It is a well-known fact that $\mathbb{Z}$ is not definable in the structure $\mathcal{R}=(\mathbb{R}, +, -, < , 0, 1)$. This follows from Tarski's quantifier elimination, and in fact we can conclude that the structure $\mathcal{R}$ is an o-minimal structure.

Another proof, suggested in the answer by Mikhail Katz, is to use the Godel's incompleteness theorem and the fact that the structure is complete.

Question. Is there a more direct proof of the above undefinability result?

I essentially mean a proof which does not use the above results of Tarski or Godel or its variants.

In general, what other different proofs of the above result exist? Providing references is appreciated.