A note on $\varepsilon$-stability

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.