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arXiv · 1504.01180

Decidable models of small theories

Alex Gavryushkin
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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.

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Alex Gavryushkin. 2015-04-06. Decidable models of small theories. https://doi.org/10.1134/s1995080215040198

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