[FOM] real numbers

[Andreas Blass]

On Fri, 20 Jun 2003, Kanovei wrote:

>
> Date: Thu, 19 Jun 2003 21:13:29 -0400 (EDT)
> From: Andreas Blass <ablass at umich.edu>
>
> >Feferman has shown how to interpret most of category theory, including
> >large categories, in a conservative extension of ZFC.
>
> Could you please give a reference ?
>
> V.Kanovei
>
The paper I had in mind is

Set-theoretical foundations of category theory. 1969
Reports of the Midwest Category Seminar. III pp. 201--247
Edited by S. MacLane. Lecture Notes in Mathematics, No. 106
Springer-Verlag, Berlin-New York 1969 iii+247 pp.

The idea is essentially to add to ZFC a constant symbol k, an axiom saying
that k is an ordinal, and an axiom schema saying that all formulas in the
language of ZFC reflect from the universe down to V_k (the collection of
sets of rank < k).  Then where category theory needs sets (or small sets)
use elements of V_k.  Subsets of V_k can play the role of proper classes,
and sets of even (far) higher rank are available if needed.  Furthermore,
the reflection schema ensures that, roughly speaking, any mathematical
assertion that you can prove about small sets (or small groups, small
topological spaces, etc.), for example by using category-theoretic
methods, automatically holds for arbitrary sets (groups, topological
spaces, etc.)

Andreas