Strong Compactness and a Global Version of a Theorem of Ben-David and Magidor

Abstract

Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ + strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, ✷ γ +ω fails but ✷ γ +ω ,ω holds. This generalizes a result of Ben-David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.Mathematics Subject Classification: 03E35, 03E55, 03E50, 03E10.In [3], Ben-David and Magidor proved the theorem Con(ZFC + GCH + κ is κ + supercompact) ⇒ Con(ZFC + ✷ * ℵω + ¬✷ ℵω ). In [2], the author and Henle were able to reduce the original hypotheses of BenDavid and Magidor and proved the theorem Con(ZFC + GCH + κ is κ + strongly compact) ⇒ Con(ZFC + ✷ * ℵω + ¬✷ ℵω ). Then, in [1], the author and Cummings proved a global version of the result of [3] by proving Con(ZFC + GCH + κ is κ +5 supercompact) ⇒ Con(ZFC + κ is κ +5 supercompact + For every singular cardinal λ < κ, there exists a stationary S ⊆ λ + such that if S = S α : α < β < cof(λ) is a sequence of stationary subsets of S, then S reflects simultaneously to cofinality δ for unboundedly many δ < λ + For every singular cardinal λ < κ, ✷ λ,cof(λ) ).The proof of this result uses the generalized form of Radin forcing given by Foreman and Woodin in [5].