Computable axiomatizability of elementary classes

The goal of this paper is to generalise Alex Rennet's proof of the non‐axiomatizability of the class of pseudo‐o‐minimal structures. Rennet showed that if is an expansion of the language of ordered fields and is the class of pseudo‐o‐minimal ‐structures (‐structures elementarily equivalent to an ultraproduct of o‐minimal structures) then is not computably axiomatizable. We give a general version of this theorem, and apply it to several classes of structures.