Yog79

Since $\mathbb{R}$ is not first order definable, the statement $\forall x \in \mathbb{R}(x = x)$ is not a first order statement and thus not provable in ZFC. Does that mean we can assume $\exists x \in \mathbb{R}(x \neq x)$? If so, would this provide us with the basis for a field with one element?