Vaughan Pratt writes: 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), ... I think the word "despite" here misses the whole point. It is precisely *because* the existence of inaccessibles cannot be proven in ZFC, that the assumption of their existence is interesting. More exactly, this is how we know that ZFC+inaccessibles is a proper (if modest) extension of ZFC itself. Now of course this is not to say that whenever ZFC fails to prove the existence of some X that we should immediately assume "X exists"; among other problems, that would quickly lead to an inconsistent theory. Inaccessibles, however, have other merits: i) If my model says there are inaccessibles, and yours says there aren't, it may just mean that your model is an initial segment of mine. In that case my model is clearly better, because it has everything in it that yours does, and more besides; moreover, we haven't lost any nice features that your model might happen to have, because they are still true when prefaced by "in your model..." . ii) Inaccessibles are an example of the intuition that says "anything we know how to do too well or too precisely, can't possibly tell the whole story; there must be more." In this case what we know too well how to do is take exponentials of cardinals and limits of sequences of ordinals, where the length of the sequence is a cardinal we already have. Strengthen this intuition to "things we can construct in L" and you start to see why 0# should exist.