I have been thinking the following problem proposed by my friends for a long time.

Let $\mathcal{L}$ be the first-order language of theory of rings and let $K$ be the class of algebraic number fields. Consider the set $T_{\exists \forall}(K)$ which collects all the $\mathcal{L}$-sentences $\exists x\forall y \varphi(x,y)$ true in all number fields.($\varphi(x,y)$ is quantifier-free). Is this set axiomatizable? That is, can we find a set of axiom $A\subseteq T_{\exists \forall}(K)$ such that (1) $A$ is recursive and (2) $A\models \exists x\forall y \varphi(x,y)$ for all $\exists x\forall y \varphi(x,y) \in T_{\exists \forall}(K) $?

I have googling for several times but I cannot found any useful methods since

most of the axiomatizable results are about elementary theory of some structures.

I try first to consider the simple case with $T_{\exists \forall}(K)$ contains $\exists \forall$ sentence with single variable $x$ and $y$ but still cannot get any answers.

Any suggestions or comments are welcomed! Thank you

isuniformly r.e. So maybe it is not that relevant. $\endgroup$Model theoretic algbebra. Since each number field is of the form $\mathbb{Q}(\alpha)$, the above sentence is true in all number fields. $\endgroup$1more comment