> Gc wrote: > > We know that every set of reals is in some sense countable according to > > Skolem`s paradox, I mean that there is a bijection between the real and > > naturals if naturals are outside of model of set theory. > > We know no such thing. By the Löwenheim-Skolem theorem any model of set > theory has a countable elementary substructure. How do you get from this > that "every set of reals is in some sense countable"?
Mathematical logic, Shoenfield, page 79, section 5.4: "We can certainly formalize enough mathematics in a countable theory to prove that the set of real numbers in uncountable. How can such a theory have a countable model? The explanation is this. 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. But this mapping is not in the model; so it doesn`t make invalid the theorem of the theory which states that there is no bijective mapping from the set of real numbers to the set of natural numbers."
> > Still Cantor`s > > theorem in set theory, which is of course valid, says that the reals > > are uncountable. To me it seems that there is a theorem in set theory > > that can`t be true in sense I understand you see it. You have said > > before that the axioms of set theory are true. How can you explain this > > - or am I missing something? > > 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.
> -- > Aatu Koskensilta (firstname.lastname@example.org) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus