> Gc wrote: > > Aatu Koskensilta kirjoitti: > > > >> Countable models of set theory are entirely irrelevant to the > >> uncountability of the reals. > > > > I don`t understand how, if what Shoenfield says is true. > > In the passage you quoted Shoenfield explains how a first order theory > in which the formalization of "there is an uncountable set" is provable > can have countable models. From this it does not follow that "every set > of reals is in some sense countable".
I understand that there is a complete set of reals in a countable model. The logic is FOL which is suffcient for set theory. Shoenfield says: "The set of real numbers in the model is indeed countable, and therefore there is a bijective mapping from it to the set of natural numbers." It`s obviously not about formalization of "there is an countable set" in set theory in a sense that the countable model doens`t include all the reals.
> -- > Aatu Koskensilta (email@example.com) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus