The $<$-relation on $\mathbb{Z}$ is not definable in $(\mathbb{Z}, 0, +)$

Question

I am a complete newcomer to logic and I'm having trouble proving the following:

The $<$-relation on $\mathbb{Z}$ is not definable in $(\mathbb{Z}, 0, +)$.

Now, I know that the $<$-relation on $\mathbb{N}$ is definable in the structure $(\mathbb{N}, 0, +)$ by the formula $$\phi(x, y) := \exists z (z \neq 0 \land z + x = y).$$ My idea was to show that if $<$ is definable in $(\mathbb{Z}, 0, +)$ then the defining formula would have to correspond to the above $\phi$ on $\mathbb{Z}^{\geq0}$ and then derive some contradiction. However, I have never done a proof of non-definability before and don't really know how one generally goes about such proofs. It is always advisable to do a proof by contradiction? And if so, is there any "standard type" of contradiction one generally looks for in such cases?

Any help is greatly appreciated!

Answer 1

The situation is invariant under negation. It follows that you can't distinguish $\lt$ from $\gt$.

Answer 2

It is easy to check that $h:\mathbb{Z}\to\mathbb{Z}$ via $x\mapsto -x$ is an automorphism and $\mathbb{Z}^+$, the set of positive integers, is not closed under this automorphism. Hence $\mathbb{Z}^+$ is not definable in $(\mathbb{Z},0,+)$. Note that the definability of $<$-relation is the same as $\mathbb{Z}^+$.