On supercompactness and the continuum function

TL;DR

This paper demonstrates how to force the continuum function to match any suitable function while preserving a supercompact cardinal, using minimal initial assumptions and extending Woodin's technique.

Contribution

It extends Woodin's surgical modification technique to handle off-range modifications, enabling control over the continuum function under weaker hypotheses.

Findings

Successfully forces the continuum function to match a wide class of functions.

Preserves $ ext{lambda}$-supercompactness of $ ext{kappa}$ under minimal assumptions.

Answers a question posed by Friedman and Honzik in 2012.

Abstract

Given a cardinal $\kappa$ that is $\lambda$-supercompact for some regular cardinal $\lambda\geq\kappa$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[\kappa,\lambda]\cap\REG\to\CARD$ satisfying $\forall\alpha,\beta\in\dom(F)$ $\alpha<\cf(F(\alpha))$ and $\alpha<\beta$ $\implies$ $F(\alpha)\leq F(\beta)$, 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.