> On 25 Jul 2006 14:13:23 -0700, Gc <Gcut667@hotmail.com> said: > > Gc kirjoitti: > > > >> Aatu Koskensilta kirjoitti: > >> > Well, there isn't. No countable model contains all the reals. > >> > >> How can the countable model then satisfy all the theorems about the > >> reals? > > > > And especially the theorem "There is uncountably many reals."! > > What that theorem really says is that there is no function from the set > of natural numbers onto the set of real numbers. In a countable model > in which that statement is true, all functions from the set playing the > role of N onto the set playing the role of R have simply been removed. > So, in the model, there is no function from the former onto the latter, > i.e., in the model, the sentence "there are uncountably many reals" is > true.
Let`s call elements playing reals in countable model kvasi-reals. From the axioms of set theory follows that there is no certain bijection, and we get "There are uncountably many kvasi-reals" by Dedekind`s definition and the law of excluded middle. BUT there really are only countably many kvasi-reals.