Is the theory of the real ordered field along with the rational numbers finitely or at least recursively axiomatizable?

Question

Consider the structure $(\mathbb{R};+,-,*,0,1,<,Q)$, where $Q$ is a unary predicate that picks out the rational numbers. Is the complete theory of that structure finitely axiomatizable, or at least recursively axiomatizable? Also, bonus question, is the complete theory of that structure countably categorical, $2^{\aleph_0}$ categorical, both, or neither?

Answer 1

Suppose for contradiction that $T = \mathrm{Th}(\mathbb{R};+,-,*,0,1,<,Q)$ is recursively axiomatizable. Since $T$ is complete and recursively axiomatizable, it is decidable. But since it interprets $T' = \mathrm{Th}(\mathbb{Q};+,-,*,0,1)$, $T'$ is decidable. This is a contradiction: $T'$ is well-known to be undecidable (e.g. since it interprets $\mathrm{Th}(\mathbb{N};+,*,0,1)$).

There are infinitely many distinct definable elements relative to $T$ (e.g. $1$, $1+1$, $1+1+1$, $\dots$), and it follows immediately from the Ryll-Nardzewski theorem that $T$ is not $\aleph_0$-categorical.

By Morley's theorem that every $\kappa$-categorical theory (for $\kappa$ uncountable) in a countable language is $\omega$-stable, $T$ is not $\kappa$-categorical for any uncountable $\kappa$. Indeed, $T$ is obviously unstable (and hence not $\omega$-stable), because it defines an infinite linear order.