Model Theory for Real-valued Structures

TL;DR

This paper extends the framework of continuous model theory to more general structures that lack uniform continuity, introducing a way to incorporate a unique distance predicate and generalizing key model-theoretic results.

Contribution

It develops a method to expand general structures into pre-metric structures with a unique distance predicate, broadening the scope of continuous model theory.

Findings

Every general structure can be expanded to a pre-metric structure.

The distance predicate is unique up to uniform equivalence.

Many model-theoretic results for metric structures extend to general structures.

Abstract

We consider general structures where formulas have truth values in the real unit interval as in continuous model theory, but whose predicates and functions need not be uniformly continuous with respect to a distance predicate. Every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, that distance predicate is unique up to uniform equivalence. We use this to extend the central notions in the model theory of metric structures to general structures, and show that many model-theoretic results from the literature about metric structures have natural analogues for general structures.