A recursive axiomatization of the theory of real addition and real multiplication

Question

This is a follow-up to my previous question, here: Are the first-order theories of real addition (respectively, real multiplication) finitely axiomatizable?. In that question, I was told that there is no finite axiomatization of either $Th(\mathbb{R};+)$ or $Th(\mathbb{R};*)$, which are the theories of real addition and real multiplication, respectively. However, is there a recursive axiomatization of either or both of those theories? And if so, can someone give an explicit recursive axiomatization of those theories?

Answer 1

This does not seem to be a verbatim homework problem, so I will give you the axiomatisations (though I'll leave the details of the proof to you).

As written in the comments, the former is a (non-trivial) divisible torsion-free abelian group (axiomatised by axioms of abelian group + axiom schema of torsion-free groups + axiom schema of divisible groups). Unless I'm missing something, the latter can be axiomatised as a commutative monoid satisfying the following conditions:

1. There is a zero element, i.e. an $x$ such that for all $y$ we have $xy=x$,
2. The set of nonzero squares forms a non-trivial divisible torsion-free abelian group,
3. Every nonzero element can be expressed uniquely as the product $xy$ where $x^2=1$ and $y$ is a square.
4. There is a unique $x\neq 1$ such that $x^2=1$.

(I believe that uniqueness in 3. actually follows from the other axioms, although it is not completely obvious.)

In particular, in both cases, there are only finely many axioms besides the axiom schemata of divisibility and torsion-freeness.

One way to see this is to observe that $(\mathbf R,\cdot)$ is interpretable from its nonzero squares (again, a nontrivial divisible torsion-free abelian group), which reduces the axioms to those which are sufficient to make the interpretation correct (i.e. isomorphic to the original monoid).