Approximate Isomorphism of Metric Structures

TL;DR

This paper introduces a unified formalism for approximate isomorphism in continuous logic, generalizing previous approaches and providing explicit Scott sentences for various metric perturbation systems.

Contribution

It presents a new formalism for approximate isomorphism that unifies prior incompatible approaches and characterizes it both syntactically and semantically.

Findings

Explicit Scott sentences for perturbation systems

Unified formalism for approximate isomorphism

Stability of theories of finite-radius $\ ext{R}$-trees and ultrametric spaces

Abstract

We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov and by Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any $\mathbb{R}$-tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson.