Is there an analog of Tennenbaum's theorem for real arithmetic?

Question

In other words, does the first order theory of $(\mathbb{R},+,\times,0,1)$ have a computable countable model? What do we know more generally about countable models of real arithmetic?

Answer 1

Real arithmetic is complete (trivially, since the theory of any single structure is complete) and decidable (by Tarski). It's a standard result in computability theory that every complete decidable theory has a computable model: basically, just run the usual Henkin argument, and note that the only points of possible non-effectiveness are ruled out by the completeness and decidability assumptions.

In the case of real arithmetic we can do better and get natural computable models, such as the real algebraic numbers as Andreas Blass observed. But the more general fact above is worth knowing separately.