FOM: ZF-based category theory
kanovei at wminf2.math.uni-wuppertal.de
Wed Jan 21 13:17:50 EST 1998
>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.
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