After seeing the volume of "fluff" that this topic had generated on Friday, I took a vow not to contribute any more to it. But Vaughan Pratt seems to be challenging the rest of us to find something wrong with his ordinal-based definition of Set, and no-one else has taken up the challenge; so I'll have to break my vow and point out its obvious shortcoming. Vaughan's definition is fine as long as you are happy, not just to assume that the axiom of choice holds, but actually to rely on it to construct codings for you. If AC doesn't hold, the Vaughan's category fails to be cartesian closed; and even if it does, you don't have the ability to point to a particular object of it and say "This is the set of real numbers" (still less to point to a particular morphism and say "This is the addition operation on real numbers"); you have to rely on God's (or "the randomizer's", as Vaughan seems to call him) ability to do it for you. I suppose only a minority of mathematicians are unhappy about AC to the extent of actually rejecting any construction that can't be done without it. But I think a very large majority would be unhappy about calling upon God to construct things for them (such as the real numbers) for which they know there is a perfectly good man-made construction. Such people are going to be in a near- permanent state of unhappiness if they are condemned to do mathematics inside Vaughan's category. By the way, the Cole--Mitchell--Osius equivalence between weak Zermelo set theory and well-pointed topos theory, which is described in my book (sorry Vaughan, but Academic Press won't reprint it) and in Mac Lane--Moerdijk, doesn't assume the axiom of choice; it's an "optional extra" which you can add to both sides of the equation if you want to. Peter Johnstone