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 theory of 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.

In the paper A dichotomy for expansions of the real field a criteria is given for the undefinability of $\mathbb{Z}$ in expansions of the real field. A natural question is if we can use this criteria and prove the theorem directly?