Testing definitional equivalence of theories via automorphism groups

TL;DR

This paper characterizes when two first-order theories are definitionally equivalent by examining automorphism groups and model category isomorphisms, settling several longstanding conjectures in the field.

Contribution

It establishes a new criterion for definitional equivalence using automorphism groups and model categories, extending prior theorems and resolving open conjectures.

Findings

Definitionally equivalent theories correspond to isomorphic automorphism groups.

Uncountably many inequivalent theories can have isomorphic model categories.

The results settle several conjectures by Barrett, Glymour, and Halvorson.

Abstract

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson.