[Kranar]

>Clearly the real numbers are a model for these axioms. But as it turns out the countable set of rational numbers is a model as well.

This wouldn't be correct, it's never the case that the set of rational numbers can satisfy a theory of real numbers, it's more subtle than that.

It's that for any theory of the real numbers, there exist subsets of the real numbers that are countable that satisfy that theory. For example the subset of all computable real numbers will satisfy any theory of real numbers despite it being countable. It's simply not possible to define a first order theory that describes the real numbers as a whole and only the real numbers as a whole.

However, there will never be any theory of real numbers that can be satisfied by the set of rational numbers. At a minimum any theory of real numbers would imply theorems that require the existence of a number that when squared was equal to 2. The rational numbers can not satisfy such a theorem.