On Borel subsets of generalized Baire spaces
Tapani Hyttinen, Miguel Moreno, Jouko Väänänen
TL;DR
This work extends Descriptive Set Theory to generalized Baire spaces $\kappa^\kappa$ without assuming $\kappa^{<\kappa}=\kappa$, replacing the bounded topology with a product-topology-inspired framework and focusing on unions of at most $\kappa$ basic opens. It develops a robust theory of $\kappa$-Borel and $\kappa$-Borel$^*$ sets, introduces $\kappa$-$\Sigma^1_1$-definability, and proves a Hierarchy Theorem for these notions. In model-theoretic applications, it shows that in the non-structure case (countable non-classifiable theories) some $\kappa$-sized model orbits and the isomorphism relation are not $\kappa$-Borel, while in the structure case (under regular $\kappa$ with $\kappa^\omega=\kappa$) certain classes yield $\kappa$-Borel orbits and, under quantitative bounds on model numbers, a $\kappa$-Borel isomorphism relation. The results thus generalize and refine prior work by removing cardinal arithmetic assumptions and connecting definability to stability-theoretic properties, with implications for the complexity of orbit equivalence in uncountable model theory. Overall, the paper provides a detailed framework for definability and complexity of orbit types in generalized Baire spaces and highlights sharp dichotomies between stability regimes.
Abstract
We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $κ^{<κ}=κ$. 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 $κ^{<κ}=κ$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $κ$ in the space $κ^κ$ are $κ$-Borel in our generalized sense. It turns out that this question depends, as is the case when $κ^{<κ}=κ$, on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.