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.