Godel and Bernays on CT.

Godel and Bernays on CT.

[Noson Yanofsky]

Recently Volume 4 of Kurt Godel=92s Collected Works came out.
Volume 4 and 5 are correspondences. There is some discussion of
category theory that I thought might be of interest. The relevant
portions have been extracted. First is Godel=92s letter to Bernays.
Then Bernays=92 response. Finally there is Solomon Feferman
discussion.

Letter 47. Godel to Bernays (9 January 1963) on page 221:
=93... I found it interesting that you speak on P. 199 of the
=93newer abstract disciplines of mathematics=94 as something lying
outside of set theory. I conjecture that you are thereby alluding
to the concept of category and to the self-applicability of
categories. But it seems to me that all of this is contained
within a set theory with a finitely iterated notion of class,
where reflexivity results automatically through a =93typical
ambiguity=94 of statements. Isn=92t that also your opinion? I=92ve
heard, by the way, that someone has formulated the axioms of set
theory with the aid of the concept of category and that this has
perhaps even been published. If you know something about it, I
would be very grateful to you for relevant information.=94



Letter 48. Bernays to Godel (23 February 1963) on pages 229-231:
=93That in the enumeration of the domains of mathematics in which
the classical methods come to be used, I named the =93newer
abstract disciplines=94 in addition to analysis and set theory was
conceived in the sense of a distinction between categorical and
hypothetical mathematics. Abstract axiomatic topology and algebra
can be pursued in such a way that one indeed uses concepts like
that of natural number and real number as well as set-theoretical
concepts and theorems, without undertaking a strict incorporation
into set theory.
As to the difficulties associate with =93categories=94, of which Mr.
Mac Lane spoke in his Warsaw address, I have only a rough idea of
the requirements in question. All the same, it seems to me that
the difficulties rest at least in part on the fact that it is
exclusively the extensional characterization of categories that
is being considered.  In an analogous way, for example, if we
characterize the number 5 by means of the class of five-element
sets, there is an impediment to forming a class with the number 5
as [an] element. To be sure, categories are also effectively
given by means of axiom systems, and the multiplicity of the
axiom systems that have to be considered can presumably be
represented as a set, and certainly as a class. Your idea that a
set theory with a finitely iterated notion of class is suitable
as a framework for the theory of categories seems very plausible
to me. If axiomatic set theory is being extended in any event,
then the other novel infinitistic investigations should also find
a place in the extended framework. Just recently I received a
manuscript of the dissertation submitted ny William Hanf, in
which the author expresses the opinion in an introductory chapter
that for his investigations a set theory with a finitely iterated
notion of class does not suffice as an axiomatic framework.
=93That someone recently formulated the axioms of set theory with
the help of the notion of category I learned for the first time
from your letter.=94



This is Solomon Feferman=92s discussion of the correspondence on
pages 59-60:
=93Foundations of category theory. The foundational aspects of the
subject of category theory came up for discussion in Godel=92s
letter of 9 January 1963 (number 47); his point of departure was
a remark in Bernays 1961a to the effect that =93the =91newer=92
abstract disciplines of mathematics=92 [are] something lying
outside of set theory.=94 Godel assumed (mistakenly as it turned
out) that Bernays was =93thereby alluding to the concept of
category and to the self-applicability of categories=94. With
reference to self-applicability, what Godel presumably had in
mind are examples such as the category of all categories, for
which there is no straightforward set-theoretical interpretation.
He went on, interestingly, to suggest that such cases of
self-reference could be handled through =93typical ambiguity=94
applied to an extension of set theory by classes of finite type.
Actually, something like that had been pursued in the
Grothendieck school of homological algebra (e.g., in Gabriel
1962), which assumes 9instead of higher types) the existence of
arbitrarily many =93universes=94, i.e., collections of sets closed
under various standard operations, such as the stages Va in the
cumulative hierarchy for a strongly inaccessible. But for typical
ambiguity, one would need that the properties (in the language of
set theory, or higher type theory as suggested by Godel) of any
universe used are the same as in any other universe. Kreisel
suggested in 1965, pp.117-118, the required applications could
just as well be taken care of by use of the reflection principle
in set theory without the assumption of inaccessible cardinals;
the idea was spelled out and verified to a considerable extent in
Feferman 1969, though not all the problems were dealt with
thereby.

=93Also in letter 47, Godel said that he had heard =93that someone
has formulated the axioms of set theory with the aid of the
concept of category and that this has perhaps even been
published,=94 and he asked Bernays if he knew anything about it.
The first publication of an axiomatic theory of the category of
sets was by F. William Lawvere in his 1964; since that appeared
in the Proceedings of the National Academy of Sciences, it is
possible that Godel had heard of its submission through one of
the other members of the academy. In any case, Bernays=92 response
in letter 48 was not really helpful in dealing further with
either question concerning category theory, and the subject was
not pursued further in the correspondence.=94