On supercompactness and the continuum function

Abstract

Given a cardinal $\kappa$ that is $\lambda$-supercompact for some regular cardinal $\lambda \ge \kappa$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[κ,λ]\cap\REG\to\CARD$ satisfying $\forallα,β\in\dom(F)$ $α<\cf(F(α))$ and $\alpha <\beta$ $⟹$ $F(\alpha )\le F(\beta )$, while preserving the $\lambda$-supercompactness of $\kappa$ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M^{\lambda }\subseteq M$ and $j(\kappa )>F(\lambda )$. Our argument extends Woodin's technique of surgically modifying a generic filter to a new case: Woodin's 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 [FH2012]. We also discuss several related open questions.

12 pages