Decidable models of small theories

TL;DR

This paper investigates conditions under which small theories have decidable models, introduces AL theories to exclude known counterexamples, and identifies properties leading to anomalies in computable models.

Contribution

It defines AL theories, a subclass of small theories, and shows they lack certain counterexamples, advancing understanding of decidability in model theory.

Findings

AL theories do not have Goncharov-Millar counterexamples.

A specific model theoretic property implies the existence of anomalies.

Decidability properties vary among models of small theories.

Abstract

Many counterexamples are known in the class of small theories due to Goncharov and Millar. The prime model of a decidable small theory is not necessarily decidable. The saturated model of a hereditarily decidable small theory is not necessarily decidable. A homogeneous model with uniformly decidable type spectra is not necessarily decidable. In this paper, I consider the questions of what model theoretic properties are sufficient for the existence of such counterexamples. I introduce a subclass of the class of small theories, which I call AL theories, show the absence of Goncharov-Millar counterexamples in this class, and isolate a model theoretic property that implies the existence of such anomalies among computable models.