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.