Generalized Descriptive Set Theory and Classification Theory

TL;DR

This paper extends descriptive set theory to uncountable spaces, revealing new properties and connections with model theory, and proposes Borel reducibility as a natural measure of complexity for uncountable structures.

Contribution

It generalizes classical descriptive set theory to uncountable spaces and links the complexity of isomorphism relations to stability theory.

Findings

Descriptive set theory behaves differently in uncountable settings.

Borel reducibility effectively compares complexities of uncountable isomorphism relations.

Connections established between model-theoretic stability and descriptive set theoretic complexity.

Abstract

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.