Unstable classes of metric structures

TL;DR

This paper establishes a non-structure theorem for certain unstable metric structures and explores how weak categoricity implies various stability notions, advancing understanding of metric class stability.

Contribution

It introduces a non-structure theorem for unstable metric classes and links weak categoricity to stability properties in metric structures.

Findings

Weak categoricity implies multiple stability variants.

Unstable pairs of formulae lead to non-structure results.

First step in studying weak categoricity and stability in metric classes.

Abstract

We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several versions of stability. This is the first step in the direction of the investigation of weak categoricity and weak stability of metric classes.