On Fri, Mar 7, 2008 at 2:52 PM, Steve Awodey <awodey@cmu.edu> wrote:
http://www.phil.cmu.edu/projects/ast/
The short answer is, it depends on how "Sets" sits in the category of classes. In fact, *any* topos can occur as a category of "Sets" satisfying replacement in a suitable category of classes constructed from the topos.
Very interesting! But I don't think that is the answer to the question I intended to ask, although perhaps I phrased the question poorly. As far as I can tell, you give a way of interpreting replacement/collection in such a way that it is satisfied in all toposes, by "constructivizing" the existential quantifier. But as you say, "In consequence, the standard arguments using Replacement that take one outside of V_\lambda(A) for \lambda non-inaccessible, are not reproducible." What I would really like to know is, can one formulate an elementary property of a topos which *does* allow one to reproduce the standard arguments of Replacement? Here's another way to phrase the same (or a similar) question. Suppose I meet a mathematician who thinks categorically enough to dislike the membership-based nature of ZF(C), but doesn't want to give up any of its consequences. In particular, he wants to be able to use transfinite induction beyond \omega+\omega. For instance, he wants Borel determinacy to be true, which is provable in ZFC but not in Zermelo set theory (ZFC minus Replacement). Is there a categorical foundation I can tell him to use? That is, is there an elementary categorical theory which is as strong as ZF(C)? Mike