On Borel subsets of generalized Baire spaces
Tapani Hyttinen, Miguel Moreno, Jouko V\"a\"an\"anen

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 . 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 . After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality in the space are -Borel in our generalized sense. It turns out that this question depends, as is the case when , on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Homotopy and Cohomology in Algebraic Topology
