Let $A$ be the set of all $q$ in $\mathbb{Q}$ such that $q\leq0$ or $1\leq q$, and let $\mathcal{A}=(A,<)$ be a structure.
I have to show that 2 is not a definable element of this structure, e.g. there doesn't exist a formula $\varphi = \varphi(x)$ such that $\mathcal{A}\models\varphi[q]$ if and only if $q=2$.
I've been trying to use induction, but it doesn't seem to work..
EDIT: the first part of the exercise is: Prove that for all formulas $\varphi=\varphi(x), (A,<)\models \varphi[2]$ iff $(A,<)\models \varphi[3]$, but how would I prove such a statement?
HINT: If $x$ is a definable element in a structure $\mathcal M$, then any automorphism of $\cal M$ must satisfy $f(x)=x$. To show that $2$ is not definable, find an automorphism of $\cal A$ such that $2\neq f(2)$.