On supercompactness and the continuum function

Abstract

Given a cardinal κ that is λ-supercompact for some regular cardinal λ⩾κ and assuming GCH, we show that one can force the continuum function to agree with any function F:[κ,λ]∩REG→CARD satisfying ∀α,β∈dom(F)α<cf(F(α)) and α<β⟹F(α)⩽F(β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding j:V→M with critical point κ such that Mλ⊆M and j(κ)>F(λ). Our argument extends Woodinʼs technique of surgically modifying a generic filter to a new case: Woodinʼs key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions.