> Gc wrote: > > Aatu Koskensilta kirjoitti: > > > >> In a model M of set theory there will necessarily be an object r which > >> in M satisfies the formula that is the formalization of "x is the set of > >> reals". In case M is countable the set of objects that according to M > >> are in the epsilon-relation to r will also be countable. But there are > >> uncountably many reals, hence some of them are not among those objects > >> (or in M at all). > > > > I don`t understand. I am no expert on set theory (and you know that), > > but model being countable means that there are only countable many > > logical symbols. > > No, you're confusing languages and models in some way. A model of set > theory is a structure <M,E> where E is a binary relation on M. Such a > model is countable if M is a countable set and uncountable otherwise.
If the language is countable, then the model is countable - it has countable many elements. This is just the result. Inside the model there is no bijection between real and naturals, but the bijection exists between the expressions of the language and the elements. Your epsilon- relation stuff doesn`t mean anything outside the model, because it doesn`t say anything about the countable language that we are dealing with.
> -- > Aatu Koskensilta (email@example.com) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus