By we I mean the world of category theorists. Twice, I have tried to publish something on categories in the so-called Mathematical Intelligencer. Not that it's relevant, but earlier this year they published an article on the philosophy of mathematics by such an ignoramus that he didn't know the difference between the axiom of choice and the axiom of pairs. The most recent one contains an article by Barwise on non-well-founded set theory. Barwise has discovered that there are no non-trivial solutions to domain equations (the one he is looking at is S = N x S, where N is the natural numbers and he really means equals) in WF set theory, because it is clear that a set satisfying it and WF could only be empty. (Although he doesn't mention it, for such an equation as S = 1 + S, there's no solution at all.) Of course, these equations are important in computer science and something has to be done etc, etc. And what does he propose? Rather than accepting what is staring him in the face, that set theory founded on the epsilon relation is irrelevant to the workings of virtually the entire world of mathematics, he proposes to compound the felony by taking as mathematical foundations Aczel's theory of NWF sets. Consider that probably not one mathematician in ten could give a coherent account of ZF axioms. Have you ever seen a complex analyst begin a paper by saying if his complex numbers are ordered pairs of reals or 2 x 2 matrices with certain properties (and what are the elements of the matrix ring anyway?) or equivalences classes of R[x] mod (x^2 + 1) or ... ? Of course not. And there is no problem with domain equations in a categorical framework. Nor for that matter, with what Barwise calls the largest solution, by which he apparently means the terminal one. Peter Freyd once asked how you know that the largest sporadic simple group isn't the smallest counter-example to the Riemann Hypothesis. Actually, I suspect that for all the constructions of the complex numbers I know of, that cannot happen. But the very answer points up the irrelevance of the elements of a set. Certainly not very many mathematicians could give you the categorical axioms for sets either. The difference is that if you explained to him the categorical axioms, he would recognize in them constructions he uses daily. It is the categorical notion of subobject that is used when she considers the integers as a subset of the rationals, the rationals of the reals and the reals of the complexes. On the other hand, it now appears that Barwise is proposing to move set theory even further from practice by introducing an even more complicated set of axioms with the anti-foundation axiom that Aczel has introduced. I can only hope that someone with enough prestige that the Intelligencer might feel constrained to accept it would write a paper pointing out these things and suggesting that the topos-based axioms (which can be easily described) lead to a much simpler and more natural approach to the same thing. As for me, I've decided to drop my subscription. They have irritated me just once too often. Michael ===================================