[pdonis]

> It's also not possible to define the natural numbers using any first order theory.

Yes, agreed.

> you were originally right that no theory of the reals can be satisfied by the rationals, but that's for a fairly unrelated reason.

Can you elaborate?

[Kranar]

At a minimum any theory of real numbers will imply a theorem of the form "There exists an x such that x * x = 2". The set of rational numbers doesn't contain any such x and hence the rational numbers will not satisfy any theory of real numbers.

[zzless]

You are correct, I should have omitted the * from the language.