Metric Scott analysis

TL;DR

This paper extends Scott analysis to metric structures and continuous logic, establishing Scott sentences, a Lopez-Escobar theorem, and analyzing Borel complexity of isomorphism relations, with applications to Gromov-Hausdorff and Kadets distances.

Contribution

It introduces a metric Scott analysis framework, proving the existence of Scott sentences and a Lopez-Escobar theorem for metric structures, and explores descriptive set theoretic implications.

Findings

Isomorphism on separable structures is Borel iff Scott rank is bounded.

Set of spaces with zero Gromov-Hausdorff distance is Borel.

Scott analysis provides tools for metric structure classification.

Abstract

We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the Lopez-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below $\omega_1$. Finally, we apply our methods to study the Gromov-Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance $0$ to a fixed space is a Borel set.