Computable Axiomatizability of Elementary Classes

TL;DR

This paper generalizes Rennet's proof showing that certain classes of topological structures, including pseudo-o-minimal structures, cannot be fully captured by computable axioms, extending the non-axiomatizability result.

Contribution

It provides a broad generalization of the non-axiomatizability of pseudo-o-minimal structures and applies this to various classes of topological structures.

Findings

Pseudo-o-minimal structures are not computably axiomatizable.

The general theorem applies to multiple classes of topological structures.

Extends previous non-axiomatizability results to broader contexts.

Abstract

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 L is an expansion of the language of ordered fields and K is the class of pseudo-o-minimal L-structures (L-structures elementarily equivalent to an ultraproduct of o-minimal structures) then K is not computably axiomatizable. We give a general version of this theorem, and apply it to several classes of topological structures.