> Gc wrote: > > How can the countable model then satisfy all the theorems about the > > reals? > > Only all first-order logic results. For the least upper bound property > (as commonly stated), you need second-order logic. Hence those results > are not true in the countable model.
Well, they are trivially not true in a FOL:s uncountable model also. And besides, at least almost every problem of analysis can be intepreted in FOL.
> Once you throw in second order > stuff you may no longer have a countable model.