I once asked Joe Shoenfield that [...] Well, Joe said in effect I can tell you everything that goes to make up the first Grothendieck universe, except I don't have time to finish telling you. Well, it's no worse than any of my existence "proofs" I suppose. I can't shoot it down with the argument that it would equally well prove the existence of cardinals we already know not to exist, since the extra time Joe would need to tell us everything that goes into making up a nonexistent ordinal might disqualify that proof without disqualifying the other. :) Further reflection on this merely seems to lead back to my original point about there not being enough room in this town for both \in and U. If your U is an inaccessible cardinal, and inaccessible cardinals *do* exist (and some set theorists seem to hope very badly they do, despite knowing it cannot be proven in ZFC), then rejecting membership is one reasonable way out. Becoming an intuitionist is another. Have the reasonable alternatives proposed to date been collected in one place and insightfully classified and compared? The flip side of this is, if inaccessible cardinals are eventually proved *not* to exist, then the Cantor-Russell paradox goes away for those working in a Grothendieck universe. But in that case the one reasonable objection I've been able to grasp so far to membership, namely its incompatibility with existence of one's mathematical home, goes away too. Is there any other argument against membership? Preferably just as short, but I'll settle for long if that's all that's possible. Or is the rejection of membership all just touchy-feely stuff based on a deep-seated feeling for something? "The von Neumann rigid-epsilon monsters" alluded to by Bill Lawvere sounds like it might be made into such an argument. But as I pointed out, when the only sets are the ordinals, one of each, the "monster" is tamed to a gentle line +1'ing steadily along, with the occasional little hiccup at each limit ordinal. That's no monster. A more convincing objection is needed. There must be something else, ideally something we are all very familiar with. The smoothness of space, for example. First off, even if physical space *is* smooth, why should this have any more bearing on our mathematical spaces than their dimension? Obviously no one can get away in these enlightened days with legislating 3, 4, or 26 as an upper bound in mathematics. Second, it is not even clear that the space we inhabit *is* smooth. Start with http://zebu.uoregon.edu/~imamura/209/may8/may8.html and look at what Vilenkin and Linde are proposing as an alternative model of space: foam at fine grain. Their idea that small distances in space are chaotic makes much more sense to me than a model based on extrapolating the smoothness of space-in-the-large down to arbitrarily fine scales as though space were a mathematical manifold. That extrapolation applied to the smoothness in the law of large numbers in statistics for example is well-known to lead to nonsensical results. With the statistical analogy in mind, my only question about the Vilenkin-Linde model is whether the appearance of chaos at fine scales would be nicely explained by populating unit volumes with only finitely many points. It is very easy to construct smooth-in-the-large manifolds from finite sets, and these will naturally look bumpy in the small. We would then have to add another physical constant to the books, the number of points of space per cubic centimeter. -- Vaughan Pratt =================================== PS. I inquired on sci.math.research about my construction of the nonstandard reals. Vladimir Pestov at U. Wellington pointed out an embarrassingly trivial oversight: I'd forgotten that the class of fields subdivides into smaller elementary classes, and had not thought to order my field (the standard part leaves no alternative, it can't be an algebraically closed field). You can't, at least not without further restrictions on the sequences. Luckily the standard part of my field *can* be ordered or I'd have to come up with some other demonstration to Peter Johnstone that one can identify the set of reals and its addition morphism without having to say which ordinals go where. More generally, any sequence can be named by its elements. There is no need to name it by whichever ordinal God picked this time around as the placeholder in the set of sequences of which that sequence is a member. I can't imagine what Peter was thinking. This sort of construction, where you have to take down the scaffolding when you're done, goes on all the time in ordinary mathematics.