FOM: ZF-based category theory

[Kanovei]

>From: Carsten Butz <butz at brics.dk>
>Date: Wed, 21 Jan 1998 14:28:06 +0100 (MET)

>(4) What ZF cannot do is to talk about classes, i.e., collections
>that are too big to be (small) sets.

This is not exactly the case: ZFC has some ways to 
speak consistently about proper classes (see e.g. 
Shoenfield's paper in "Handbook on Mathematical Logic"). 

However a more appropriate way would be to employ a 
suitable class theory which conservatively extends ZFC. 
(For instance NBG will work as long as you consider only 
classes of sets. Classes of classes need the next level.) 
By conservativity we are still in ZFC in the 
sense that every statement about sets proved in such a 
class extension is a theorem of ZFC. 

We can get in trouble only if they essentially 
need sometimes quantifiers over categories (say to define 
sets), which cannot be handled this way, at least directly 
--- but I do not know whether they really need this. 

Vladimir Kanovei