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.