Gc wrote: > Aatu Koskensilta kirjoitti: > >> In that case you're just mistaken. There is no countable model that >> includes all reals. > > http://help.com/wiki/944267/skolems-paradox/ > > "Using the Löwenheim-Skolem Theorem, we can get a model of set theory > which only contains a countable number of objects. However, it must > contain the fore-mentioned uncountable sets, which appears to be a > contradiction. However, the sets in question are only uncountable in > the sense that there does not exist within the model a bijection from > the natural numbers onto the sets. It is entirely possible that there > is a bijection outside the model." > > The reals are uncountable inside the model, but not outside, just like > the Shoenfield says.
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).
> And the quote from the Shoenfields book that I posted earlier doesn`t > make sense if it the countable model doens`t include all the reals, > btw.
-- Aatu Koskensilta (email@example.com)
"Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus