On the first Banach problem, concerning condensations of absolute $\kappa$-Borel sets onto compacta

TL;DR

This paper explores the conditions under which certain large, complex sets called absolute ppa-Borel sets can be condensed onto compact metric spaces, revealing consistency results related to set-theoretic assumptions and model theory.

Contribution

It establishes new consistency results about the condensation of absolute ppa-Borel sets onto compacta, depending on set-theoretic and model-theoretic conditions.

Findings

Consistency that no absolute ppa-Borel set condenses onto a compactum when the continuum is large.

Consistency that certain absolute ppa-Borel sets containing specific subspaces do condense onto compacta.

Existence of forcing extensions where condensation properties depend on a subset of natural numbers.

Abstract

It is consistent that the continuum be arbitrary large and no absolute $\kappa$-Borel set $X$ of density $\kappa$, $\aleph_1<\kappa<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large and any absolute $\kappa$-Borel set $X$ of density $\kappa$, $\kappa\leq\mathfrak{c}$, containing a closed subspace of the Baire space of weight $\kappa$, condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any $A\subseteq \mathbb{N}$ with $1\in A$, there is a forcing extension in which every absolute $\aleph_n$-Borel set, containing a closed subspace of the Baire space of weight $\aleph_n$, condenses onto a compactum if, and only if, $n\in A$.