> Gc wrote: > > Aatu Koskensilta kirjoitti: > > > >> What, if anything, do you mean by there being "a complete set of reals > >> in a countable model"? > > > > It`s universe includes all the reals. > > In that case you're just mistaken. There is no countable model that > includes all reals.
"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. 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