On supercompactness and the continuum function

Brent Cody*, Menachem Magidor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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{long rightwards double arrow}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 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 [5]. We also discuss several related open questions.

Original languageEnglish
Pages (from-to)620-630
Number of pages11
JournalAnnals of Pure and Applied Logic
Volume165
Issue number2
DOIs
StatePublished - Feb 2014

Keywords

  • Continuum function
  • Forcing
  • Large cardinal
  • Supercompact cardinal

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