[Kranar]

You're right that it is not possible to define the reals using the language he started with, but it's worse than that. It's also not possible to define the natural numbers using any first order theory. There is no way to extend a first order theory so that it defines the natural numbers and only the natural numbers and furthermore there is no way to define a first order theory that defines the reals and only the reals.

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

[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.