A non-axiomatizable class of structures whose theory is computably infinitely axiomatizable

Question

Does there exist a finite signature $L$ and a class $K$ of $L$-structures which is closed under elementary equivalence, such that $K$ is not first-order axiomatizable, $Th(K)$ is not finitely axiomatizable, but $Th(K)$ is computably axiomatizable? If possible, I would prefer a "familiar" signature $L$, like the signature of groups, or rings, or partial orders.

Answer 1

Yes, this can happen.

Let $T$ be any consistent computably axiomatizable non-finitely-axiomatizable deductively closed theory such that no finite extension $T\cup\{\varphi\}$ is consistent and complete; for example, we can take $T$ to be Robinson arithmetic. Now let $\mathbb{S}$ be a "dense codense" set of completions of $T$; that is, for each sentence $\varphi$ consistent with $T$ there should be complete consistent theories $A,B\supseteq T\cup\{\varphi\}$ with $A\in\mathbb{S}$ and $B\not\in\mathbb{S}$.

If we let $K$ be the class of models of elements of $\mathbb{S}$, then $K$ has the desired properties; in particular, $Th(K)=T$ since $\mathbb{S}$ is dense, but $K$ is not first-order axiomatizable since for any $T'\supsetneq T$ letting $\varphi\in T'\setminus T$ there are structures in $K$ not satisfying $T$.