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.
Categories: math.GN math.LO