TITLE: Is there a theory in a finite language that is computably axiomatizable but not by a finite number of axiom schemas?
QUESTION [9 upvotes]: I was told to ask this question on mathoverflow. I asked on math stack exchange whether there is a computably axiomatizable theory that can't be axiomatized by a finite number of axiom schemas. I got an answer, but it was a theory in an infinite language. Now, I am asking whether there is an example in a finite language.

EDIT by non-OP: this is the above-mentioned MSE question, and this answer gives the definition of "scheme" being used.

REPLY [8 votes]: This answers complements Fedor Pakhamov's, who provided an example of a computable theory that is not axiomatizable by finitely many schemas.
Following up on the comment by Andreas Blass to the question: Vaught proved that if a theory $T$ is computable and has "a modicum of coding", then $T$ is axiomatizable by a scheme. Vaught's result was improved by Albert Visser, in the paper below, where "the modicum of coding" used by Vaught is reduced to the modest demand that $T$ interprets a non-surjective unordered pairing, where pairing need not be functional.
A. Visser, Vaught's theorem on axiomatizability by a scheme, The Bulletin of Symbolic Logic, vol. 18 (2012), pp. 382-402.
A preprint of Visser's paper can be found here.