Vaughan Pratt says <<What arguments exist in *support* of the existence of Grothendieck <<universes? I see the "cogito ergo sum" argument, what else?>> I once asked Joe Shoenfield that -- not quite in those terms -- and his answer certainly didn't seem to me "cogito ergo sum". That seems to me, assuming I understand what argument you mean, not radically better than Anselm's proof of the existence of God: we can imagine the best thing in the world, and if it is the best it must exist (otherwise one that existed would be even better), so it exists, QED. Now I don't suppose you mean a White Knight's sort of argument -- the universe imagined me, therefore it exists. You mean I imagine Grothendieck's universe, therefore IT exists. 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. It's the null set, and the singleton of the null set, and [and on. Not just countably, of course; we can describe \omega very satisfactorily, and the union of an omega-sequence of ordinals, and on. This differs from Anselm's word game in being a string of constructions. The first Grothendieck universe is rather large, so it is a fairly formidable kit of constructions. Pass to a second Grothendieck universe, and you used at least one miracle, to produce an individual from the first-universe construction. John Isbell