Combining Finite Combination Properties: Finite Models and Busy Beavers

TL;DR

This paper explores the combination of finite model properties and stable finiteness in theories, providing examples using the Busy Beaver function to understand their relationships and existence.

Contribution

It introduces the construction of theories with combined properties, including those with finite model property but not finitely witnessable, using the Busy Beaver function.

Findings

Existence of theories with combined properties demonstrated.

Examples provided with the simplest signatures.

Utilization of the Busy Beaver function for theory construction.

Abstract

This work is a part of an ongoing effort to understand the relationships between properties used in theory combination. We here focus on including two properties that are related to shiny theories: the finite model property and stable finiteness. For any combination of properties, we consider the question of whether there exists a theory that exhibits it. When there is, we provide an example with the simplest possible signature. One particular class of interest includes theories with the finite model property that are not finitely witnessable. To construct such theories, we utilize the Busy Beaver function.