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CONTINUOUS LOGIC AND BOREL EQUIVALENCE RELATIONS

Part of: Set theory Model theory

Published online by Cambridge University Press:  22 June 2022

ANDREAS HALLBÄCK ,
MACIEJ MALICKI  and
TODOR TSANKOV
ANDREAS HALLBÄCK
Affiliation:
INSTITUT DE MATHÉMATIQUES DE JUSSIEU–PRG UNIVERSITÉ PARIS CITÉ 75205 PARIS CEDEX 13, FRANCE E-mail: superand007@hotmail.com
MACIEJ MALICKI
Affiliation:
INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES UL. SNIADECKICH 8, 00-656 WARSAW, POLAND E-mail: mamalicki@gmail.com
TODOR TSANKOV*
Affiliation:
INSTITUT CAMILLE JORDAN UNIVERSITÉ CLAUDE BERNARD LYON 1 43 BOULEVARD DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX, FRANCE and INSTITUT UNIVERSITAIRE DE FRANCE PARIS, FRANCE
*
E-mail: tsankov@math.univ-lyon1.fr
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Abstract

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

Keywords

Borel equivalence relationsinfinitary continuous logiclocally compact structures

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03C75: Other infinitary logic
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 4 , December 2023 , pp. 1725 - 1752
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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