Can there be an $S \subseteq \mathbb{R}$ closed under multiplication and addition with $|\mathbb{Q}| < |S| < |\mathbb{R}|$?

Question

In $ZFC+\lnot CH$, is the statement that there is an $S \subseteq \mathbb{R}$ closed under multiplication and addition with $|\mathbb{Q}| < |S| < |\mathbb{R}|$ true, false, or independent?

Answer 1

Yes. Take $S$ which is a counterexample to the Continuum Hypothesis, and consider the field generated by $S$ given by Löwenheim–Skolem theorem. It has the same cardinality as $S$, say $\aleph_1$, and it is a field. So it is closed under addition and multiplication.