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 language | English |
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Pages (from-to) | 620-630 |
Number of pages | 11 |
Journal | Annals of Pure and Applied Logic |
Volume | 165 |
Issue number | 2 |
DOIs | |
State | Published - Feb 2014 |
Keywords
- Continuum function
- Forcing
- Large cardinal
- Supercompact cardinal