Two Upper Bounds on Consistency Strength of ¬□ℵω and Stationary Set Reflection at Two Successive ℵn

Abstract

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a ${\kappa }^{+}$-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ${\aleph }_{\omega }$ and make the principle ${\square }_{{\aleph }_{\omega },<\omega }$ fail in the generic extension. We also show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ${\aleph }_2$ and arrange in the generic extension that simultaneous reflection holds at ${\aleph }_2$, and at the same time, every stationary subset of ${\aleph }_3$ concentrating on points of cofinality $\omega$ has a reflection point of cofinality ${\omega }_1$.