Colin is exactly right on all points. I tend to look at sets from the perspective neither of a set theorist nor a category theorist but a combinatorialist. As long as people agree on the cardinalities of the homsets between sets, particularly the finite ones, I figure they must be talking about essentially the same objects. Infinite domains are problematic for everyone, infinite codomains much less so (we understand the homset N^2 much better than 2^N). The remark in my post about the self-evident being merely a convenient long-held hypothesis (which I put on my "sayings" website http://boole.stanford.edu/dotsigs.html less than a month ago) applies in spades to membership as characteristic of sets, the premise for ZF. Those who identify acceptance of the category Set with acceptance of ZF have not only not accepted but not even grasped that the wholesale replacement of the binary relation of membership by the (partial) binary operation of composition, with a set of axioms radically different from those of ZF, is a foundational move. ZF is so deeply ingrained in their thought processes that they have no idea how to think about mathematical structures without falling back on its axioms. Borrowing from Hilbert, they are unable to replace "set," "function," and "composite" by "table," "chair," and "beermug." If you find it hard to imagine how anyone could find it hard to imagine mathematics without ZF, just read Steve Simpson on 2/25/98 (almost exactly a decade ago) at http://cs.nyu.edu/pipermail/fom/1998-February/001228.html The bit "I totally repudiate every syllable of every word of every subclaim of every claim that McLarty has ever made about what he is pleased to call `categorical foundations'" made abundantly clear back then that Steve could not begin to concieve of replacing membership by composition as the basis for an alternative foundation of mathematics. While I can't speak for Steve today, this remains a stumbling block for those raised to believe that rigorous mathematics would not be possible in a world where propositions such as "for all x and y there exists z such that x is a subset of z and y is a member of z" did not hold. How could x U {y} fail to exist and the walls of mathematics not come tumbling down? Vaughan Colin McLarty wrote:
Vaughan Pratt <pratt@cs.stanford.edu> Wednesday, March 5, 2008 8:32 am
wrote, with much else:
On a related note, a careful reading of Max Kelly's "Basic Concepts of Enriched Category Theory" reveals that it is thoroughly grounded in Set,as I pointed out in August 2006 in my initial Wikipedia article on Max. I gave some thought to how one might eliminate Set from the treatment,without much success, and concluded that Max's judgment there was spot on.
Without addressing this particular issue I want to say I appreciate the phrase in the article: "the explicitly foundational role of the category Set." I take it this is Vaughan's?
Various people including Sol Feferman promote the view that if you use "sets" then you are admitting that you use ZF and not some categorical foundations. Vaughan's phrase goes aptly against that: If you use sets, then you use sets, but there is no reason it cannot be on categorical foundations. He does not say it *is* on categorical foundations, and that is fine in the context. He reminds people that it *could* be.
best, Colin