Let me introduce my response by saying that although I was more-or-less familiar with Lawvere's thoughts on the subject, I had never thought too much about them until some years ago, more than 5, less than 10, when I found myself teaching a course in set theory. For the first time, I came to realize what a complex horror set membership really is. In every other type of mathematics I had ever studied, the objects were some kind of sets with some kind of structure and the arrows were the functions (or at worst equivalence classes of such) that preserved them. Mostly, the structure was given by operations, or at least partial operations, although Top was something of an exception, but even there, continuous maps are those that preserve ultrafilter convergence. But of course, Sets are an exception. Here are sets defined in terms of these elaborate epsilon trees and this structure is invariably ignored. It seemed to me intuitively, confirmed by Makkai, that the ONLY arrows between sets that actually preserved all that structure were inclusions of subsets. So that obvious category is just a poset. But of course the truth is that that epsilon structure is invariably ignored. So why is it taken as the basis of mathematics. Much better to simply define sets as the objects of a category and then an element is just a global section, or rather an equivalence class thereof. My whole experience with category theory convinces me that membership and the closely related idea of equality is an intrinsically obscure notion Or rather, not that it itself is obscure, but it obscures anything it touches. Like, the idea of embedding Z into Q by taking the eqivalence classes of fractions and then removing those that include a fraction of the form n/1 and replacing them by the corresponding integer. Yes, it can be done and we certainly want to say that Z is a subring of Q, but it is such a mess and so unnecessary. Some would say, but not I, that you should just work in your favorite topos. I take a platonic view that there really are sets, but membership and equality are not the simple concepts we think they are. In particular, I would like to say that a fraction is a pair of integers m/n, n > 0 and that m/n = p/q when mq = np. I don't know about pointers to the literature. Cheers, Mike