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

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 $$\mathbb{Z}$$.

• To clarify, you're adding a unary predicate symbol to the language? – Reveillark Jun 1 at 22:20

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