TITLE: Set of formulas that define uncountable subsets of $\mathbb{R}$.
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: Your question is answered in an old paper of Abraham Robinson, Solution of a Problem of Tarski (Fundamenta Mathematicae, 1959). Robinson proved (see Theorem 4.1) that the theory of real closed fields with a predicate $S$ naming a dense proper real closed subfield is complete.
As a consequence (since any subfield of $\mathbb{R}$ contains $\mathbb{Q}$ and hence is dense in $\mathbb{R}$), for any two proper real closed subfields $R_1$ and $R_2$ of $\mathbb{R}$, $(\mathbb{R};+,\times,0,1,<,R_1)$ and $(\mathbb{R};+,\times,0,1,<,R_2)$ are elementarily equivalent. In particular, when $R_1$ is countable and $R_2$ is uncountable, there is no set of first-order sentences that holds in $(\mathbb{R};+,\times,0,1,<,R_2)$ but not in $(\mathbb{R};+,\times,0,1,<,R_1)$.
This result was extended by Angus Macintyre in his PhD thesis "Classifying pairs of real closed fields" in 1967. Among other things, Macintrye showed that the theory of real closed fields with a predicate $S$ naming a real closed subfield (but not necessarily dense) has $2^{\aleph_0}$-many completions.
By the way, the "problem of Tarski" referenced in the title of Robinson's paper was to give a decision procedure for the theory of the structure $(\mathbb{R};+,\times,0,1,<,A)$, where $A$ is the field of real algebraic numbers. Robinson solved this problem by giving a complete recursive axiomatization of the theory of this structure (just the axioms of of real closed fields with a predicate $S$ naming a dense proper real closed subfield).