Games and Scott sentences for positive distances between metric structures

TL;DR

This paper introduces Ehrenfeucht-Fra"{sse9} games for measuring distances between metric structures, utilizing positive bounded logic and Scott sentences to characterize fixed distances and approximate isomorphisms.

Contribution

It develops new game-based and logical tools for analyzing distances between metric structures, extending Scott sentences to positive distances and infinitary logic.

Findings

Scott sentences characterize 0-distances in _1 logic.

Scott sentences for positive distances are in _2 logic.

Ehrenfeucht-Fra"{sse9} games effectively measure distances between metric structures.

Abstract

We develop various Ehrenfeucht-Fra\"{\i}ss\'{e} games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{\omega_1\omega}$ for 0-distances and in $\mathcal{L}_{\omega_2\omega}$ for positive distances.