Prove $\{1\}$ cannot be defined by any formula in structure $(R, +)$

Question

Consider the structure structure $(R, +)$ of real numbers for the language consisting of $∀, +$ (no multiplication nor constants). The addition operation is the usual operations. Prove that the set $\{1\}$ cannot be defined by any formula. Could someone please give me some hints? Much appreciated. Thanks.

Answer 1

Use the fact that $x\mapsto 2x$ is an isomorphism of structures to prove that if a formula defined $\{1\}$ then it would also define $\{2\}$.

Answer 2

I think that you can try to prove by induction on formula structure that the only definable sets are {}, {$0$} , {$x : x != 0$} or $R$.