"Herman Jurjus" <firstname.lastname@example.org> wrote in message news:<c5o47o$3ei9o$1@ID-110648.news.uni-berlin.de>... > "Rupert" <email@example.com> wrote in message news://firstname.lastname@example.org... > > email@example.com (David Petry) wrote in message news:<firstname.lastname@example.org>... > > > email@example.com (Rupert) wrote > > > > > > > firstname.lastname@example.org (David Petry) wrote > > > > > > Could you explain why V can't be countable? Certainly you can't > > > > > prove that in any consistent first order formalism for which the > > > > > Loewenheim Skolem theorem applies. > > > > > > > > It's a triviality to prove V can't be countable in ZFC. The > > > > Loewenheim-Skolem theorem says that if ZFC is consistent, it has a > > > > countable model. But that's not V. > > > > > > > > Philosophically speaking, if we are discorsing in the first-order > > > > language of set theory and uttering theorems of ZFC, we can always > > > > suppose, without making any of our theorems false, that our discourse > > > > is relative to a countable model. But as I say, that's different to > > > > saying V can be countable. > > > > > > Somehow, I don't think you explained anything. You merely restated > > > what you said the first time. > > > > > > If you can prove V exists, you can prove that it is countable > > > relative to some meta-language. But you claim otherwise. Why? > > > > > > I suppose I should admit that I don't know a great deal about > > > this topic, but you have hardly increased my knowledge of it. > > > > If ZFC is consistent, and we are uttering theorems of ZFC in the > > first-order language of set theory, we can always entertain the > > possibility, without making any of our theorems false, that our > > discourse, let's call it the object language, is relative to some > > countable model. Then a metalanguage will be available in which what > > we referred to as V in our object language, will now in fact be > > countable. This possibility *may* hold of our language, I don't think > > it *has* to hold, this may be a point of difference between us. > > > > But in any case, you have to decide what language you're talking in. > > Whether you're talking in the object language or the metalanguage, it > > won't be true that V is countable. In both the object language and the > > metalanguage, all theorems of ZFC are true, and it's a trivial theorem > > of ZFC that V is not countable. > > But then, just like the original thing only _seemed_ uncountable, > but in reality countable, your 'meta world' could also be only > seemingly uncountable, but be 'meta-meta' countable. > Or do i miss something? > > Cheers, > Herman Jurjus
We can always entertain the *possibility* of any language that its semantics is relative to a model which is correctly called "countable" in some metalanguage. I don't think this possibility always has to hold.