The relation $<$ is not definable in structure $\langle\mathbb R,+,0\rangle$ as a first order formula

Question

the question is:

The relation $<$ is not definable in structure $\langle\mathbb R,+,0\rangle$ with a first order formula.

I asked my professor for help and he told me to consider $\langle\mathbb R,+,0\rangle$ as a vector space and work on its automorphism.

But even now I can't solve this problem...

Can someone please give me some help to understand the clue?

And also why I can't instead prove: the relation $<$ is not definable in the structure $\langle\mathbb N,+,0\rangle$ as a first order formula?

Answer 1

Note that $x\mapsto -x$ is both an automorphism and an order-reversing function.

But I guess your professor instead thought about taking a Hamel basis for $\Bbb R$ over $\Bbb Q$, then any permutation of this basis induces an automorphism of the structure, since it is an automorphism of the abelian group. So by switching just two basis elements, we necessarily don't preserve the usual ordering on $\Bbb R$.

(Or maybe he meant the first suggestion, by wanting you to note that $x\mapsto qx$ is an automorphism when $q\in\Bbb Q\setminus\{0\}$, and then consider $q=-1$, I can't quite know since I'm not your professor...)