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Easton’s theorem in the presence of Woodin cardinals

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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 2γ = 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 2γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin.

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Correspondence to Brent Cody.

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This work was carried out while the author was a student under the advisement of Joel David Hamkins. The author wishes to thank Professor Hamkins for his guidance, as well as for many helpful conversations regarding the topics contained in this paper. The author also wishes to thank Arthur Apter for his helpful comments regarding Lemma 14 below.

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Cody, B. Easton’s theorem in the presence of Woodin cardinals. Arch. Math. Logic 52, 569–591 (2013). https://doi.org/10.1007/s00153-013-0332-0

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