Polish G-spaces, the generalized model theory and complexity

TL;DR

This paper develops a generalized model theory for Polish G-spaces, analyzing the Borel and algorithmic complexity of definable sets, and extends previous work with new corrections and insights.

Contribution

It introduces a new framework for analyzing the complexity of sets in Polish G-spaces within a generalized model theory context, improving upon prior results.

Findings

Framework for Borel complexity analysis of definable sets

Extension and correction of previous model theory results

Insights into the structure of Polish G-spaces

Abstract

Given Polish space ${\bf Y}$ and continuous language $L$ we study the corresponding logic $\mathsf{Iso}({\bf Y})$-space ${\bf Y}_L$. We build a framework of generalized model theory towards analysis of Borel/algorithmic complexity of subsets of ${\bf Y}^k_L \times (\mathsf{Iso} ({\bf Y})^l$. The paper substantially develops (and corrects) Section 5 of arxiv:1304.5135. The second version is extended by Remark 3.4.