On some questions concerning strong compactness

Archive for Mathematical Logic 51 (7-8):819-829 (2012)
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Abstract

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We get a negative answer to the first of these questions and positive answers to the second of these questions for a supercompact cardinal κ in the context of the absence of the full Axiom of Choice.

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References found in this work

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
[Omnibus Review].Akihiro Kanamori - 1981 - Journal of Symbolic Logic 46 (4):864-866.
On a problem of Woodin.Arthur W. Apter - 2000 - Archive for Mathematical Logic 39 (4):253-259.
Adding closed cofinal sequences to large cardinals.Lon Berk Radin - 1982 - Annals of Mathematical Logic 22 (3):243.

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