Continuous Logic and Borel Equivalence Relations

Journal of Symbolic Logic 88 (4):1725-1752 (2023)
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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.

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Citations of this work

Isomorphism of Locally Compact Polish Metric Structures.Maciej Malicki - 2024 - Journal of Symbolic Logic 89 (2):646-664.

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References found in this work

Model theoretic forcing in analysis.Itaï Ben Yaacov & José Iovino - 2009 - Annals of Pure and Applied Logic 158 (3):163-174.
Omitting types for infinitary [ 0, 1 ] -valued logic.Christopher J. Eagle - 2014 - Annals of Pure and Applied Logic 165 (3):913-932.

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