Subject: categories: Re: G�del and category theory

From: [email-masked] (Colin McLarty)
Date: Wed, 11 Nov 1998 16:41:48 -0500 (EST)

Perez Garcia Lucia <[email-masked]> wrote, among other things:

>- In what sense the self-applicability of categories transcends the concept
>  of set?. (It is obvious that categories transcend the concept of well-
>  founded set but, what's the matter with non-well-founded sets?.

        Well-founding is an irrelevant detail. Take any non-wellfounded set
theory which includes the axiom of choice, such as Aczel's AFA. Then every
set is isomorphic to an ordinal, that is to a well-founded set. Since
categorical methods are all isomorphism invariant, any categorical structure
available in this set theory is also available in well-founded sets. I have
discussed this in an article "Anti-foundation and self-reference" Journal of
Philosophical Logic 22 (1993) 19-28. There is no real chance that abandoning
the axiom of choice will help either--say by adopting AFA without Axiom of
Choice. 

        Rather, the apparent issue is existence of a universal set--a set of
all sets, so that you make a category of all categories. If you want to use
membership based set theory this will require non-wellfounding, but again
the details of membership and wellfounding are irrelevant. 

        Anyway, the problem here is that functions are hard to work with in
set theory with a universal set. I have shown that in any such set theory
meeting a few weak conditions there is a category of all categories, and it
is not cartesian closed. The result is clear from the more particular case
"Failure of cartesian closedness in NF" Journal of Symbolic Logic57 (1992)
555-56. Working with such a poor 'category of all categories' is much more
difficult than just doing without. 

        I think a more promising approach is to use Benabou's theory of
fibrations and definability as in Benabou J. (1985). "Fibered categories and
the foundations of naive category theory". Journal of Symbolic Logic 50,
10-37. I have discussed this briefly in "Category theory: Applications to
the foundations of mathematics" Routledge Encyclopedia of Philosophy (1998);
and in "Axiomatizing a category of categories" Journal of Symbolic Logic56
(1991) 1243-60. 

        I see no good arguments that there SHOULD be a genuine "category of
all categories" in any strong sense. But it seems an interesting question.

>- In what sense do you think G?del proposed distinguishing different levels
>  of categories?. Would it be possible that G?del was thinking of something
>  like type theory?.

        More likely he was thinking of Eilenberg and Mac Lane's use of
Goedel-Bernay's set theory as a foundation in "The general theory of natural
equivalences", so there are set categories and class categories.

        To study Goedel's claim here, you should look at any of Mac Lane's
papers on foundations that Goedel might have seen by this time. Maybe the
foundational parts of "The general theory of natural equivalences" are all
he could have seen, I don't know. Then it would be good to know what people
around Princeton were saying about category theory at this time--and that
might be very hard to find out.

Colin