3.1 Quine's Original Theory
W. v. Quine's article [30], "New Foundations for Mathematical Logic," appeared in
the February 1937 edit ion of The American Mathematical Monthly. In the article,
Quine st arts by stating his view that in Whitehead and Russell's Principia Math-
43
ematica we have "good evidence that aIl mathematics is translatable into logic."
Quine then notes that there is possible a further translation of the logic of the Prin
cipia into a logic whose language contains expressions built only from an infinite list,
Xo, Xl,· .. ,xn ,· .. , of variable symbols and the three primitives 1 (alternative denial),
V and E, together with parentheses. This is the language Quine adopts for his New
Foundations l . NaturaIly, the atomic formulas of the language are expressions of the
form X E y, for variable symbols X and y, and formulas are defined as usual. It is the
stated purpose of the remainder of Quine's article to outline this further translation
of the Principia. To this end, Quine provides a standard list of definitions for truth
functional connectives, the existential quantifier, "x being a subset of y," etc.; ordered
pairs are defined in the usual (Kuratowski) set-theoretic fashion2 , relations are defined
as sets of ordered pairs, and so on. Quine does not take equality to be a primitive,
giving the following "Leibnizian" definition: x = Y <r?def Vz(x E z +--+ Y E z).
Quine also adopts as an axiom the following "princip le of extensionality."
Extensionality. VxVy(Vz(z E x +--+ Z E y) ---+ x = y).
To this "principle," Quine adds five "rules" (actuaIly, two axioms and three rules of
inference). These include one theorem of propositional logic, the rule of inference
modus ponens and standard rules of universal generalization and universal specifi
cation, together sufficient for deriving aIl theorems of standard first-order predicate
logic. Thus far, there is nothing particular about Quine's logic. Rounding out his
rules, however, is an axiom of set formation. Quine points out that unrestricted set
formation, in the form of an axiom schema such as
Unrestricted Comprehension. ~xVy(y E x +--+ cp),
where cp is a formula not containing x free, leads to RusseIl's paradox, and that Rus-
lQuine does not actually use the symbol 't:/; he symbolises the universal quantification of an
expression such as x = x by (x)(x = x) rather than by 't:/x(x = x). We shall make use of the latter
notation, with the same meaning, of course.
2I.e. (x,y) =def {{x}, {x, y}}.
44
sell's Theory of Types overcomes this difficulty by assigning a "type" (i.e. a natural
number) to each variable symbol and barring all expressions of the form x E y, except
in the case where x is of type one less than is y. However, as Quine points out, "in
all contexts [i.e., formulas] the types appropriate to the several variables are actually
left unspecified; the context remains systematically ambiguous, in the sense that the
types of its variables may be construed in any fashion conformable to the require
ment that E connect variables only of consecutively ascending types." Any formula,
cp, of untyped set theory, that has a meaningful interpretation in type theory, i.e., for
which it is possible to assign types (natural numbers) to its variables such that the
resulting expression is a formula of TT, will in fact have infinitely many meaningful
interpretations in TT, obtained by repeatedly "shifting" the type of each variable
up by one. In type theory, there are endless "cleavages and reduplications" of sets,
relations and even arithmetic, which Quine says are not only "intuitively repugnant,
but they continually call for more or less elaborate technical manoeuvres [ ... ]."3
To avoid this, and since there is no apparent difficulty with considering expressions
of the form x tJ. x as meaningful, only with considering the existence of a set composed
of exactly the members satisfying it, Quine proposes that we reject the strict Principia
separation of types, and sim ply restrict the axiom of set formation in an appropriate
way. To this end, we may define Quine's notion of stratification as follows.
Definition 3.1. A formula, cp, (of NF) is said to be stratified if there exists an in
dexing, (J', (by natural numbers) of the variable symbols in cp such that the resulting
formula, cp*, is a formula of TT. The indexing, (J', will be called a stratification as
signment for cp and cp* will be called the TT-formula obtained from cp by (J', or simply
a TT-formula obtained from cp.
3Interestingly, Russell rejects Quine's solution (if indeed it is one) of the paradoxes as "created ad
hoc and not to be such that even the cleverest logician would have thought of if he had not known
of the contradictions." ([24], p.SO)
45
As a final axiom, then, Quine adopts the following revised axiom schema of com
prehension.
Stratified Comprehension. :JxVy(y Ex+-> cP), where cP is a stratified formula not
containing x freé.
Note that the definition of stratified formula applies equally well if we adopt the
equality symbol, = , as a primitive. That is, it follows from Quine's "definition" of
equality that if cP is a formula such that the expression x = y occurs in cP, then any
stratification assignment for (the formula abbreviated by) cP assigns the same index
to x and to y, as required for the result to be a legitimate formula of TT. Below we
adopt equality as a primitive.
3.1.1 A Concise Statement of Quine's New Foundations
To recap the above introduction to New Foundations we give a more concise presen
tation of Quine's theory. We will denote the set theory described below by "NF."
NF is a theory in first-order predicate logic (with identity). Its signature contains
the single binary predicate symbol E, and its only axioms are Extensionality and the
Schema of Stratified Comprehension, stated ab ove , and repeated here for convenience:
Axiom of Extensionality. VxVy(Vz(z Ex+-> Z E y) ---+ x = y).
(Note that the right-Ieft implication is here assumed an axiom of predicate logic.)
Axiom Schema of Stratified Comprehension. :JxVy(y Ex+-> cP), where cP is a stratified
formula not containing x free.
4The Axiom Schema of Stratified Comprehension can be replaced by an equivalent finite list of
comprehension axioms. See [14] and [15].
46
3.2 The Consistency of NF and Variants
The consistency of NF relative ta any more thoroughly studied or accepted set theory
remains essentially an open question. We will present and sketch a pro of of a famous
consistency result of Specker's ([28]) ta the effect that a model of NF exists iff there
exists a model of TT with a "shifting automorphism," a.k.a. a "Specker model." The
importance of this result for our purposes is that it can be easily modified ta prove
that the set theory NFU (defined below) is consistent if there is a model of TTU
(also defined below), which in turn can be constructed from a model of weak Zermelo
set theory. NFU and weak extensions thereof will then provide the framework for an
interpretation of GG.
3.2.1 Specker Models and Models of NF
In this section we will indicate the free variables Xl, X2,' .. ,Xn , of a formula cp as in
CP(Xl' X2, . .. ,xn ). If CP(Xl, X2, . .. ,xn ) is a stratified NF-formula, we will denote by
cp*(X~l, X~2, . .. ,x~n) the TT-formula obtained from cp by a stratification assignment,
(J, such that O'(Xj) = ij for 1 ~ j ~ n.
We now show that the existence of a model of NF is equivalent to the existence
of a model of TT for which there exists a "shifting automorphism." In this, we follow
essentially Jensen's presentation in [20].
If M = (Ui , Ei)iEN is a model of TT, i.e., M 1= TT, then we will denote by
M+ the model (Ui , Ei)iEN\{O} 5 of TT. That is, M+ 1= TT is the model obtained
by "dropping" the bottom level of M. Now, a shijting automorphism6 of M is an
isomorphism 0' : M --+ M+ such that, for each i, 0' restricted ta Ui is a bijection ta
Ui+l . In other words, 0' is a bijection on the underlying sets, with O'(Ui ) = Ui+1 and
5 Really, M+ should be defined as (Vi, EniEN with Vi = Ui+l and Ei=Ei+l for i E N, but l take
it the meaning is clear.
6Technically not an automorphism, but the terminology is standard.
47
Definition 3.2. A Specker model, M, is a model of TT such that there exists a
shifting automorphism of M.
With these definitions in place, we can prove the foIlowing.
Theorem 3.3. (Specker) NF is consistent iff there exists a Specker model.
Proof. Let M = (Ui , Ei)iEN be a Specker model, and let (J be a shifting automorphism
for M. Define N = (U, EN) by taking U = Ua and EN such that, for u, v E Ua,
U EN V {::}def U Ea dv). We show that N 1= NF.
We first note that for any stratified NF-formula cfy(XI, X2, . .. ,xn), any TT-formula
cfy* (X~l , X~2, ... , x~n) obtained from cfy and any UI, U2, ... , Un in Ua, the foIlowing equiv
alence holds: (*) N 1= cfy[UI, U2, ... ,un] {::} M 1= cfy* [(Jil (ud, (Ji2 (U2), ... , (Jin (Un)].
(*) is shown by induction: If cfy is x E y, then cfy* is necessarily Xi E yi+l
for some natural number i. Now if u, v E Ua, then by definition of EN, we have
U EN v {::} U Ea (J( v). It foIlows from the fact that (J is a shifting automorphism that
U Ea (J(v) {::} (Ji(U) Ei (Ji+1(v) for every i. So we have
NI= cfy[u,v] {::} MI= cfy*[(Ji(U),(Ji+I(V)],
as required. If cfy is x = y, there is nothing to prove. Induction on the struc
ture of cfy is straightforward, although the case of quantifiers requires some care.
Let us consider the case where cfy(XI' X2, . .. ,xn) is Vy1jJ(y, Xl, X2,' .. ,xn) for some
1jJ. Then cfy* is Vyj 1jJ* (yj , X~l , X~2 , ... , x~) for some appropriate indexing of the vari
ables. Let UI, U2,· .. ,Un E Ua· Now, if N 1= cfy[UI, U2, . .. ,un] then, for aIl v E Ua,
N 1= 1jJ[v, UI, U2, ... ,un]' By the induction hypothesis,
N 1= 1jJ [v, UI, U2, ... ,un] {::} M 1= 1jJ* [(Jj ( V ), (Jil ( UI), (Ji2 ( U2), ... , (Jin ( Un)],
SO we have that (t) (for aIl v E Ua, M 1= 1jJ* [(Jj (v), (Jil (UI), (Ji2 (U2), ... , (Jin (un)]).
But (Jj is a bijection from Ua to Uj , so every w E Uj is w = (Jj (v) for some v E Ua
and therefore (t) implies that
48
for aH w E Uj , MF 'l,Ii* [w, (ji! (Ul), (ji2 (U2), . .. ,(jin (un)],
i.e. that MF q'>*[(ji!(ud,(ji2(U2)"" ,(jin(Un)].
The reverse implication follows similarly.
So (*) holds. But now we need only note that the axiom of Extensionality and aH
instances of the Axiom Schema of Stratified Comprehension are stratified formulas
of NF and any TT-formula obtained from one of these axioms is an axiom of TT.
Since M F TT, M of course models all such axioms, and by (*), N therefore models
Extensionality and Stratified Comprehension. I.e. NF NF.
The converse is much simpler. For suppose N = (U, EN) F NF. Define M =
(Ui , Ei)iEf\! by letting Ui = U and Ei=EN for all i. It's easy to verify that MF TT
and that the identity function on U de fines a shifting automorphism of M. 0
3.2.2 Jensen's New Foundations with Urelements
Just over thirty years after Quine introduced NF, Ronald Bjorn Jensen wrote an
article titled "On the Consistency of a Slight(?) Modification of Quine's New Foun
dations" ([20]). The "slight" modification Jensen considers is a weakening of the
axiom of extensionality to allow "Urelements," i.e. non-sets. Thus, in addition to an
empty set, the resulting theory, denoted by "NFU," would allow for many different
objects that have no members. Formally, NFU is the theory obtained from NF by
replacing the Axiom of Extensionality with the following:
Axiom of Weak Extensionality.
\fx\fy[3z(z E x) ~ ((\fw(w E x <-t W E y) ~ x = y))].
Note that any object which is a non-empty set, i.e., an object which has at least one
member, is subject to the fulllaw of extensionality. On the face of it, this modification
indeed seems slight; this was certainly Quine's own opinion. For in his 1937 article,
49
he suggests that for non-sets, a, we may interpret expressions of the form x E a as
stating the equality of x and a. In a footnote, he points out that "this interpretation,
along with the subsequent postulate Pl [(Full) Extensionality], results in the fusion
of every individual with its unit class; but this is harmless." ([30], p. 71) He repeats
this opinion in his 1940 book, Mathematical Logic, where he also mentions that this
"fusion" of an individual with its unit class is warned against by Frege, but in the
case of non-sets or "individuals," the "retention of the pre-Fregean attitude leads to
no trouble." ([31], p.136). But, as we have seen, Frege himself was taken by the same
impulse for uniformity among the objects in his theory. Quine wanted all objects
in his theory to be interpretable as sets and Frege wanted all objects in his to be
interpretable as courses-of-values. But both felt that their theories should, at least in
principle, be applicable to aU objects. That being the case, it seems hard to provide
a reasonable justification for their interpretations of non-sets as sets and non-courses
of-values as courses-of-values, respectively, other than the desire for uniformity, to
"tidy up the picture.,,7 In any case, the modification proposed by Jensen is quite
natural.
Here we are primarily concerned with the effect this modification has on the
consistency of the theory. We will now show that NFU is in fact consistent relative
to a weak form of Zermelo set theory. This result indicates that the fusion of a
non-class with its unit class is, in terms of consistency at least, far from "harmless."
Before defining weak Zermelo set theory, denoted "WZ," the first step in our pro of
is to record an immediate corollary of Theorem 3.3. Let us denote by "TTU" the
modified theory of types obtained from TT by replacing each instance of Extension
ality with the corresponding instance of Weak Extensionality. We define a shifting
automorphism of a model of TTU exactly as for models of TT, and a Specker model
7 If, on the other hand, we wish our Iogic to apply exclusively to mathematicai objects, perhaps
one couid make the case for these interpretations.
50
of TTU as a model of TTU for which there is a shifting automorphism. Then, simply
noting that the Axiom of Weak Extensionality is also a stratified NFU -formula, the
pro of of Theorem 3.3 can be carried out exactly as above to give
Corollary 3.1. NFU is consistent if there exists a Specker model of TTU.
In fact, since formulations of the axioms of Infinity and Choice are also stratified,
we may give a refined version of this corollary:
Corollary 3.2. NFU (+ Infinity, + Infinity + Choice) is consistent if there exists a
Specker model of TTU (+ Infinity, + Infinity + ChoicejB.
In the next section, it is actually a Specker model of TTU that we construct from
a model of WZ. We can then define a model of NFU in the same way that we defined
a model of NF from a Specker model (of TT).
3.3 Constructing Models of NFU
Below, we define the theory WZ, some equiconsistent extensions, and construct a
Specker model of TTU from a model of one such extension. The construction also
shows the consistency of NFU + Infinity (+ Choice) relative to WZ + Infinity (+
Choice). Finally, we will consider the addition to NFU of an "Axiom of Counting,"
introduced by Rosser and discussed by Randall Holmes in his work on mathematics in
NFU-style set theories. Holmes is primarily interested in much st ronger extensions of
NFU and gives a detailed account of related consistency results in [17]. As our ai ms
8The same statement could technically have been made with NF and TT replacing NFU and
TTU, respectively; however Specker [27] has shown the NF disproves Choice, and therefore proves
Infinity. So every Specker model of TT models Infinity + ...., Choice.
51
are more modest here, we will discuss specifically only the addition of "Counting"
to NFU, or rather to NFUT (see 4.2.1), which will be the theory we finally use to
interpret a modification of GG in the next chapter. NFUT with Counting appears
to be sufficient for Fregean arithmetic.
3.3.1 Weak Zermelo Set Theory and Extensions
Below, a formula cp is said to be bounded if aIl occurrences of V in cp are within expres
sions of the form Vx(x E y ---+ ..• ) and aIl occurrences of :3 are within expressions of
the form :3x(x E y 1\ ... ). WZ is the first-order set theory axiomatized by (universal
closures of) the following.
1. Extensionality: V Z (z E x <--t Z E y) ---+ x = y.
2. Empty set, 0: :3xVy(y ~ x).
3. Pair set, {x, y}: :3zVw(w E z <--t (w = X V W = y)).
4. Union, Ux: :3yVz(z E y <--t :3w(w E x 1\ Z E w)).
5. Power set, Px: :3yVz(z E y <--t Vw(w E Z ---+ W EX)).
6. Schema of Bounded Comprehension: :3xVy(y E x <--t cp), where cp is a bounded
formula not containing x free.
In his article, Jensen also outlines a proof of
Lemma 3.1. TT( + Infinity, + Infinity + Choice) is consistent if! WZ( + Infinity, +
Infinity + Chaice) is consistent.
We give here a pro of of the following theorem of Jensen's.
Theorem 3.4. (Jensen) There is a Specker model of TTU (+ Infinity, + Infinity +
Choice) if WZ (+ Infinity, + Infinity + Choice) is consistent.
52
It follows then, by Corollary 3.2, that
Corollary 3.3. NFU (+ Infinity, + Infinity + Choice) is consistent if WZ (+ Infin
ity, + Infinity + Choice) is consistent, equivalently, by Lemma 3.1, if TT( + Infinity,
+ Infinity + Choice) is consistent.
Pmof. We sketch a proof of Theorem 3.4 to give the reader a picture of an NFU
model. For further details ofthe proof, we refer the reader to Jensen's original article
[20].
Assume WZ is consistent. We add Skolem functions to WZ to obtain the theory
WZ*. That is, WZ* is the theory WZ plus the axioms
Skolem axioms
\Ix 1 , X2,···, xn[:Jy(<p(y, Xl, X2,···, Xn) ~ <P(J</J(X1, X2,···, Xn), Xl, X2,···, Xn))],
where f<P is a new function symbol for each WZ* formula <p. WZ* is consistent if WZ
lS.
N ext, consider the following three axioms for a set of constant names {Ci hEM.
2. <p(CilJ Ci2'.··' Cin) ~ <p(CjlJ ch,···, Cjn) for il < i 2 < ... < in, j1 < j2 < ... < jn,
and any WZ* formula <p.
3. Ci is infinite.
Let W Zl denote the theory obtained from WZ* by adding the constant symbols
{Ci hEM and the axioms 1 and 2 above. Let W Z2 denote the theory obtained from
W Zl by further adding axiom 3.
Let M = (U, EM, L;) 1= WZ* (where L; interprets the Skolem functions). We show
that any finite subset of the W Zl axioms is interpretable in M. Let Ua E U. Define
Ui, i E M, by M 1= Ui+1 = Ui U PUi. Clearly, M 1= Ui U PUi ç Uj for i < j. Let
53
(h, tP2' .. . ,tPn be any finite sequence of WZ* formulas. Byan application of Ramsey's
theorem, we obtain an infinite subset U* = {Uij hEN of {UdiEN such that, for each
<pi,i S; n,
M ~ <Pi ( Ujl' Uh, ... ,Ujmi) +-+ <Pi ( Uh 1 , Uh2' ... ,Uhmi ) for Ujk' Uhk E U*, k S; mi, and
j1 < j2 < ... < jmil hl < h2 < ... < hmi · Taking Cj to be Uij for j E M, we therefore
have that M interprets any given finite subset of the W Zl axioms. By Compactness,
WZ1 is therefore consistent if WZ* is. Furthermore, if M ~ lnfinity, then we may
take Uo to be an infinite set and axiom 3 will be satisfied by each Ci. If WZ + Choice
is consistent, we may also choose M such that M ~ Choice. So if WZ (+ lnfinity,
+ lnfinity + Choice) is consistent, then so is WZ1(WZ2, WZ2 + Choice).
Now, if there exists a model of WZ1 , then there exists a term model. Suppose
M = (U, EM, l:) ~ WZ1 is a term model. Then an automorphism, CY, of (U, EM) is
uniquely determined by CY(Ci) = CH1, i E M. Let us define, for i E M:
Ui =def {x E UI M ~ x E Ci} and x Ei y {:}def M ~ (y ç Ci 1\ x E y). Setting
N = (Ui , Ei)iEN, we wish to show that N ~ TTU; it's clear that CY restricted to
UiEN Ui is then a shifting automorphism for N.
N ~ Weak Extensionality by definition of Ei. To show Comprehension, one
proves by induction that for any TTU formula tP( X~l , X~2 , ... , x~n), there is a bounded
WZ1 formula <P*(X1, X2, ... ,xn) such that
N ~ tP[U1, U2,···, un] {:} M ~ tP*[U1, U2, ... , Un]. Comprehension for N then follows
from Bounded Comprehension for M. So N is a Specker model of TTU. See [20] for
details. If M models Infinity (Choice), then N models Infinity (Choice). 0
We then obtain the desired model, (P, EP ), of NFU from a Specker model of TTU
as described previously, namely P =def Uo and for any u, v E Uo,
U EP V {:}def U Eo cy(v).
54
3.4 Arithmetic in NFU
We take a brief look now at arithmetical concepts in NFU. The basic definitions are
essentially those originally given by Frege, transferred to a set-theoretic context.
Let us assume that ordered pairs are defined and satisfy (*) (x, y) = (z, w) +-t
(x = z 1\ y = w). (x, y) may be defined as usual as {{x}, {x, y}}, but in the next
chapter we discuss and adopt an alternative definition; for now we assume only that
ordered pairs exist and satisfy (*). Then relation, function, bijection, etc., are defined
as usual and we introduce also the following definitions.
Definition 3.5. Two objects a, b are said to be equinumerous if there exists a bijection
between a and b (i.e., a bijective correspondence between the elements of a and the
elements of b; this is well-defined for urelements also). We write a rv b for "a and b
are equinumerous."
Note, in particular, that all urelements are equinumerous to the empty set.
Definition 3.6. For any object a, the cardinality of a is defined as
Card(a) =def {xix rv a}.
It's easy to verify that an NFU formula defining x rv y is stratified (assuming, for
instance, that (x, y) is defined as {{x}, {x, y} }, or as we define it in Chapter 4).
Definition 3.7. Zero is defined as 0 =def Card({xlx =1= x}}.
Definition 3.8. One is defined as 1 =def Card( {O}).
We define now the three important notions of successor, ancestral and natural
number.
Definition 3.9. Y is the successor of x, written xSy, iff
:Jz:Jw(w E z 1\ y == Card(z) 1\ x = Card(z - {w})).
55
Note that this formula is stratified and therefore the successor relation, S, exists
as a set in NFU.
Definition 3.10. Given any binary relation (set of ordered pairs) X, we define the
ancestral, ~ x, of X as follows:
x ~x y {=}def Vz[(VwVu((w E z 1\ wXu) -t u E z)) -t (x E z -t y E z)J9.
Definition 3.11. A natural number is any object x such that 0 ~s x, i.e., such that
"0 is an ancestor of x in the S-relation."
o ~s x is (an abbreviation for) a stratified formula of NFU and so the set of
natural numbers exists in NFU.
3.4.1 The Axiom of Counting
The problem with the above NFU definition of natural number is that we do not know,
for a general formula cjJ( x), that Mathematical Induction holds with respect to cjJ( x).
If cjJ(x) is a stratified formula, then the set {xlcjJ(x)} exists and it is straightforward to
deduce from the definition of natural number that cjJ(O) and cjJ(n) -t cjJ(n + 1) jointly
imply Vn(n EN -t cjJ(n)). (See 2.2.3; the proof that induction holds in NF/NFU for
stratified formulas parallels the proof given there for GG.) However, if cjJ(x) is unstrat
ified, we cannot (in general) know that {xlcjJ(x)} exists, and so we cannot conclude
from cjJ(O) and cjJ(n) -t cjJ(n + 1), that Vn(n E N -t cjJ(n)). For instance, NFU cannot
prove the following: 1o
9Note that this definition doesn't quite mirror Frege's definition of ancestral: for that we would
have to replace x E z --> y E z in the above formula with (Vv(xXv --> VEz)) --> y E z. The
difference is that, using Definition 3.10 ab ove , which is equivalent to Quine's ([31], p.216), we have
y :s;X y for any binary relation X and any y, whereas on Frege's definition this will generally not be
the case. Of course, we could define <x here to properly mirror Frege's definition, but the above
definition is slightly more efficient for our purposes.
lOIn fact, Orey [21J has shown that the Axiom of Counting is independent of NF.
56
Definition 3.12. Axiom of Counting. For aIl n E N, {O, 1, ... , n} En + 1.
Extending NFU by the addition of this axiom was suggested by Rosser ([22])11
This is in fact a very weak extension. Jensen [20] has shown that in ZF we can
construct w-standard models of NFU, in which Mathematical Induction holds for the
natural numbers with respect to any formula, stratified or unstratified. In particular,
Counting is true in such models. We consider this axiom here because its analogue in
GG is an important theorem in Frege's Grundgesetze, and our interpretation in the
next chapter of a modified form of GG will be unable to prove this theorem of Frege's
for basically the same reason that NFU cannot prove Counting. This will provide us a
chance to bring out the parallels between arithmetic in GG and in NFU, culminating
in a proof that adding Counting to NFU makes the analogous GG theorem true in
our interpretation of GGree , the modified form of GG that we present in the next
chapter.
llIn the form n E N -+ {1, ... ,n} E n. Holmes suggests stating this axiom as "AU finite sets
are strongly Cantorian." For a definition of "Cantorian" and "Strongly Cantorian," see [17], p.7.
Holmes also discusses several considerably st ronger extensions, ultimately settling on a theory he
caUs "NFUM," which he proves to be equiconsistent with Morse-KeUey set theory plus the axiom
"there is a K-complete nonprincipal ultrafilter on the proper class ordinal K"; see [17], p.30 for
definitions of these terms.
57
58
Chapter 4
Interpreting GGrec in NFUT
We give in this chapter an interpretation in NFUT of a modified form, denoted by
"GGree" ("GG reeonsirueied"), of GG; NFUT is defined below in 4.2.1. The most
noticeable modification to GG will be the elimination of function variables. We first
give a presentation of the language of GGree and define (contextually) the GGree
primitives in NF UT. We then define a notion of F-siraiijicaiion for terms of GGree
and prove that courses-of-values corresponding to F-stratified terms exist. We can
then examine to what extent Frege's original axioms are satisfied by GGree and how
far his developinent of arithmetic can be pushed. We will see that basic arithmetic in
GGree (and GG) is interpreted in essentially the same way as in NFU frameworks.
4.1 The Language of GGree
Variables
Variables in GGree are untyped and correspond to those of object type in GG. We
will again distinguish between free and bound variables, so we have the following lists
of variables.
59
• Free variables: xo, Xl, X2,· .. ,xn ,· ...
• Bound variables: xo, Xl, X2,· .. ,Xn ,· ...
GGree has nine primitive functions. Application, X ~ y, must be taken as primi
tive because its definition in GG makes use of a quantification over a function variable.
In Chapter 2, we showed that given an application operator as defined in GG, we can
replace quantification over function variables with quantification over object variables,
but this result clearly cannot be used to eliminate the quantifier in the definition of
X ~ y. The second major difference between the primitives of GG and those of GGree
is the presence, among the latter, of two abstraction operators (unary and binary)
and two application operators (unary and binary). The reason for this is discussed
in 4.2.2.
The primitives of GGree
The nine primitive functions of GGree are
l. truth function: --x,
2. negation: -, x,
3. implication: x ~ y,
4. equality: x ~ y,
5. universalization over objects: VXf(X)I,
6. application: x ~ y,
7. binary application: xy~z,
1 A unary (first-order) function, such as indicated here by f(x), is represented in GGrec by a term
and a specified free variable, binary functions by a term and two specified free variables.
60
8. abstraction: if(x),
9. binaryabstraction: fYf(x, fj).
Terms
GGree term and the set of free variables, sfv, of a GGree term are defined inductively
as follows.
1. Each free object variable, Xi, is a term with sfv {Xi};
2. if rp is a term with sfv X,p, then each of -- rp and ~ rp, is a term with sfv X,p;
3. if rp and 1j; are terms with sfv X,p and Xlp, respectively, then each of rp ~ 1j;,
rp ~ 1j; and rp ~ 1j; is term with sfv X,p U XI/;;
4. if rp, 1j; and X are terms with sfv X,p, X'Ij; and Xx, respectively, then rp1j;;:::"'X is a
term with sfv X,p U X'Ij; U Xx;
5. if rp is a term with sfv X,p, X, y are free object variables and X, fj are bound
object variables not occurring in rp, then each of \';"xrp[~l and irp[~l is a term
with sfv X,p - {x}, and fYrp[~, ~l is a term with sfv X,p - {x, y};
6. nothing else is a term.
Statement, formal deduction, theorem, etc. are defined for GGree exactly as for
GG.
4.2 NF UT, Interpreting GGrec Primitives, and F-
stratification
Before we can examine the basic arithmetical definitions of GG as given in GGree ,
we need to interpret the primitives above in NF UT. For, as might be expected, this
61
translation will place a restriction on which courses-of-values we can assume to exist.
This restricted form of abstraction will be the biggest difference between the GG
axioms as recovered in our interpretation of GGree and the original axioms (see 1.2.3).
Note that it will not be possible, in general, to translate a term of GGree into the
primitive set-theoretic notation of NFUT, for the GGree functions of universalization
and abstraction are interpreted as second-order functions. That is, since these second
order functions take first-order functions as arguments, and the latter are not first
class existents in NF UT, we cannot give explicit definitions of universalization and
abstraction. \Vhat is possible, for any GGree-term 1, is to translate the expression
y = 1 into the language of NF UT, where y is a variable not occurring in 1.2 As we
shall see, this will be sufficient to translate any expression of the form 1 = Tinto the
language of NFUT and therefore to determine which GGree statements are true on
our interpretation. First, we need to fix the objects T and L
4.2.1 Extending the Language of NFU: NFUT
If we are to define the functions --x, -.:, x, x ~ y and x ~ y (and interpret
y = Vx1[~]), then clearly we need to fix two distinct objects as T and _L Intuitively,
these truth values are "concrete" objects, in that they seem to be neither sets nor
courses-of-values. For reasons of philosophical purity, Quine would have us construe
these objects so that T = {T} and .-L = {.-L}, while Frege takes them to be courses
of-values as described in Chapter 2. However, these interpretations seem unnatural,
especially in the present NFU context in which we are in any case allowing urele
ments. Therefore, the obvious interpretation to give of T and .-L is that they are
2Note that y will be a variable of the language of NF UT, while all variables in cp are of the
language of GGree, so y is a priori different from all variables in cp. However, in practice, we will
use the same symbols for variables of NFUTand free variables of GGree , so the statement above
indicates simply that we choose a variable symbol y not appearing in cp.
62
two distinct urelements. Do we know that such elements exist? In fact, Holmes has
shawn (NFU + Infinity + Choice) that "there are many cardinals between IVI [the
cardinality of the universel, and IP(V)I [the cardinality of the set of all sets]" ([17]),
i.e., that the number of urelements in any model of the theory is quite enormous. In
arder ta select and fix two of them, we will add the names T and ..l to the language
of NFU, together with the axiom
Axiom A. T =1= ..l.
Instead of an axiom stating that nothing is a member of either T or ..l, we will
also add the unary predicate symbol 'Set' to the language, with the axioms
Axiom B. ::Jx(x E y) -+ Set(y),
Axiom C. -,Set(T) 1\ -,Set(..l).
Stating the axioms of extensionality and stratified comprehension in the following
forms then allows us to distinguish the empty set from other urelements, including
T and 1...
Extensionality. (Set(x) 1\ Set(y)) -+ ('v'z(z E x ~ Z E y) -+ x = y).
Schema of Stratified Comprehension. ::Jx(Set(x) 1\ 'v'y(y E x ~ CP)), where cP is a
stratified formula not containing x free.
Note, however, that the notion of stratification now requires sorne clarification in
the case of formulas containing T and/or ..l. In this regard, we take T and 1.. quite
63
naturally to be constant terms, or O-ary functions, as we would a name for the empty
set. For suppose we wish to introduce the name 0 for "the set x such that nothing is
an element of x", i.e., we add 0 to the language and the axiom Set(0) /\ Vy(y ~ 0).
What stratification restrictions should we place on formulas containing 07 Note
that any formula containing 0 is of course equivalent to one not containing 0. For
instance, the atomic formula 0 E x is equivalent to (*) 3y(Set(y)/\ Vz(z ~ y)/\y EX).
Similarly for x E 0, x = 0 and 0 = x, and from the translations of these we can
easily define, for any formula cp containing 0, an equivalent formula cp* not containing
0. Note that if cp is a formula containing the atomic formula 0 E x (and no other
atomic formulas containing 0) and cp* is a formula obtained from cp by replacing each
occurrence of 0 E x by (alphabetic variants of) (*), then we may of course choose the
bounded variables in (*) so that these do not occur in cp. Suppose now that there is
an indexing, a, of the variables of cp such that a is a stratification assignment for an
atomic formulas other than 0 E x. Then it should be clear that a can be extended
to a stratification assignment, a* for CP*. 3 So we see that if we add 0 as a name for
the empty set, the atomic formulas 0 E x, x E 0, 0 = x and x = 0 are free from
stratification restrictions. More precisely, we define a stratification assignment for a
formula, cp, which may contain occurrences of 0, as an indexing, a, of the variables
of cp such that the restriction of a to the variables of any atomic formula, 'Ij;, of
cp, not containing an occurrence of 0, is a stratification assignment, in the original
sense, for 'Ij;. This definition of an extended notion of stratification is motivated by
our discussion ab ove , which shows that if cp, possibly containing occurrences of 0, is
stratified in this extended sense, then there is a formula cp*, not containing occurrences
3Specifically, if, in the formula fjJ, x also occurs outside of occurrences of the atomic formula
o Ex, then we take a* = aU {(y,a(x) - 1), (z,a(x) - 2)} (where we might need to replace each
a( u) by a( u) + 2 to insure that a* (u) ::::: 0 for ail variables u in fjJ). If, in fjJ, x occurs only in the
occurrences of the atomic formula 0 Ex, then we take a* = aU {(x,n +2), (y,n+ 1), (z,n)}, for
some arbitrary n E N.
64
of 0, such that </J* is equivalent to </J and </J* is stratified in the original sense.
This then also motivates our definition of stratification assignment for a formula
of NFUT. Namely, to repeat, for T and -.l, the conditions stated above for 0: a
stratification assignment for a formula, </J, of NFUT is an indexing, cr, of the variables
of </J such that the restriction of cr to the variables of any atomic formula, 'ljJ, of </J,
not containing an occurrence of T or -.l, is a stratification assignment, in the original
sense, for 'ljJ.
The discussion above concerning the addition of names to the language of NFU
is closely related to the use of definite description terms, which we will now briefly
describe. As with 0 for the empty set, we may wish to use the name Au B for the
x such that Vy(y E x ~ (y E A V Y E B)). We can handle all such cases in which a
formula (possibly with parameters) defines a unique object by introducing an analogue
to Frege's unit class operator \x, such as Russell's unit class operator iX</J for definite
descriptions </J. Suppose, for a formula </J, we know that :Jx(</J /\ Vy(</J[~l --+ y = x)).
We can introduce the term iX</J and define its use in any context from the following
definitions.
• U = iX</J {::}dej :Jx( </J /\ Vy( </J[~l --+ y = x) /\ U = x);
• same definition for iX</J = u;
• u E iX</J {::}dej :Jy(y = iX</J /\ u E y);
• iX</J E U {::}dej :Jy(y = iX</J /\ yEU).
If we allow a formula, </J, to contain such definite description terms, we could
adopt the following stratification conventions: We assign types not only to variables
of </J, but also to the terms, iX'ljJ, and require, in addition to the usual stratification
requirements, that type(ix'ljJ) = type(x). To see that, e.g. x E (iy(Set(y)/\ Vz(z ~ y)))
is stratified in this extended sense, we may consider the stratification assignment:
65
type(x) = 0, type(ty(Set(y) 1\ 'iz(z ~ y))) = 1 = type(y) , type(z) = O. It's not hard
to verify that if a formula containing definite descriptions is stratified in this new
sense, then the NFU formula which it abbreviates is stratified in the original sense
(although we may have to take alphabetic variants of the definite description terms).
We don't bother proving this here as we will make no technical use of the notion of
stratification extended to formulas possibly containing definite descriptions.
Ordered Pairs and Infinity
The axioms we officially adopt for this chapter are the five above (A, B, C, Weak
Extensionality and the Schema of Stratified Comprehension), together with an ax
iom of infinity. Choosing a form for the statement of in finit y actually raises a point
concerning ordered pairs, and we digress briefly to discuss this point. If we adopt
the usual definitions of the ordered pair, (x, y), as { {x}, {x, y}} and functions as sets
of ordered pairs, then we will have a type difference of 3 between a function and its
arguments and values. This is not unmanageable, but it is awkward, especially wh en
we consider that the ordered triple, (x, y, z), cannot be defined as a special case of
ordered pair if we want x, y and z to be assigned the same type. Quine suggests
(x,y,z) =def ({{x}}, (y,z)) = {{{{x}}},{{{x}}, {{y},{y,z}}}} (!),
so as to recover uniformity between x, y and z. We would then have a level difference
of 5 between a binary function and its arguments and values. Holmes suggests ([17],
p.4) rather that we introduce as primitives the projections 1fl and 1f2 with the same
stratification requirements as equality, e.g. X1flY is to be read "y is the first projection
of x", and any stratification assignment for a formula containing X1fiY must assign
the same level to x and y. Definitions of n-tuple, relation, function, etc. are given
as usual in terms of ordered pairs and we have the more natural level difference of 1
between a function and its arguments and values, for functions of all arities. So we
add the binary predicate symbols 1fl and 1f2 to the language of NFUT and state the
66
axiom of infinity as follows.
In relation to the above we quote the following (NFU) results of Holmes: "If
the Axiom of Infinity were given in the more usual form asserting the existence of
an infinite set, and the Axiom of Choice were assumed as well, it would be possible
to prove the existence of a type-Ievel ordered pair operation [1['1 and 7f2 ab ove] as a
theorem [ ... ]; proving that there is an infinite set, given a type-Ievel pair, is quite easy."
And: "In the absence of Choice, the existence of a type-Ievel ordered pair implies but
is not equivalent to the existence of an infinite set, but NFU with the axiom 'there is
an infinite set' interprets NFU with a type-Ievel pair in a straightforward manner."
([17], pp.4-5)
As it is not necessary for the purposes of this essay, we do not explicitly adopt
the Axiom of Choice, but we note that if the reader assumes it as an acceptable
(or necessary) axiom to add to Zermelo set theory, then from Chapter 3, it is also
acceptable in the context of NFU. We adopt the Axioms A, B, C, Extensionality,
Infinity and the Schema of Stratified Comprehension, as listed ab ove , as the axioms
of our set theory, which we label "NFUT" ("New Foundaiions with Urelements and
Truth Values"). Later, we will consider the addition of Counting.
4.2.2 Translating GGree into NFUT
Given any GGree term rp, we now define inductively the translation of y = rp into a
formula rp~FUT of NFUT. In what follows, rp, 'ljJ, X and pare GGree-terms, x, Z, u are
free variables of GGree or variables of NF UT, and i, z are bound variables of GGree .
1. rp is x: y = X Bde! Y = x;
67
2. cP is -'ljJ:
(y = -'ljJ) Bdef [('ljJ - T 1\ Y = T) V ('ljJ -::1 T 1\ Y = ..l)];
3. cP is -.:, 'ljJ:
(y = -.:, 'ljJ) Bdef [('ljJ -::1 T 1\ Y = T) V ('ljJ = T 1\ Y - ..l)];
4. cP is 'ljJ ~ x:
(y = ('ljJ ~ X)) Bdef
[('ljJ = T 1\ X -::1 T 1\ Y = ..l) V (('ljJ -::1 T V X = T) 1\ Y = T)];
5. cP is 'ljJ ~ x:
(y = (1/J ~ X)) Bdef
:3X:3Z[(X = 'ljJ 1\ z = X) 1\ ((x = z 1\ Y = T) V (x -::1 z 1\ y = ..l))];
(y = (1/J ~ X)) Bdef
:3x:3Z[(X = 'ljJ 1\ z = X) 1\ (( "z is a unary F - function" 1\ (x, y) E z)
V (--,("zisaunaryF - function") 1\ y = 0))];
7. cP is 'ljJx~p:
(y = (1/Jx ~ p)) Bdef
:3x:3z:3w[(x = 'ljJl\z = Xl\w = p) 1\(("wisabinaryF- function" 1\ (x,z,y) E w)
V (--,("wisabinaryF - function") 1\ y = 0))];
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
74
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
75
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.
76
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 .
77
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.
78
\- (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
79
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 .
Definition 4.4. I(p) =def \ix\iY[(xy~p) ~ (\iz(xz~p) ~ (Y ~ z))].
Note that the RHS is F-stratified. Note also that any F-stratification assignment
for a term containing l (p) will assign the same type to p and to l (p), while the bound
variables in I(p) are unimportant in F-stratification assignments, as they can always
be changed to obtain an alphabetic variant.
The following three definitions involve "binary courses-of-values" and are accord
ingly slightly different from their GG counterparts.
Definition 4.5. For any p, we define
)p =def f[;(I(p)  Vu(u ~ x ~ :3v(v ~ y  uv~p))).
Note that I(p) 1\ \iu(u ~ x ~ :3v(v ~ y  uv~p)) is F-stratified, and therefore
)p exists as a binary course-of-values, i.e., a binary F-function, for any p. Any F
stratification assignment, Œ, for a term containing )p will assign types to subterms
su ch that ŒOp) = Œ(p) + 1.
80
For objects a, b, we read ab::::-')p = T as "the p-relation maps the a-concept into
the b-concept", i.e., as the GGree equivalent of a ~ (b ~)p) = T in GC. (See 2.2 for
the CG originals of an definitions given here and a more complete discussion of their
meanings.)
Definition 4.6. Inverse relation. U(p) =def fY(f;x::::-,p))).
Note that yx::::-'p is clearly F-stratified.
Definition 4.7. Given objects a, b, the concepts --(x ~ a) and --(x ~ b) are
said to be equinumerous if there exists a p such that (ab::::-')p 1\ ba::::-')U(p)) = T,
i.e., p maps the a-concept into the b-concept and U(p) maps the b-concept into the
a-concept.
We now arrive at the definition of the cardinal of the a-concept, without so far
having to make any major changes to the definitions as given in 2.2.
Definition 4.8. The cardinal of the u-concept.
Given any u, we define the cardinal of the u-concept as
CardF(u) =def i[~fj(ux::::-')U(f;) À (xu::::-')fj))].
Note that ~fj(ux::::-')U(fj) À (xu::::-')fj)) , which we will abbreviate x rvF U, is F
stratified, and that any F-stratification assignment, Œ, for a term containing CardF(u)
will be su ch that Œ(CardF(u)) = Œ(U) + l. We define the cardinals zero and one as before.
Definition 4.9. Zero. OF =def CardF(i(~ (x == x))).
Clearly, ~ (x == x) is F -stratified.
Definition 4.10. One. IF =def CardF(i(x == OF )).
Again, x == OF is clearly F-stratified (for an alphabetic variant of the definition of
OF given ab ove ).
81
Successor, Ancestral and the N atural N umbers
We come now, as before, to the definitions of successor, ancestral and natural number.
We have thus far had to make only minor changes to Frege's core definitions, replacing
iterated abstraction and application in GG with binary abstraction and application
in GGrec . But the definitions are otherwise the same, indicating, for one thing, the
extent to which Frege himself did without quantification over function variables. We
can also define successor essentially as in GG, but then we come to the definition of
ancestral, which will require more attention.
Definition 4.11. Successor.
Sp =dej iY(~û~v(x ~ Cardp(i(z ~ Û A ~ (z ~ v))) A v ~ Û A f) ~ Cardp(û))).
We write zSpw for zw;::::"Sp.
As before, m is the successor of n if m is the cardinal of a concept under which
exactly one more object falls than under a concept for which n is the cardinal.
We come now to the definition of ancestral, the first definition from 2.2 containing
a function variable. In the context of GG, Frege could have given an equivalent defi
nition in which the quantification over a unary variable is replaced by a quantification
over an object variable, as we have discussed before. This is, in fact, the definition we
now give; of course, given the restricted Comprehension axiom of GGrec , this modified
definition no longer carries the same weight as that of GG. This last statement will
be made precise below in discussing Induction in GGree .
Definition 4.12. Ancestral, modified.
<q=dej iY('ifz([(\iÛ\iv(û ~ z) A (ûv;::::"q)) ---.:..... (v ~ z)]
---.:..... [(\iw(xw;::::"q) ---.:..... (w ~ z)) ---.:..... (f) ~ z)])).
82
Given a GGree term 4> and variable x, let us caU the concept, --4>(x), hereditary
in the q-relation if
(*) ('iuVv((4>[~l A uv:::-'q) ~ 4>[~])) = T.
If 4> is F-stratified, then x(4)[~]) is a unary F-function (course-of-values) and, setting
a =def x( 4>[~]), (*) above is equivalent to . . .
(t) (VuVv(u ~ a !\ uv:::-'q) ~ v ~ a) = T.
If 4> is not F-stratified, then we cannot know that x(4)[~]) is the desired course-of
values, in particular, we cannot know that there is any object a such that (*) is
equivalent to (t). Clearly, this will have an effect on Induction, as we discuss after
giving the two final definitions of this section.
The foUowing definition is still straightforward.
Definition 4.13. Equal to or finitely preceding in the q-relation.
~q=def fY(x <q fJ V x ~ fJ)·
We write x ~q y for xy:::-' ~q.
Note that any F-stratification assignment, T, for a term containing x ~q y is such
that T(X) = T(Y).
The definition of natural number as equal to zero or finitely preceded by zero in
the SF-relation is essentially unchanged.
Definition 4.14. An object nF is a natural number if OF ~SF nF = T.
Given a GGree term 4> and variable x as before, suppose now that 4>(OF) = T and
that --4>(x) is hereditary in the SF-relation. If 4> is F-stratified, then (t) above
holds for q = SF and x ~ a = 4>(x) for all x (again setting a =def x(4)[~])). Using the
modified definition of Ancestral above and an argument entirely analogous to that
given at the end of 2.2.3, we see that Induction holds for 4>(x), i.e., that nF ~ a = T,
and therefore 4>( n F) = T, for all natural numbers n F.
83
However, if </J is not stratified, we cannot, in general, prove that Induction holds for
</J( x). This is the sense in which the modified definition of ancestral "does not carry
the same weight" as the original. To determine whether Frege's theorems and proofs
regarding the natural numbers are still possible requires a case-by-case inspection.
4.3.3 The Axiom of Counting
As it turns out, there is no difficulty in proving most of Frege's main results. In par
ticular, Frege's proofs of I(SF) and I(U(SF)) (i.e., there is at most one SF-successor
and at most one SF-predecessor for any object, respectively) do not make use of
courses-of-values corresponding to un-F-stratified terms and can be reproduced in
GGree . We seem to run into no difficulty until we reach the following important the
orem, mentioned earlier in 2.2.3, and here slightly modified for the GGree context:
(*) f- (OF ::;SF nF) ~ (nFSFCardF(i(OF ::;SF i ::;SF nF))),
where OF ::;SF i ::;SF nF abbreviates OF ::;SF i A i ::;SF nF.
The above term is not F-stratified. For suppose T is an F-stratification assignment
for (*). Then r(i(OF ::;SF i ::;SF nF)) = r(i)+l = r(nF)+l and T(CardF(i(OF ::;SF
i ::;SF nF))) = r(i(OF ::;SF i ::;SF nF)) + 1 = r(nF) + 2. But the above formula (*)
also requires r(CardF(i(OF ::;SF i ::;SF nF))) = T(nF), so we have the contradiction
r(nF) = r(nF) + 2.
That the development of arithmetic runs into the same obstacle in GGree as in
NFU is perhaps not surprising, given that we are interpreting GGree in NFUT and
that the notion of F-stratification is built on that of stratification. However, given
that our GGree definitions are basically those of the Grundgesetze, this do es bring out
quite clearly the extent to which arithmetic in NFU is essentially arithmetic as Frege
envisioned it, though transferred to a set-theoretic context. Our aim in the remainder
of this chapter is to prove that the truth of the above GGree statement (*) follows if
84
we add the Axiom of Counting to NFUT. The steps we take to demonstrate this will
also make more explicit the parallels between GGrec and NFU.
Unravelling the GGrec definition of natural number
For the following discussion, it will be useful to define two unary functions. As it
happens, these functions are defined for all x E V, i.e. they are unary F-functions, as
is clear from their stratified defining formulas. However, we introduce them mainly as
notational abbreviations and the fact that they are F-functions will not be essential.
Definition 4.15. Y = x* {::}def Vw(w E y <--t W ~ X = T).
(I.e., x* = {wlw ~ x = T}.)
Definition 4.16. 13
y = Char(x) {::}def ["y is a unary F - function" /\
VzVw((z, w) E Y <--t ((z E x /\ w = T) V (z ~ x /\ w = ..1)))J.
Note that Char(x) is itself a unary function, and that any stratification assignment
for a term containing x* will assign the same type to x and x*. The same is true of
any stratification assignment for a term containing Char(x). It's immediate from the
definitions of these two functions that y E x* <--t Y ~ x = T and y E x <--t
Y ~ Char(x) = T for any x and y. It's also easy to verify that for any x,
Set(x) -+ (Char(x))* = x. (Note that it's not true that Set(x) -+ (Char(x*)) = x.
We have rather "x is an F-concept" -+ (Char(x*)) = x, where an F-concept is a
unary F-function equal to T or ..1 for every argument.)
13It's important here that ((z E x 1\ w = T) V (z ~ x 1\ w = -.1)) is stratified, for otherwise we
cannot know that y exists as described (in particular, as a unary F-function) in the above formula.
That Char(x) exists for each x and that CHAR =def {(x,y)ly = Char(x)} exists are two different
(though related) facts.
85
In discussing the natural numbers, we will have to be careful to distinguish between
the objects which are natural numbers on the GGree definition and those which satisfy
theNFU(T) definition. The former we will call GGree , or Frege, natural numbers, and
label the individuals nF and the set NF, and the latter we call NFU natural numbers,
n for individuals and N for the set. To every object a is associated a Frege cardinal,
as defined above: CardF(a) =def i(i l'VF a).
CardF(a) is a unary F-function such that, for each object x, (x, T) E CardF(a) iff
there is a y such that ax;:::~·-)U(y) = T and xa;:::::")y = T, which is to say that y* is a
function such that the restriction of y* to x* is a bijection between x* and a*. And
if there exists any bijection z between x* and a*, then setting y = Char(z), we see
that ax;:::::")U(y) = T and xa;:::::")y = T, so that x l'VF a. So we have the following
"NFUT definition" of the Frege cardinal CardF(a), equivalent (in NFUT) to the
original definition given in GGree notation.
Definition 4.17. For any object a, CardF(a) =def {xlx* l'V a*}.
Note, e.g., that OF = {xlx* = 0} and IF = {xlx* l'V {OF} n. Next, we define a function f on the NFU(T) natural numbers, in the aim of
eventually showing that f is a bijection between N and NF.
Definition 4.18.
(x,y) E f {=}def [x E NI\\t'z\t'u((z,u) E y <-+ ((z* E xl\u = T)V(z* ~ xl\u = ..l)))].
So, for n E N, (J(n))* = {zlz* En}.
We must be careful that the formula on the RHS is in fact stratified, to ensure the
existence of f. Inspection of the definitions involved shows that it is indeed stratified.
Now, it is clear from the above that, for any n E N, f(n) is indeed sorne Frege
cardinal (which is not to say that it is clearly a Frege natural nurnber!). We wish to
86
show that, in fact, f(N) = NF' We state first the following lemma and give a proof,
making explicit use of some of the simple properties of Char(x) and x*. We will not
always go into such detail in subsequent proofs.
Lemma 4.2. \ix\in[n E N --> (x E n <-+ (Char(x) ~ f(n) = T))].
Pro of. Note that Char(x) .---.. f(n) = T is equivalent to Char(x) E (f(n))* and that
(*) x E n <-+ (Char(x) ~ f(n) = T) is a stratified formula, so Induction holds in
NFUT relative to (*).
Suppose n = O. Now, \iy(y E 0 <-+ \iz(z ~ y)), so x E 0 implies x is in bijection
with the empty set. By definition, then, Char(x) is such that
\iy(y.---.. Char(x) =.1) and therefore \iy(y ~ (Char(x))*), which implies
(Char(x))* E 0 and therefore Char(x) ~ f(O) = T.
So x E 0 --> (Char(x) ~ f(O) = T). The reverse implication is similar.
Assume that (*) holds for k ~ n and suppose x E n + 1. By definition of successor
(in NFUT), :3y:3z(y E zl\n+l = Card(z)l\n = Card(z-{y}))14. Let y and
z be such that y E z 1\ n + 1 = Card(z) 1\ n = Card(z - {y}). Then x rv z.
Let G : x"'::""z be a bijection. Then G(y) E x. But :3w(w E x) --> Set(x) and
Set(x) --> (Char(x))* = x. Therefore (Char(x))* E n + 1, so, by the definition
of f, Char(x) ~ f(n + 1) = T. Conversely, if Char(x) .---.. f(n + 1) = T, then
(Char(x))* E n+l and, just as for x above, we may deduce that :3w(w E (Char(x))*),
which means :3w(w .---.. Char(x) = T), which in turn means :3w(w E x) and therefore
x = (Char(x))* En + 1.
So x E n + 1 <-+ (Char(x) .---.. f(n + 1) = T).
By Induction, (*) holds for aU n; clearly no property of x was used in the proof
and the proof is complete. D
14Note that Card(x) = '" +--> xE", is (almost) immediate from the definition of cardinal, as is
(CardF(x) ~ a ~ x '" a) = T from the definition of Frege cardinal.
87
We next prove
Lemma 4.3. Vn(n E N --+ (J(n) SF f(n + 1) = T)).
Praof. Let n E N. Now, (J(n))* = {ulu* E n} and (J(n + 1))* = {ulu* E n + 1}.
Let y and z be such that y E z /\ (n + 1 = Card(z)) /\ (n = Card(z - {y})).
Then Char(z) E (J(n + 1))* and Char(z - {y}) E (J(n))*. We have that, for any
w, w ~ Char(z - {y}) = T f-+ (w ~ Char(z) A':"" (w ~ y)) = T. Therefore,
Char(z - {y}) = 1ÎJ(w ~ Char(z) A':"" (w ~ y))15. So Char(z) and y are such that
[y ~ Char(z) A f(n) ~ CardF(1ÎJ(w ~ Char(z) A':"" (w ~ y)))
A f(n + 1) ~ CardF(Char(z))] = T,
so :3u:3v(u ~ v A f(n) ~ CardF(1ÎJ(w ~ v A':"" (w ~ u))) A f(n + 1) ~ CardF(v)),
i.e., (J(n) SF f(n + 1) = T). 0
We consider now the characteristic function, Char(J(N)), of f(N). Char(J(N))
is itself a unary F-function, and we may write Charf(N)(x) for x ~ Char(J(N)), its
value for any object x. Thus, we have Charf(N)(x) = T iff xE f(N).
Lemma 4.4. VX(OF ~SF x = T --+ x ~ Char(J(N)) = T). I.e.) NF ç f(N).
Proof. RecaU from the GGree definitions of ancestral and natural number that, for
any object w, OF ~ W = T and --(x ~ w) being hereditary (taking --(x ~ w)
as a function of x) in SF jointly imply that nF ~ w = T for aU nF (i.e. for aU x such
15We use here that (i) w '"""' Char(z-{y}) equals T or ..1 for all w and (ii) w '"""' Char(z) A:' (w ~ y)
is F-stratified. (ii) is actually cheating a little, since we haven't defined Char(z) in GGree primitives
and the lemma on F-stratification does not technically apply. But it is clear from that lemma and the
definition of Char(z) that the translation of u = (w '"""' Char(z) A:' (w ~ y)) into NFUT primitives
is stratified, which is what we really need for the existence of 1Î!(w '"""' Char(z) A:' (w ~ y)) as a
course-of-values.
88
that OF ::;SF X = T). We show that Char(f(N)) satisfies the two conditions given for
wabove. First, since OF = f(O), OF ,----.. Char(f(N)) = T is clear. Now, suppose that
X ,----.. Char(f(N)) = T, i.e. that X E f(N). Then x = f(n) for some nE N. Now, from
Lemma 4.3, the SF-successor of f(n) is f(n+1)16. So for any y, x SF Y --) Y = f(n+1),
and, of course, f(n + 1) ,----.. Char(f(N)) = T. So x ,----.. Char(f(N)) is hereditary in
the SF-relation. So, finally,
\fX[(OF ::;SF x = T) --) (x ,----.. Char(f(N))) = T] and NF ç f(N), as desired. D
Lemma 4.5. \fx(x E N --) (OF ::;SF f(x) = T)). I.e., f(N) ç NF.
Proof. This proof, like the last one, is by Induction, only this time the induction is on
the NFUT natural numbers rather than the GGree natural numbers. Note that (the
translation into NFUT primitives of) (*) OF ::;SF f(x) = T is stratified. So Induction
holds, in NF UT, relative to (*). Clearly, OF ::;SF f(O) = T, since f(O) = OF. If
n E N and OF ::;SF f(n) = T, then, since f(n)SFf(n + 1) = T, it follows that
OF ::;SF f(n + 1) = T. So by Induction, \fx(x E N --) (OF ::;SF f(x) = T)) and we
have f(N) ç NF. D
The previous two Lemmas show that f(N) = NF. In fact, recalling Lemma 4.3,
we may state the following.
Theorem 4.19. f is an isomorphism (N, S)""::'(NF, SF).
The following lemma is straightforward.
Lemma 4.6.
\fx\fn[n E N --) (Char(x) E (f(n + 1))* --) (f(n) SF CardF(Char(x)) = T))].
16We assume here that I(SF) = T, i.e., that any object has at most one SF-successor. As
mentioned ab ove , there seems to be no difficulty in reproducing Frege's pro of of this in GGree .
89
Pro of. Assume Char(x) E (J(n + 1))*. Now, for any GGree cardinal "'F and any
x, we have that x E ("'F)* <-+ CardF(x) = "'F. Since f(n + 1) is a GGree natural
number (and therefore a GGree cardinal) for any n E N, Char(x) E (J(n + 1))*
implies CardF(Char(x)) = f(n + 1). From Lemma 4.3, f(n) SF f(n + 1) = T, so
f(n) SF CardF(Char(x)) = T. 0
We can now state the main result of this section.
Theorem 4.20. NFUT + Counting proves:
f- (OF ~SF nF) ~ (nF SF CardF(i(OF ~SF X ~SF nF ))).
Proof. We need to show that in NF UT, given the axiom of Counting, we can deduce
that OF ~SF nF = T -+ (nF SF CardF(i(OF ~SF X ~SF nF ))) = T. From results
above, we have that OF ~SF nF = T is equivalent to nF E f(N), which in turn
is equivalent to n E N (with n = f-l(nF )). Also, since i(OF ~SF x ~SF nF) =
Char( {OF, IF, . .. , nF}), we have that (nF SF CardF(i(OF ~SF X ~SF nF)) = T) is
equivalent to (nF SF CardF(Char( {OF, IF, ... , nF})) = T).
So we need to show that n E N -+ (nFSFCardF(Char({OF,tF, ... ,nF}))
T). Assume that n E N. Then nSF CardF(Char({OF, IF, . .. ,nF})) = T follows
from Lemma 4.6 if Char( {OF, IF,· .. , nF}) E (J(n + 1))*. But, from Lemma 4.2,
{O,l, ... ,n} E n+ 1 <-+ (Char({O,l, ... ,n}) ~ f(n+ 1) = T) for all nE N. So,
given the axiom of Counting, Char( {O, 1, ... , n}) ~ f(n + 1) = T for n E N, and by
definition of X*,
(Char({O,l, ... ,n}) ~ f(n+ 1) = T) <-+ Char({O,l, ... ,n}) E (J(n+ 1))*. Fi-
nally, sin ce f restricted to {O, 1, ... , n} is an obvious bijection between this set and
{OF, IF, ... , nF} (and therefore Char(J) is a bijection in the sense of GGree between
Char( {O, 1, ... , n}) and Char( {OF, IF, ... , nF} )), we have that
Char({O, 1, ... ,n}) E (J(n+ 1))* <-+ Char({OF, IF, ... ,nF}) E (J(n+ 1))*. As men-
tioned, it now follows from Lemma 4.6 that n SF CardF(Char( {OF, IF, . .. , nF})) =
90
T, i.e., that nF SF CardF(i(OF :s;SF X :s;SF nF)) = T, and the proof is complete. 0
91
92
Conclusion
In this essay, we have attempted to shed light on sorne of the connections between
various logical foundations of arithmetic. Our princip le goal in this vein was to
interpret a modified form of Frege's Grundgesetze theory in an NFU framework and
to sketch the parallel development of arithmetic in this interpretation and in NFU
proper. This we have done in the last chapter, with Theorem 4.20 as our main
result in this direction, establishing the derivation of a central arithmetical theorem
of the Grundgesetze 17 from its corresponding statement, as the Axiom of Counting,
in NF UT. In proving 4.20, we make use of connections between the GGree natural
numbers and the NFU natural numbers, and between mathematical induction as
applied to each of these two. The connections clearly run deep and it may prove
interesting to pursue them for more general results. A study of the GG definitions of
the rationals and reals (see [10]) in the context of our interpretation may also prove
fruitful.
We have also sought in this essay to trace historical reasons for similarities between
GG and NFU. Clearly, the conceptual ties between GG, TT, NF and NFU have been
our main focus, but these cannot be entirely separated from the historical progress
of research in mathematical logic. Of course, we had to restrict our focus to a very
limited number of historical developments, indicating in certain places important
points that have been left aside or only briefly touched upon in this essay. As with
17To the importance of which Frege himself called attention [11], §46.
93
historical analysis in any field, there remains much more to be said and many more
connections to be traced; perhaps this is especially true in the fields of mathematics
and logic, where historical understanding is not generally considered a priority.
Ultimately, one can say, in the case of the Grundgesetze, that while Frege's formaI
system was inconsistent, his construction of arithmetic was not. In fact, it is this con
struction of arithmetic that is developed in NFU-style set theories. The paradoxes
don't block Frege's arithmetical definitions, but they clearly complicate the develop
ment of an arithmetic that uses these definitions. As compared to ZFC and variants,
technical complications may indeed make NFU-style set theory a less natural setting
for interpreting mathematics. However, it is certainly not an uninteresting setting,
nor one without a history and motivation, for one can trace its roots, as with so much
of modern logic, to Frege's own groundbreaking work.
94
Appendix A
Frege's Cumulative Type Theory
In this appendix, we present a theory based on Frege's informaI discussion of concept,
or function, levels at the beginning of the Grundgesetze (esp. §§1-4 and §§21-25). We
believe this theory is an extremely plausible interpolation from what Frege actually
says. Frege himself does not give any formaI presentation of the "Cumulative Type
Theory" (CTT) below and we don't claim the details we furnish are specifically
"what Frege had in mind." For instance, the use in presentation of both "levels"
and "types," which seems to us necessary to describe CTT, Frege does not explicitly
mention. However, it seems likely that had Frege described a formaI theory for his
concept levels, it would be equivalent to CTT as we present it below.
As mentioned in Chapter 1, CTT is rather messy and is used neither in the
Grundgesetze nor here for the purposes of actually developing arithmetic. Frege's
discussion of CTT serves to motivate the formaI language GG that he ultimately
adopts. Our main purpose in giving a formaI presentation of CTT is to show that,
given two simplifying assumptions, CTT reduces to TT.
Types of variables in CTT will be given by a pair, (n, m), of natural numbers and
to each pair (n, m) E [(N - {O}) x N] U{(O, On corresponds a distinct CTT type. Each
type, other than type (0,0), is a function type with range (a subset of) the entities
95
of type (0,0) (Le. "objects"); the type of a function is determined by the types of its
variables. Type (0,0) is Frege's object type and may be thought of as a 0 - ary or
constant function type. n is the ZeveZ of the type (n, m) and for each n > 0 there are
infinitely (though countably!) many distinct types of level n. For each n > 0, we will
have to fix an ordering, (Jn, of the types of level n. Let us first describe the levels
informally.
• LeveZ O. These are the objects, or constant functions. There is only one type of
level 0, that is, type (0,0).
• LeveZ 1. These are all the functions taking objects as arguments. These func
tions may be of any finite arity, so there is one type for each arity n EN, and
we may sim ply take the ordering of level 1 functions to be given by their arities.
That is, "f is a function of type (1, n)" is then equivalent to "f is an n + 1-ary
level 1 function." (Note that no level 1 type corresponds to O-ary functions, so
type (1,0) corresponds to unary functions, type (1,1) to binary functions, etc.)
• LeveZ 2. These are all the functions taking as arguments functionsjobjects of
level ::; 1, such that at Zeast one argument is of ZeveZ 1. Since again these
functions may be of any finite arity, we have one type of level 2 for every finite
sequence (al, a2,· .. ,am) of types ai = (ail, ai2) of level ::; 1 (i.e. such that
ail::; 1), such that for some i ::; m, ai is of ZeveZ1 (i.e. ail = 1). Let 52 be the
set of all such sequences and let (J2 : 52 --> N be a bijection ordering 52. Given
any kEN, we write (aik, a~k, ... ,a;!:J for (J:;l(k). Then "f is a function of type
(2, k)" is equivalent to "f is a level 2 function of the variables (h, 12,···, fmk)'"
where fi is of type ark for i ::; mk.
96
Level n. These are all the functions taking as arguments functionsjobjects of
level ::::; n - 1, su ch that at least one argument is of level n - 1. These
functions may be of any finite arity and we have one type of level n for every
finite sequence (al, a2, ... ,am) of types ai = (ail, ai2) of level ::::; n - 1 (i.e. such
that ail ::::; n - 1), su ch that for some i ::::; m, ai is of level n-l (i.e.
ail = n - 1). Let Sn be the set of all such sequences and let crn : Sn ----t N be a
bijection ordering Sn- Given any kEN, we write (a'lk, a'2k, ... , a~k) for
cr;;l(k). Then "f is a function of type (n, k)" is equivalent to "f is a level n
function of the variables (h, h, ... ,fmk)'" where fi is of type ark for i ::::; mk.
We now give a more formaI presentation of CTT. We will not write the full details
of term formation and substitution rules, as we don't intend on actually working
within CTT; it should be clear by analogy with GG how such details may be fiUed
lll.
We use the foUowing notations and definitions. Types of level 1 are ordered by
their arity as indicated above: (1, n) is the type of an n + l-ary level 1 function. We
define inductively, for n E N - {O}, (i) the set, Sn, as the set of aU finite sequences,
for sorne (at least one) i ::::; m, as weU as (ii) the bijection crn : Sn ----t N, ordering Sn
(In fact, we don't actuaUy define this bijection here, but it is clear that sorne such
ordering of Sn exists, for each n E N, and we assume that such a crn is fixed for each
n EN.) Given any n > 0, kEN, we have cr;;l(k) E Sn and we denote cr;;l(k) by
( nk nk nk ) . d· t d b al ,a2 , ... , amk ' as ln 1ca e a ove.
N ow, the presentation of CTT. CTT has the foUowing variable symbols: An
. fi ·t 1· t (0,0) (0,0) (0,0) f· bl f t (0 0) d f h ( k) III m e 1S X o , Xl , ... , X n , . .. 0 vana es 0 ype , ,an, or eac n, E
97
(N - {O}) x N, an infinite list f~n,k), fin,k) , ... , f;::,k) , . .. of variables of type (n, k).
As for G G, we assume corresponding lists of bound variable symbols.
CTT has the following primitive functions:
• truth function: --x,
• negation: -, x,
• implication: x ~ y.
• for each CTT type (n, k), equality of type (n, k) entities: f(n,k) ~(n,k) g(n,k) ,
• for each CTT type (n, k), universalization over f(n,k): v}<n,k)G(j(n,k») 1.
A CTT term and its set of free variables, sfv are defined inductively as follows.
1. Each free object variable, x~o,ü), is a term with sfv {x~O,O)};
2. if f(n,k) and g(n,k) are variables of type (n, k), then f(n,k) ~(n,k) g(n,k) is a term
with sfv {f(n,k), g(n,k)};
3. if mEN, 4>i is a term with sfv Xq,i for i :::; m + 1, and f is a variable of type
(1, m), then f(4)l, 4>2,' .. ,4>m+l) is a term with sfv Ui::om+l Xq,i U {f};
4. if n, mEN, ai is a type of level ail:::; n - 1 for each i :::; m, ail = n - 1 for
sorne i :::; m, f is a variable of type (n, k) with k = O'n(al, a2, . .. ,am), and fi is
a variable of type ai for each i :::; m, then f(h, f2,' .. ,fm) is a term with sfv
{f, h, 12,· .. ,fn};
5. if 4> is a term with sfv Xq" then each of -- 4> and":" 4>, is a term with sfv Xq,;
1 A function of a variable of type (n, k) is given in CTT by specifying a term, cp, and a variable,
j(n,kl, of type (n,k).
98
6. if <p and 'ljJ are terms with sfv Xr/> and X1j;, respectively, then each of <p -.:...... 'ljJ
and <p ~(ü,ü) 'ljJ is term with sfv Xr/> U X1j;;
7. if <p is a term with sfv Xr/>, j(n,k) is a free variable of type (n, k) and j(n,k) is a
bound variable of type (n,k) not occurring in <p, then vj(n,k)<p[~~:::~l is a term
with sfv Xr/> - {j(n,k)};
8. nothing else is a term.
With the exception of Basic Law VI, we may adopt a set ofaxioms which are very
close to those of GG. We state these as follows.
• Basic Law 1.
a)f-x-':"""(y-':"""x)
b)f-x-':"""x,
where x, y are variables of type (0,0).
• Basic Law II. For each type (n, k),
f- 'ij(n,k)g(j(n,k») -.:...... gu(n,k»),
where j(n,k) is a variable of type (n, k) and 9 is a variable of level n + 1 and of
type corresponding to the sequence ((n, k)) E Sn+1, i.e. 9 is a (variable for a)
function of the single variable j(n,k).2
• Basic Law III. Substitutability of equal terms. For each type (n, k),
f- h(l,ü)Uin,k) ~(n,k) jJn,k») -.:...... h(l'Ü)(Vg(guJn,k») -.:...... gUin,k»))),
where 9 is a function of the same type as in the statement of Basic Law II
above. Note that f- Uin,k) ~(n,k) jJn,k») -.:...... Vg(guJn,k») -.:...... gUin,k»)) is a
special casé.
2Note that the application of this law to functions given in CTT by a specified term and variable
j(n,k) is determined by the substitution rules.
3Note also that, e.g. h(l,O)Uin,k) ~(n,k) jJn,k)) is in fact a term by clause 3 in the definition of
"term," since jin,k) ~(n,k) jJn,k) is a term by clause 2.
99
• Basic Law IV
f- (~ (-x ~(O,O)~ y)) ~ (--x ~(O,O) -y) .
• Basic Law V Extensionality. For each type (n, k),
L (f(n,k) --.:.. j(n,k») --.:.. ,- 1 -(n,k) 2 -(0,0)
. _ark . _a~k (n,k) _ark _a~k --.:.. (n,k) _ark _a~k [\191 ... \l9mk (f1 (91' ... ,9mk ) -(0,0) f2 (91 , ... , 9mk ))],
h k - (nk nk) w ere - (Jn al , ... , amk .
At first glance, it seems that Frege might object to the statement of Extensionality
given here as Basic Law V For Frege thinks that we cannot talk about equality of
functions - unsaturated entities - in the same sense that we talk about equality of
objects. Frege indeed never talks about the equality of functions, but rather of the
courses-of-values corresponding to functions. But clearly the identity of courses-of
values corresponding to two functions is sorne kind of statement of equality relating
to the two functions, and it is exactly this notion of equality, namely extensional
equality, that we capture in our statement of Basic Law V ab ove , and that we denote
by the typed equality symbol, ~(n,k), for each CTT type (n, k). If someone were to
really insist that this notion of "extensional equality" cannot properly be called an
equality, then we would gladly calI it an extensional equivalence instead.
Nonetheless, whether Frege fully realized it or not, he may have had a very good
technical reasons for keeping expressions of the form fin,k) ~(n,k) fJn,k) out of his
formaI language, or at least not giving them the status of terrn. The reason is that
Frege takes a function of any specifie type to be formally given by a term and speci
fied variable(s) of the appropriate type(s), and he therefore allows as an instance of
universal specification any substitution of a function, given in this way as term and
specified variable(s), for a universally quantified function variable (of the correspond
ing type). But it is essential for Frege's notion of substitution (both as he spelled it
100
out and as we reconstructed it in Chapter 1) that with any occurrence of a function
variable, f, the variables of f be expressed, e.g. as f(x, y) if f is a first-order binary
function variable. This is not the case in an expression such as fin,k) ~(n,k) fJn,k)
and substitution, as Frege and we have defined it, of a function given by a term and
specified variable(s) for a function variable in this expression do es not make sense
(unless n = k = 0). Because we don't have courses-of-values in CTT, we no longer
have any means of stating the (typed) equality of two "extensionally equivalent" func
tions, as given by terms and specified variables. If, for instance, c/J(!I, ... ,fm) is the
function of type (n, k) given in CTT by the term c/J and the variables !I, . .. ,fm and
'ljJ(ft, ... , fm) is the function (of type (n, k)) given by the term 'ljJ and the variables
!I, ... , fm, and we have that vll ... Vlm(c/J[~~, ... , ~:] ~ 'ljJ[~~, ... , ~:]) = T, then we
have no way of expressing the equality of c/J(ft,· .. ,fm) and 'ljJ(!I, . .. ,fm) in CTT,
since, e.g., c/J(fl, ... ,fm) ~(n,k) 'ljJ(!I, . .. ,fm) is not a legitimate term. This situation
we remedy by the addition of the following axiom schema:
Axiom Schema of CTT-Comprehension. For every CTT term c/J and any variables,
!I, ... ,fm, of types al, . .. ,am, respectively,
f- ~-(n,k)~f- ~f- (-(n,k)(f- f- ) ~ -+.[iI lm]) ::J9 VI· .. V m 9 l, ... , m -(0,0) 'f' fI"'" fm '
where k = O"n(al,' .. ,am)'
Now, if c/J(ft, . .. ,fm) and 'ljJ(!I, . .. ,fm) are two functions given in CTT as men
tioned above, their equality (of type (n, k)) may be formally stated as follows:
( ) ~-(n,k)~-(n,k)['f- ~f- (-(n,k)(f- f- ) ~ -+.[11 lm] * ::J9l ::J92 VI· .. V m 91 l,· .. , m -(0,0) 'f' !l''''' fm - -
. -(n,k)(f- f- ) ~ OI,[!I fm]/\' -(n,k) ~ -(n,k)) 1\ 92 l, ... , m -(0,0) 'f' fI"'" fm 91 -(n,k) 92 ,
(assuming again k = O"n(al,"" am))'
By two instances of the Axiom Schema of CTT-Comprehension, we can be sure
that the (*) is in fact equivalent to the statement of "(n, k )-equality" between
101
The need for a comprehension principle in the absence of courses-of-values is in
teresting. As we have mentioned often, a function in GG and in CTT is represented
by a term and specified variable(s). (Note that this of course includes the case where
the term is simply the specified variable.) In GG, we can meaningfully substitute
a function so given for a function variable (of the appropriate type) in any context
in which the latter meaningfully occurs. Given Basic Law V of GG, we state the
extensional equivalence of two functions by stating the equality of the correspond
ing courses-of-values. The existence of a function represented by a term and specified
variable(s) is essentially given by the fact that the term can be formed in GG; however
the existence of courses-of-values for functions is required in order to make statements
of extensional equivalence between these functions, or, rather, of the uniqueness (in
terms of courses-of-values) of extensionally equivalent functions. The positing of a
corresponding course-of-values for every function is built into the grammar of GG. In
CTT, given the absence of courses-of-values, we must state the uniqueness of exten
sionally equivalent functions directly as in the Basic Law V of CTT ab ove. But this
means that there are contexts in which we cannot meaningfully substitute a function
as given by a term and specified variable(s) for a function variablé. In particular, if
we wish generally to make statements of extension al equivalence between functions
represented by terms and specified variables, this forces us to directly posit the ex
istence, for any term and specified variable(s), of the corresponding function, as we
do in the Axiom Schema of CTT-Comprehension above. In both GG and CTT,
statements of equality, or equivalence, between functions requires an assumption of
existence, an explicit postulate on the existence of functions in CTT and an implicit
assumption of the existence of courses-of-values in GG. Of course, Russell's paradox
4That is, even when the function variable is of the appropriate type, i.e., the type of the function
represented by the term and specified variable( s).
102
shows us that if we assume the existence of courses-of-values as in GG, that is, en
tities all of the same type (object type in GG), corresponding to functions that take
entities of this type as arguments, then we must place some restrictions(s) on our
existence assumption. E.g., we may adopt a "limitation of size" doctrine as in ZF,
or we may adopt a stratification requirement as in NF. On the other hand, if we
postulate a comprehension principle on the existence of functions as in CTT, while
maintaining a strict separation of types, then no paradox seems to arise from unlim
ited comprehension. This is analogous to the unlimited comprehension principle for
sets in TT.
We come finally to the rules of CTT. As for GG, we may adopt modus ponens, uni
versal generalization and universal specifications (see Chapter 1). For CTT, however,
we need a more elaborate notion of substitution. For the statement of universal speci
fication in GG, we defined substitution of terms for object variables, and of first-order
unary and binary functions, represented in GG by a specified term and one or two
specified object variables, for unary and binary function variables, respectively. The
statement of universal specification in CTT requires a general notion of substitution
of a function of type (n, k), as given by a specified CTT term and mk variables of
types a~k, a'2k, ... , a~k' respectively, for a variable of type (n, k). For cases in which
the variable f(n,k) occurs with its variables expressed, this generalization is straight
forward from the notion of substitution in GG, if rather tedious to spell out. For
both these reasons, we skip the details here. For occurrences of function variables
with their own variables unexpressed, we do not allow substitutions. CTT as we
have presented it to this point is our formaI interpretation of Frege's discussion in the
Grundgesetze. We turn our attention now to simplifying assumptions for CTT.
We consider first simplifying CTT by restricting the range of each function to
a set of exactly two distinct objects. Specifically, we would like these objects to be
T and 1-, which we may represent in CTT as Vx(x ~(O,O) x) and -.:, Vx(x ~(O,O) x),
103
respectively, where x is a variable of type (0,0). This restriction can be achieved with
the following axiom schema.
Axiom Schema of Restriction to Concepts. For every CTT type (n, k) and func
tion f of type (n, k),
. . . f- '::/91 ... '::/9m(J(91 , ... ,9m) ~(O,O) '::/x(x ~(O,O) x)
V f(91,"" 9m) ~(O,O)":" vx(x ~(O,O) x)),
where k = (Jn(a~k, ... , a;:k) and gi is of type ark for i ~ mk·
Let us abbreviate f(gl,' .. ,gm) ~(O,O) '::/x(x ~(O,O) x) by (gl,"" gm) E f, and
f(gl,'" ,gm) ~(O,O)":" '::/x(x ~(O,O) x) by (gl,'" ,gm) tI. j. Then we may restate the
above axiom as: ., .
f- '::/91 ... '::/9m((91 , ... ,9m) E f V (91, ... ,9m) tI. 1).
In this way, the functions of CTT are restricted essentially to predicates. (Note
that each of the primitive functions of CTT is assumed to satisfy this axiom.) Clearly,
if we adopt the above axiom, then every function may be interpreted as a set or n
ary relation, for sorne n E N. Let us denote CTTR the theory obtained from CTT
by the addition of this axiom. Roughly speaking, CTTR is TT extended to allow
not only sets at each level n, but k-ary relations for kEN, and in such a way that
these relations may be of mixed levels, so long as they are restricted to sets and
relations of levels ~ n - 1 and any tuple (gl, ... ,gm) contained in such a relation, f,
((gl, ... ,gm) E 1) contains sorne gi of level n - 15 .
Therefore, if we further restrict CTTR by allowing only unary functions, we re
cover essentially the theory TT. That is, we would have a single type for each level n,
5To state this last condition more precisely, we should say that any tuple (91, ... ,9m) for which
the expression (91, ... ,9m) E f is grammatical (i.e. a CTT term) is such that 9i is of level n - 1 for
sorne i :::; m.
104
corresponding, for n > 0, to functions of a single variable of level (or type, since we
may then identify types with their levels) n - 1 and to objects for n = O. Any such
function of level n > 0 is in fact the characteristic function for a set of entities of type
n - 1. By Extensionality (Basic Law V) ab ove , any two functions of the same level
are equal iff they take the same value for every argument of level n - 1, which means
they are equal iff they are characteristic functions for the same set (if we assume also
the usual set-theoretic axiom of extensionality). Basic Law V reduces to:
fn+ 1 ~(n,k) gn+l ~ ';/hn(jn+l(hn) ~(O,O) gn+l(hn)), for n E N.
In CTTR, this may also be written as:
r+l ~(n,k) gn+l ~ Vhn(hn E r+1 ~ hn E gn+l).
We also have that the following holds in CTTR: for any CTT term 4> and variable
f of type n EN,
3gn+lvj(gn+l(j) ~(O,O) Vx(x ~(O,O) x) ~ 4>[tl ~(O,O) vx(x ~(O,O) X)).6
This might also be written as:
3gn+lVj(j E gn+l ~ 4>[tl ~(O,O) T),
where we have abbreviated 'ï/x(x ~(O,O) x) by T.
But now it should be quite dear that we can interpret TT in CTTR restricted
to unary functions and vice versa. This brings precision to our daim that, in the
Grundgesetze, Frege starts from a kind of Cumulative Type Theory, to which he adds
abstraction and application (via the unit dass operator). This result also demon
strates our further daim that CTT reduces to TT with the above simplifying as-
sumptions.
6With the substitution c;b[tl appropriately defined.
105
106
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