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Consistency Strength of ¬ ℵω and Reflection at Two Successive ℵ N

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 using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into אω and make the principle אω ,<ω fail in the generic extension. We also show that using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into א2, and arrange that in the generic extension simultaneous reflection holds at א2 and at the same time every stationary subset of א3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. We present two models built using modest large cardinal hypotheses. In the first model the principle אω,<ω fails. In the second model any family of size ω1 of stationary subsets of ω2 concentrating on ordinals of cofinality ω has a common reflection point, and at the same time every stationary subset of ω3 concentrating on ordinals of cofinality ω reflects at a point of cofinality ω1. It is of course known that constructing models for these combinatorial situations requires large cardinals, and all models previously known build on the existence of large cardinals of some degree of supercompactness. The natural long-standing open problem is to determine the exact consistency strength. In the past there has been a considerable amount of work done along these lines, and it seems that determining the lower bounds is significantly more demanding than determining the upper bounds. Using relatively simple forcing techniques, we give upper bounds which seem to be not too far from the actual consistency strength, and significantly below any variant of supercompactness.