2013/06/03 by Brent Cody, Menachem Magidor, Cody, Brent +1
Mathematics · #03E35 #03E55 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E35 #msc:03E55
paper · pdf · doi:10.48550/arxiv.1306.0449
12 pages
arxiv created 2013/09/11 · arxiv updated 2013/09/12
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 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.