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TESTING DEFINITIONAL EQUIVALENCE OF THEORIES VIA AUTOMORPHISM GROUPS

Part of: General logic Philosophical aspects of logic and foundations Algebraic structures Model theory

Published online by Cambridge University Press:  17 July 2023

HAJNAL ANDRÉKA Open the ORCID record for HAJNAL ANDRÉKA [Opens in a new window] ,
JUDIT MADARÁSZ ,
ISTVÁN NÉMETI  and
GERGELY SZÉKELY
HAJNAL ANDRÉKA*
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA STREET 13–15 H-1053 BUDAPEST, HUNGARY E-mail: madarasz.judit@renyi.hu E-mail: nemeti.istvan@renyi.hu
JUDIT MADARÁSZ
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA STREET 13–15 H-1053 BUDAPEST, HUNGARY E-mail: madarasz.judit@renyi.hu E-mail: nemeti.istvan@renyi.hu
ISTVÁN NÉMETI
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA STREET 13–15 H-1053 BUDAPEST, HUNGARY E-mail: madarasz.judit@renyi.hu E-mail: nemeti.istvan@renyi.hu
GERGELY SZÉKELY
Affiliation:
UNIVERSITY OF PUBLIC SERVICE 2 LUDOVIKA SQUARE H-1053 BUDAPEST, HUNGARY E-mail: szekely.gergely@renyi.hu
*
E-mail: andreka.hajnal@renyi.hu
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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.

Keywords

definitional equivalence of theoriesfirst-order logicautomorphism groupultraproductcategorical equivalence of theoriesmany-dimensional definitional equivalenceMorita equivalencecategory of models

MSC classification

Primary: 03A10: Logic in the philosophy of science 03B10: Classical first-order logic 03C20: Ultraproducts and related constructions 03C40: Interpolation, preservation, definability 08A35: Automorphisms, endomorphisms
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 22
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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