Isomorphism between direct sum of $\mathbb{Z}$ and $\mathbb{Z}$ Given the language $\{+,0\}$, I would like to prove that $\mathbb{Z} \oplus \mathbb{Z}$ is not isomorphic to $\mathbb{Z}$, but I am not sure where to start?

HINT: Let $\varphi(x)$ be the formula $\exists y\,(y+y=x)$, and consider the sentence
$$\forall x\forall y\Big(\neg\varphi(x)\land\neg\varphi(y)\to\varphi(x+y)\Big)\;.$$
