On Roitman's principles $\mathsf{MH}$ and $\hat{I}”$

Abstract

The Model Hypothesis (abbreviated $\mathsf{MH}$) and $\hat{I}”$ 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 $\ddot{I}{‰}^{\ast }$ and is therefore not a theorem of $\mathsf{ZFC}$; (2) $\mathsf{MH}$ also fails in the side-by-side Sacks models; (3) as $\hat{I}”$ holds in these models, this implies $\hat{I}”$ is strictly stronger than $\mathsf{MH}$; (4) furthermore, $\hat{I}”$ holds in a large class of forcing extensions in which it was not previously known to hold.

18 pages, 1 diagram