The generalized continuous model theory, Borel complexity and stability

TL;DR

This paper develops a generalized model theory framework to analyze Borel complexity and stability properties of families of subsets in Effros spaces within continuous logic.

Contribution

It introduces a new framework connecting continuous model theory with Borel complexity analysis, focusing on stability in Effros spaces.

Findings

Framework for analyzing Borel complexity of model-theoretic properties

Application of the framework to stability analysis in Effros spaces

Enhanced understanding of the descriptive set-theoretic complexity in continuous logic

Abstract

Given Polish space $\mathcal{Y}$ and a continuous language $L$ we study the corresponding logic $\mathsf{Iso}(\mathcal{Y})$-space $\mathcal{Y}_L$. We build a framework of generalized model theory towards analysis of Borel complexity of families of subsets of Effros spaces $\mathcal{F}(\mathcal{Y})^k_L\times \mathcal{F}(\mathsf{Iso} (\mathcal{Y}))^l$ corresponding to standard model-theoretic properties. In this paper we mainly apply this approach to stability.