Archive for Mathematical Logic 52 (5-6):569-591 (2013)

Abstract
Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous results for supercompact cardinals [Menas in Trans Am Math Soc 223:61–91, (1976)] and strong cardinals [Friedman and Honzik in Ann Pure Appl Logic 154(3):191–208, (2008)], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin
Keywords Woodin cardinals  Continuum function  Easton’s theorem
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DOI 10.1007/s00153-013-0332-0
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References found in this work BETA

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
[Omnibus Review].Akihiro Kanamori - 1981 - Journal of Symbolic Logic 46 (4):864-866.
Cardinal Invariants Above the Continuum.James Cummings & Saharon Shelah - 1995 - Annals of Pure and Applied Logic 75 (3):251-268.
Perfect-Set Forcing for Uncountable Cardinals.Akihiro Kanamori - 1980 - Annals of Mathematical Logic 19 (1-2):97-114.

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Citations of this work BETA

Large Cardinals Need Not Be Large in HOD.Yong Cheng, Sy-David Friedman & Joel David Hamkins - 2015 - Annals of Pure and Applied Logic 166 (11):1186-1198.
Easton's Theorem for the Tree Property Below ℵ.Šárka Stejskalová - 2021 - Annals of Pure and Applied Logic 172 (7):102974.
Woodin for Strong Compactness Cardinals.Stamatis Dimopoulos - 2019 - Journal of Symbolic Logic 84 (1):301-319.
An Easton Like Theorem in the Presence of Shelah Cardinals.Mohammad Golshani - 2017 - Archive for Mathematical Logic 56 (3-4):273-287.

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