On Borel subsets of generalized Baire spaces

TL;DR

This paper extends Descriptive Set Theory to generalized Baire spaces without assuming a key set-theoretic condition, exploring how topological and model-theoretic properties influence the Borel complexity of model orbits.

Contribution

It develops a modified Descriptive Set Theory framework for generalized Baire spaces without the assumption ^{<\u1e7b}=, and applies it to analyze the Borel complexity of model orbits based on stability theory.

Findings

The basic topological concepts require modification without ^{<7b}=.

The Borel complexity of model orbits depends on the stability properties of the underlying theory.

The theory aligns with classical results when ^{<7b}=.

Abstract

We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $\kappa^{<\kappa}=\kappa$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to obtain a meaningful theory. This modification has no effect if $\kappa^{<\kappa}=\kappa$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $\kappa$ in the space $\kappa^\kappa$ are $\kappa$-Borel in our generalized sense. It turns out that this question depends, as is the case when $\kappa^{<\kappa}=\kappa$, on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.