The Borel partition spectrum at successors of singular cardinals

TL;DR

This paper explores the structure of Borel partitions of the real line at certain large cardinal levels, showing that under specific set-theoretic assumptions, the number of Borel sets in a partition can be increased.

Contribution

It establishes a new consistency result relating Borel partitions at different levels of the cardinal hierarchy under the assumption that $0^\

Findings

If there is a partition of $\\mathbb R$ into $\aleph_\omega$ Borel sets, then there is also one into $\aleph_{\omega+1}$ Borel sets.

The result depends on the non-existence of $0^\dagger$, linking descriptive set theory with large cardinal hypotheses.

Abstract

Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_\omega$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{\omega+1}$ Borel sets.