Computable axiomatizability of elementary classes (Q2793906)

An elementary class is \textit{computably axiomatizable} if there is a computable axiomatization of its theory. A structure is \textit{pseudo-o-minimal} if it is elementarily equivalent to an ultraproduct of o-minimal structures. \textit{A. Rennet} proved in [J. Symb. Log. 79, No. 1, 54--59 (2014; Zbl 1338.03071)] that the class of pseudo-o-minimal structures is not computably axiomatizable. Sinclair adopts Rennet's argument to prove a general theorem with several applications to other classes, in particular to C-minimal structures defined in [\textit{D. Haskell} and \textit{D. Macpherson}, Ann. Pure Appl. Logic 66, No. 2, 113--162 (1994; Zbl 0790.03039)] and P-minimal structures defined in [\textit{H. Schoutens}, J. Symb. Log. 79, No. 2, 355--409 (2014; Zbl 1337.03055)]. Other examples include topologically totally transcendental first order topological structures of \textit{A. Pillay} [J. Symb. Log. 52, 763--778 (1987; Zbl 0628.03022)], and Zariski structures of \textit{E. Hrushovski} and \textit{B. Zilber} [J. Am. Math. Soc. 9, No. 1, 1--56 (1996; Zbl 0843.03020)].