Are infinite subsets of the real field definable by a single formula?

Question

Consider the structure $(\mathbb{R},+,*,0,1,<)$. We adjoin to it a subset $S$ of $\mathbb{R}$. Is it possible to give a single formula $F$ in the expanded language such that $F$ is true precisely when $S$ is an infinite subset of the reals? I know it is certainly possible by replacing $\mathbb{R}$ with $\mathbb{N}$ or by $\mathbb{Z}$. We can just say that $S$ has no upper bound, in the case of $\mathbb{N}$, or that $S$ has either no upper bound or no lower bound, in the case of $\mathbb{Z}$.

Answer 1

Yes. $S$ is infinite if and only if either $S$ is unbounded or $S$ contains pairs of elements arbitrarily close together: $\forall {\epsilon>0}\,\exists x\in S\,\exists y\in S(0< (x-y)^2<\epsilon)$.