On Roitman's principles $\mathsf{MH}$ and $\Delta$

TL;DR

This paper investigates the set-theoretic principles $ ext{MH}$ and $ ext{Delta}$ introduced by Roitman, establishing their logical strength, independence from ZFC, and behavior in various models and forcing extensions.

Contribution

It demonstrates that $ ext{MH}$ implies the existence of $P$-points, is not provable in ZFC, and compares its strength to $ ext{Delta}$ across different models and extensions.

Findings

$ ext{MH}$ implies the existence of $P$-points in $eta abla$.

$ ext{MH}$ fails in side-by-side Sacks models.

$ ext{Delta}$ is strictly stronger than $ ext{MH}$ in these models.

Abstract

The Model Hypothesis (abbreviated $\mathsf{MH}$) and $\Delta$ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1) $\mathsf{MH}$ implies the existence of $P$-points in $\omega^*$ and is therefore not a theorem of $\mathsf{ZFC}$; (2) $\mathsf{MH}$ also fails in the side-by-side Sacks models; (3) as $\Delta$ holds in these models, this implies $\Delta$ is strictly stronger than $\mathsf{MH}$; (4) furthermore, $\Delta$ holds in a large class of forcing extensions in which it was not previously known to hold.