logic – Set of formulas that define uncountable subsets of $mathbb{R}$.

This is a follow-up to my previous question about a formula in the language of ordered fields that can distinguish infinite subsets of $mathbb{R}$. My current question is this. Consider the structure $(mathbb{R};+,*,0,1,<)$. We adjoin to it a new unary predicate $S$, that picks out a certain subset of the reals. Is there a single formula, or if not, an infinite set of formulas, in the expanded language that holds precisely when $S$ is an uncountable subset of the reals? I think there is not, but I would like a proof.