On perturbations of continuous structures

TL;DR

This paper develops a framework for understanding how continuous structures in logic behave under small perturbations, establishing conditions for approximate isomorphisms and characterizing theories with unique models up to such perturbations.

Contribution

It introduces a general perturbation framework for continuous logic and proves isomorphism results for structures approximately saturated under these perturbations.

Findings

Separable, elementarily equivalent structures are isomorphic up to small perturbations.

Characterization of theories with unique models up to small perturbations.

Framework allows specifying which parts of the logic can be perturbed.

Abstract

We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $\aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations (where the notion of perturbation is part of the data). As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.