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

Question

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.

Answer 1

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.

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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.

Answer 2

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.