Continuous logic and Borel equivalence relations

TL;DR

This paper explores the complexity of isomorphism relations in metric structures using continuous logic, establishing conditions under which these relations are essentially countable, and providing new proofs for existing theorems.

Contribution

It introduces a model-theoretic criterion for classifying the complexity of isomorphism relations in Borel classes of metric structures, generalizing previous results.

Findings

If the isomorphism relation is potentially Σ^0_2, then it is essentially countable.

Provides an easy-to-check model-theoretic condition for classifying complexity.

Offers a new proof that orbit equivalence relations of Polish locally compact group actions are essentially countable.

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.