Testing definitional equivalence of theories via automorphism groups

Judit Madarász, István Németi, Gergely Székely, Hajnal Andréka

Publication date: 25 November 2022

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.

Mathematics Subject Classification ID

Classical first-order logic (03B10) Automorphisms and endomorphisms of algebraic structures (08A35) Interpolation, preservation, definability (03C40) Ultraproducts and related constructions (03C20) Logic in the philosophy of science (03A10) Categories and theories (18Cxx)

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