# Are the field reducts of real closed fields first-order axiomatizable?

We know that the class of real closed fields $$(F,+,-,*,0,1,<)$$ is first-order axiomatizable. Is the $$\{+,-,*,0,1\}$$ class of reducts of real closed fields axiomatizable? This is different from my previous question on the axiomatizability of ordered fields, because I am asking about real closed fields, rather than merely ordered fields.

## 2 Answers

The ordering on a real closed field is determined by the field structure alone: we set $$a\le b$$ iff the equation $$a+x^2=b$$ has a solution. This automatically gives axiomatizability, since it says that the reducts of real closed fields are exactly the fields in which the relation $$R(a,b):\equiv\exists x(a+x^2=b)$$ satisfies a particular property.

More generally, suppose we have languages $$\Sigma_0\subseteq\Sigma$$, an axiomatizable class $$\mathfrak{K}$$ of $$\Sigma$$-structures, and for each symbol $$R\in\Sigma\setminus\Sigma_0$$ a $$\Sigma_0$$-formula $$\varphi_R$$ of appropriate arity such that for each $$\mathcal{A}\in\mathfrak{K}$$, the structure $$\mathcal{A}'$$ gotten by first taking the reduct of $$\mathcal{A}$$ to $$\Sigma_0$$ and then expanding it to $$\Sigma$$ by interpreting $$R\in\Sigma\setminus R_0$$ as $$\varphi_R$$ is also in $$\mathfrak{K}$$. Then the set of reducts to $$\Sigma_0$$ of elements of $$\mathfrak{K}$$ is again axiomatizable: letting $$\Theta$$ axiomatize $$\mathfrak{K}$$, consider the $$\Sigma_0$$-theory $$\Theta'$$ gotten by replacing each $$R\in\Sigma\setminus\Sigma_0$$ with $$\varphi_R$$ in every sentence in $$\Theta$$.

In the case of real closed fields things are particularly nice: real closed fields are uniquely orderable. But this uniqueness (that is, $$\mathcal{A}=\mathcal{A}'$$) isn't a necessary condition for the argument above: for example, let $$\Sigma=\{U\}$$ with $$U$$ a unary relation symbol, $$\Sigma_0=\emptyset$$, and $$\mathfrak{K}$$ be the class of $$\Sigma$$-structures where $$U$$ holds of at most one element. While we cannot recover $$\mathcal{A}\in\mathfrak{K}$$ from its underlying set (= $$\mathcal{A}\upharpoonright\Sigma_0$$), the formula $$\varphi_U:\equiv x\not=x$$ satisfies the condition of the above paragraph.

The $$(+, -, \cdot, 0, 1)$$ reducts of real closed fields are precisely those in which the relation defined by $$x < y$$ iff $$\exists z \neq 0(y = x + z^2)$$ gives a structure for the signature $$(+, -, \cdot, 0, 1, <)$$ that satisfies the axioms of a real closed field. This gives the axiomatization that you are looking for.

• Not finite axiomatizability, only infinite axiomatizability. The axioms for real closed fields require an infinite set of axioms. – user107952 Mar 2 at 0:32
• I really don't know how "finite" crept in there. I've deleted the offending word. – Rob Arthan Mar 2 at 0:33