Are the first-order theories of real addition (respectively, real multiplication) finitely axiomatizable?

Question

I know that the theory of the real field is not finitely axiomatizable. But what about the theories of just addition or just multiplication? That is, is $Th(\mathbb{R};+)$ finitely axiomatizable? And also, is $Th(\mathbb{R};*)$ finitely axiomatizable? If either is finitely axiomatizable, can someone provide a finite set of axioms for the theory?

Answer 1

I'll prove that $Th(\mathbb{R};+)$ is not finitely axiomatizable; a similar argument will work for $Th(\mathbb{R};*)$.

Let $\mathcal{U}$ be a nonprincipal ultrafilter on $\mathbb{N}$, let $p_n$ denote the $n$th prime, and let $C_{p_n}$ be the group with $p_n$ elements. By Los' Theorem, the ultraproduct $$\mathcal{G}:=\prod_{n\in\mathbb{N}}C_{p_n}$$ is a divisible group with no elements of finite order (other than the identity); by a quick counting argument, $\vert\mathcal{G}\vert=2^{\aleph_0}$.

But this pins down $\mathcal{G}$ entirely up to isomorphism: it's a vector space over $\mathbb{Q}$ with dimension continuum. The same is true for $(\mathbb{R};+)$, so we've just "accidentally" built a copy of the reals.

Why is this relevant? Well, suppose $Th(\mathbb{R};+)$ were axiomatized by a single sentence $\varphi$. Then we would obviously have $C_{p_n}\models\neg\varphi$ for each $n$ (since no finite group is elementarily equivalent to the reals), leading to a contradiction with Los' theorem. More generally, no structure elementarily equivalent to a nontrivial ultraproduct of finite structures is finitely axiomatizable.