THE RELATIVE STRENGTH OF ZERMELO'S SET THEORY
AND QUINE'S NEW FOUNDATIONS
J.
BARKLEY
ROSSER
The present note is a summary of some of the results that have so far been
obtained in an investigation that is still proceeding. Many of the technical
points involved are as yet in the process of being published, having appeared
only in the unpublished doctoral theses of Firestone and Orey (see [1] and [2]).
We shall refer here generically to any of the considerable variety of systems
that stem from the original Zermelo concept as a Zermelo set theory (we shall
use merely "ZST" henceforth). In [3], Chap. I l l and Chap. IV, is given a
survey of the better known ZST's, with references to the literature. We shall
focus attention on three ZST's as being sufficiently representative for the
present summary, and shall present them in forms convenient for the present
exposition without regard whether these are their familiar forms.
The term "Quine's new foundations" (we shall use merely "NF" henceforth) usually refers to a specific system, namely that proposed in [5] and
expounded at length in [6]. However, one consequence of the comparison of
NF with the ZST's is the discovery of a variety of systems which parallel the
main kinds of ZST's while retaining the characteristic features of NF. Specifically, corresponding to each of the forms of ZST which we shall consider, there
is a form of NF] these forms of NF differ among themselves in a way quite
analogous to that of their corresponding ZST's] also, in each of the NF's one
can form a model of the corresponding ZST, so that if some of these NF's are
consistent, then so are their corresponding ZST's. The study (if any) of relative
consistency proofs in the reverse direction has not met with any success. Most
likely, we do not yet have the best forms of NF corresponding to the various
forms of ZST, and a search is continuing for more satisfactory forms of NF.
In some forms of ZST, there are two categories of classes, those which are
also members (that is, the "sets") and those which are not (see [4]). Wang
(see [7]) has pointed out that Quine's Mathematical Logic (see [8]) can be
considered as consisting of NF with the addition of classes that are nonmembers. This suggests that ZST's with two categories of classes should be
compared with Mathematical Logic, rather than with NF. In [2], such a
comparison is carried out, and proceeds quite analogously to the simpler comparison of a single-category ZST with NF. Accordingly, we shall here restrict
289
attention to ZST's in which all classes are also members. The question of finding the appropriate forms of NF to compare with ZST's containing individuals
has not been considered.
We herewith consider the most uncomplicated version of ZST, in which
there is a single category of object, namely that of sets, and a single predicate,
namely that of class membership. There are roughly two sorts of axioms: those
that specify the existence of certain particular sets, and those that permit the
generation of new sets from those already given. Seeking convenience rather
than elegance, we list the primary generation axioms:
The pair axiom: Given two sets, a and ß, there exists a set of which a and ß
are the sole members. (It is not required that a and ß be distinct).
The sum axiom: Given a set a, there exists the set consisting of the sum
of all the members of a.
The power axiom: Given a set a, there exists the set consisting of all the
subclasses of a.
The subclass axiom: Given a set a and a statement F(x), there exists the
subset of a consisting of all x's in a which satisfy F(x).
Since each generation axiom merely produces a new set from some given
set (or sets) these axioms alone do not suffice to prove the existence of any set,
and they must be supplemented by some axiom that explicitly states the
existence of some specific set (from which one then derives many other sets by
the generation axioms). By taking different existence axioms, one can get
different ZST's.
One possibility is to assume an axiom stating the existence of the null set.
Then the stated generation axioms suffice to generate only sets of finite cardinality. Consequently the resulting ZST is of limited interest. A much more
popular course is to assume an axiom stating the existence of what is in effect
the set of non-negative integers. By repeatedly applying the power-class axiom,
one can generate classes of high enough cardinality for many mathematical
purposes. However, one cannot infer the existence of a class with the cardinality
corresponding to an infinite number of applications of the power-class axiom to
the set of non-negative integers.
If one wishes to deal with very high cardinalities, one can either assume an
axiom explicitly stating the existence of some set of adequately high cardinality
or one can introduce a new generation axiom, namely:
The replacement axiom: Given a set a and a statement F(x, y) such that
for each x in a there is at most one y such that F(x, y) is satisfied, there exists
the set of all y's for which there is an x in a such that F(x, y) is satisfied.
With this added generation axiom and the assumption of the existence of
the set of non-negative integers, one can develop a full-fledged theory of
290
cardinals and ordinals. One could get still stronger systems, or variations o
those already indicated, but these suffice for our present purpose. Indeed, let us
denote by Sx, S2, and S 3 the three following systems. For Sx, we use the four
primary generation axioms and assume the existence of the null set. For S2,
we add to S! the assumption of the existence of the set of non-negative integers.
For S3, we add to S2 the replacement axiom.
For NF, there is also a single category of object, namely that of sets which
are also members, and a single predicate, namely that of class membership.
Sets are generated by the single principle
(P)
(Ey)(x)(xey = F(*)) f
where F(x) is required to be stratified. For the technical definition of "stratified," see [5] or [6]. However, in effect, it signifies that one can assign types
to the variables of F(x) in such a way that F(x) would become a legitimate
formula of simple type theory. One can replace the single principle (P) by
an existence axiom and a set of generation axioms having no reference to
stratification (see [9]), but these lack the naturalness of the Zermelo formulation, and it is the principle (P) which is really characteristic for NF.
By letting F(x) contain the names of sets already given, one can use the
principle (P) to generate new sets from given ones. Alternatively, one can use
the principle (P) to prove the existence of various specified sets. Thus, by (P)
one easily infers the existence of the null set, and one can even infer the
existence of the set of non-negative integers (see [10]).
If one tries to identify the sets of NF as sets of a ZST in a straightforward
way, one immediately runs into difficulty. For example, in NF there is a
universal set. Not only is there no universal set for a ZST, but the universal set
of NF fails to have a number of properties that sets are expected to have in
mathematics and which sets do have in the various ZST's. We indicate three
such properties.
(1) The power class of a set a (that is, the set of all subsets of a) should
have higher cardinality than oc. However, the power class of the universal set of
NF is just the universal set itself.
(2) A set a should have the same cardinality as the set of all its unit subsets. Indeed, intuitively one sets up the one-to-one correspondence by letting
each member x of a correspond to the unit subset containing just x. However,
this correspondence is not stratified, and so one is not entitled to infer its
existence by invoking principle (P). Moreover, in NF, one can prove that the
universal set has higher cardinality than the set of all its unit subsets.
(3) It is very popular among mathematicians to assume that every set can
be well ordered. One can add such an assumption to a typical ZST without risk
291
(see [4]). However, Specker has shown that it is a theorem of NF that the
universal set cannot be well ordered (see [10]).
One interpretation for this state of affairs is that the principle (P) is too
generous, and supplys us with various pathological sets as well as with various
sets that we wish. Accordingly we seek a criterion for distinguishing the
pathological sets. Then we can seek a correspondence between the non-pathological sets of NF and the sets of a ZST.
We shall indicate one such possible criterion, and summarize some results
of adopting it. We wish to emphasize that there are other possible criteria, and
we invite attention to the problem of devising such other criteria. Much remains
to be learned about the present criterion, and it is too early to decide whether
it is wholly adequate.
To indicate the criterion that has been studied so far, let us follow [6] and
say that a set a is strictly Cantorian if a can be put into one-to-one correspondence with its set of unit subsets in such a way that each member # of a is
made to correspond with the unit set which contains just x. Let us agree to
consider a set as pathological if it is not strictly Cantorian. If attention be
restricted to strictly Cantorian sets, one gets in NF a theory of cardinals and
ordinals that follows the classical theory much more satisfactorily than when
one allows sets to appear that are not strictly Cantorian.
We now seek a correspondence between the sets of some ZST and the
strictly Cantorian sets of NF. Thus we exclude from consideration such pathological sets as the universal set of NF. However, there still arise difficulties.
For strictly Cantorian sets the power axiom holds (see [6], Thm. XIII.2.7),
but the sum axiom fails to hold, as one can see by taking a to be the unit set
whose sole member is the universal set. It thus appears that if we wish a strict
correspondence, we must use same stronger criterion than that of being strictly
Cantorian. So far, attention has been focussed on an alternative approach.
In seeking a correspondence between sets of a ZST and some collection of
sets of NF, let us abandon the requirement that the relation of class membership in the ZST must correspond to the relation of class membership in NF.
If this is done, it is possible to get a correspondence between the sets of S x and
some (or all) of the non-negative integers of NF. This is done as follows. One
can enumerate all finite sets of non-negative integers (see [6], Thm. XI.4.18).
Let 0O, 0X, . . . be the finite sets of non-negative integers. Define sx by saying
that for non-negative integers m and n, mexn holds if and only if m is a member
of 0n. Then with non-negative integers playing the role of sets and ex playing the
role of class membership, the axioms of Sx are readily verified as theorems of
NF. Thus the sets whose existence is guaranteed in Sx must correspond to some
(or all) of the non-negative integers in NF.
292
To model S2 or S 3 in NF in an analogous manner certainly requires that
one be able to discover in NF sets of high cardinalities. Accordingly it seems
likely that we must add to NF axioms insuring the existence of suitable sets
of high cardinalities before we can model S2 or S 3 . However, we may as well
make use of a device for keeping the cardinalities involved relatively low.
In [4], a model A is set up for a certain strong ZST. Suitable portions of A
can be used as models for S2 and S 3 . As A is well ordered, we can readily set up a
model A0 in the ordinal numbers which is isomorphic (in a loose but obvious
sense) to A. By using the relation e2 between ordinals, that serves in AQ as the
image of class membership in A, we can get models for S2 and 5 3 by using
suitable fragments of A0. We observe that by paralleling the definition of A,
we can make a definition of e2 by transfinite induction over the ordinals. Thus
we can get models for S2 and S 3 in any formal logic which:
(a) contains a sufficiently developed theory of ordinals to carry out the
definition of e2 by transfinite induction;
(b) contains sufficiently large segments of the ordinals so that adequately
high cardinalities are available.
Let us first consider point (a) in NF. If we restrict attention to the
ordinals of those well ordered sets which are strictly Cantorian, we get a fairly
decent ordinal theory. In [2], Orey has shown that by adding a special axiom
dealing with transfinite induction for such ordinals, one gets the full classical
theory of ordinals in its familiar form. However, this axiom is almost certainly
stronger than is needed for the definition of e2. More likely, we can define e2
without addition of any extra axioms to NF, but this point needs further study
before a firm statement is justified.
Suppose we have added to NF an axiom (or perhaps no axiom) sufficient
for the definition of e2. Then let us consider point (b). Here is where we find a
strong analogy with the ZST's. To get S 2 from Sx, we added the assumption
that there is an infinite set. Correspondingly, to extend NF (supposed adequate
for ordinal theory) enough to have a model for S2, we add the assumption that
there is an infinite strictly Cantorian set. With NF extended as indicated above
to contain a model of S2, we can insure a model for S3 by adding an axiom
identical in wording with the replacement axiom except that a is required to be
strictly Cantorian.
Looking at S2 and 5 3 as successive extensions of Sx got by adding specified
axioms, we see that exactly analogous axioms dealing with strictly Cantorian
sets of NF will suffice to enlarge a suitable ordinal theory in NF to provide
models for S2 and 5 3 .
Actually, the version of the replacement axiom indicated for use in NF is
stronger than necessary. Since S3 is to be modelled in the ordinals, it suffices in
293
the replacement axiom to require that a be a strictly Cantorian class of ordinals.
This indeed can be still further weakened (see [2]). Probably the weakest axiom
that would suffice would be a simple assumption that there is a well-ordered
strictly Cantorian set of sufficiently high cardinality.
Probably one cannot get a model of S2 with any weaker assumption than
that there is an infinite set which is strictly Cantorian. However, this assumption can be given various alternative forms. For example, it is equivalent to the
axiom of counting (see [6], p. 485).
To summarize, it appears that NF is considerably stronger than the weak
ZST which we have called Sx. However, the extra strength of NF appears in its
ability to generate sets like the universal set, which are pathological, though
useful and interesting. If we restrict attention to the non-pathological sets of
NF, then we find NF to be about on a par with Sx. If we strengthen Sx by
adding an axiom of infinity to get S2, then we can keep NF at least on a par by
adding to it an axiom of infinity for non-pathological sets. Moreover, the same
situation holds for the axiom of replacement. This does highlight the fact
(which is sometimes discounted) that it is primarily the presence of the strong
axiom of replacement that makes 5 3 a very powerful system of logic.
REFERENCES
[1]
C. D. F I R E S T O N E , Sufficient conditions for the modelling of axiomatic set theoryDoctoral Thesis at Cornell University, September, 1947.
[2] STEVEN O R E Y , Formal development of ordinal number theory and applications
to consistency proofs. Doctoral Thesis at Cornell University, September, 1953.
[3] H. W A N G and. R. MCNAUGHTON, Les systèmes axiomatiques de la théorie des ensembles. Gauthier-Villars, Paris, 1953.
[4] KURT GöDEL, The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. Princeton University Press,
Princeton, 1940.
[5] W. V. QUINE, New foundations for mathematical logic. The American Mathematical
Monthly, vol. 44 (1937), pp. 70-80.
[6] J. BARKLEY ROSSER, Logic for mathematicians. McGraw-Hill Book Co., New
York, 1953.
[7] H A O WANG, A formal system of logic. The Journal of Symbolic Logic, vol. 15 (1950),
pp. 25-32.
[8] W. V. QUINE, Mathematical Logic. Third edition. H a r v a r d University Press, Cambridge 1951.
[9] THEODORE H A I L P E R I N , A set of axioms for logic. The Journal of Symbolic Logic,
vol. 9 (1944), pp. 1-19.
[10] E. P. S P E C K E R , The axiom of choice in Quine's New Foundations for mathematical
logic. Proceedings of the National Academy of Sciences, vol. 39 (1953), p. 972-974.
CORNELL UNIVERSITY.
294
SECTION
VII
PHILOSOPHY, HISTORY AND EDUCATION
STATED ADDRESSES