Borel Reductions and Cub Games in Generalized Descriptive Set Theory

TL;DR

This paper explores the embedding of complex set-theoretic structures into Borel equivalence relations on generalized spaces, revealing deep connections between set theory and descriptive set theory for uncountable cardinals.

Contribution

It demonstrates that the power set of an uncountable regular cardinal can be embedded into the Borel reducibility order of equivalence relations, extending classical results to the uncountable setting.

Findings

Embedding of the power set of $ppa$ into Borel equivalence relations

Connections established between non-stationary ideals and Borel reducibility

Extension of classical descriptive set theory results to uncountable cardinals

Abstract

It is shown that the power set of $\kappa$ ordered by the subset relation modulo various versions of the non-stationary deal can be embedded into the partial order of Borel equivalence relations on $2^\kappa$ under Borel reducibility. Here $\kappa$ is uncountable regular cardinal with $\kappa^{<\kappa} = \kappa$.