I'm having trouble with the following exercise from the book Mathematical Logic by H.D. Ebbinghaus, J. Flum, and W. Thomas.
Show:
(a) The relation $<$ ("less-than") is elementarily definable in $(\mathbb{R},+,\cdot ,0)$, i.e, there is a formula $\varphi \in L_2^{(+,\cdot ,0)}$ such that for all $a,b\in \mathbb{R}$, $(\mathbb{R},+,\cdot ,0)\vDash \varphi [a,b]$ iff $a.
(b) The relation $<$ is not elementarily definable in $(\mathbb{R},+,0)$. (Hint: Work with a suitable automorphism of $(\mathbb{R},+,0)$, i.e, a suitable isomorphism of $(\mathbb{R},+,0)$ onto itself.)
It's the first time I see the term "elementarily definable". Still I was able to solve (a):
$\varphi \text{:=}\exists _x\left(x\neq 0\land v_0+x^2=v_1\right)$
But I have not been able to solve (b). Thanks.