UC Irvine

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 κ+-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 Nω and make the principle □Nω, <ω 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 N2 and arrange in the generic extension that simultaneous reflection holds at N2, and at the same time, every stationary subset of N3 concentrating on points of cofinality ω has a reflection point of cofinality ω1.