Undefinability of $\mathbb{Z}$ in the reals 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?
 A: This is not a real answer but rather an observation. The undefinability of $\mathbb{Z}$ follows from the fact that every infinite definable set in such a structure has uncountable cardinality. This property is strictly weaker than both o-minimality and quantifier elimination. Nevertheless, I do not know any proof of this fact that does not use neither of those. I guess this simply induces a nice sub-question of the original one.  
A: The theory of real closed fields is complete and if the integers were definable in $\mathbb R$ this would contradict Goedel's incompleteness result.
A: This is very similar to the answer of Mikhail Katz, but we can avoid the incompleteness theorem by using the halting problem instead. 
That is, since the theory of real-closed fields is computably axiomatizable and complete, it is decidable. So if $\mathbb{Z}$ were definable in $\langle\mathbb{R},+,\cdot,0,1,<\rangle$, then arithmetic truth would be decidable, contradicting the undecidability of the halting problem. 
This argument still relies, however, on Tarski's quantifier-elimination.
