Re: Challenge from Harvey Friedman

Being a newcomer to the category list, I have a really naive and stupid question (concerning H. Friedman's challenge). Namely, I was always under the impression that you had to know what a set was before you could talk about what a category was (in particular a topos). Is it possible to talk about toposes without knowing what a set is? This seems somewhat related to a question that has been bugging me for some time, namely how to talk about a ``category'' which is enhanced over itself, but not necessarily having any functor to or from Sets. The very first part of the structure would be a class of objects O together with a function (x,y)\mapsto H(x,y) from O\times O to O, but I can't get beyond that. ---Carlos Simpson