Definition or Axiom for less than for real numbers I was curious as to what the definition for $<$ would be concerning real numbers. I have looked at Peanos axioms which discusses inequalities for every $a, b\in\mathbb{N}$, $a<b$ iff there is some $c\in \mathbb{N}$ such that $a+c=b$. I am curious as to what it would be for real numbers. I could try to guess what it might be, but I am wanting to be exact. Any help or resources would be appreciated.

One can define it as follows: $x\leq  y \iff (\exists z)(z^2 = y-x)$
Since someone voted to delete my answer, but did not leave a comment as to why this answer is incorrect, I will explain my answer more thoroughly. 
My answer consider's the reals as a field in the ring language $\mathcal{L} =\{1,0,\times,+, -\}$. The reals as a field are naturally a structure in this language. My answer shows that $\leq$ is definable (in the model theory sense) in this structure in this particular language. 
