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A note on $\varepsilon$-stability

Nicolas Chavarria

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.

A note on $\varepsilon$-stability

Abstract

We study -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 -stability and briefly discuss finitely satisfiable types. We then do a short survey of -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.

Paper Structure

This paper contains 5 sections, 13 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Continuous Model Theory
  3. Stability in a Model
  4. Stability in a theory
  5. Global stability

Key Result

Proposition 3.3

Suppose $\varphi(x,y)$ is $\varepsilon$-stable in $M$. Fix $\gamma>\delta>0$ and let $p\in S_{\varphi(x)}(M)$. There are $a_0,\ldots,a_{n-1}\in M^x$ such that, if $b,c\in M^y$ satisfy $\mathfrak d(\varphi^M(a_i,b),\varphi^M(a_i,c))<\delta$ for all $i<n$, then $\mathfrak d(\varphi(p,b),\varphi(p,c))<

Theorems & Definitions (32)

  • Definition 3.1
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.6
  • proof
  • Definition 3.7
  • Proposition 3.8
  • proof
  • ...and 22 more