A note on $\varepsilon$-stability

TL;DR

This paper explores $\varepsilon$-stability in continuous logic, providing new definability results, establishing forking symmetry, and analyzing the stability degree as a seminorm within a first-order framework.

Contribution

It introduces improved definability of types, proves forking symmetry for $\varepsilon$-stability, and characterizes the stability degree as a seminorm in continuous logic.

Findings

Enhanced approximation for definability of types.

Forking symmetry established for $\varepsilon$-stability.

Stability degree forms a seminorm in the theory.

Abstract

We study $\varepsilon$-stability in continuous logic. We first consider stability in a model, where we obtain a definability of types result with a better approximation than that in the literature. We also prove forking symmetry for $\varepsilon$-stability and briefly discuss finitely satisfiable types. We then do a short survey of $\varepsilon$-stability in a theory. Finally, we consider the map that takes each formula to its "degree" of stability in a given theory and show that it is a seminorm. All of this is done in the context of a first-order formalism that allows predicates to take values in arbitrary compact metric spaces.