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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The author’s research was partially supported by PSC-CUNY grants.
The author wishes to thank Brent Cody for helpful conversations on the subject matter of this paper.
The author also wishes to thank the second referee for helpful corrections and suggestions which were incorporated into the current version of the paper.
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Apter, A.W. On some questions concerning strong compactness. Arch. Math. Logic 51, 819–829 (2012). https://doi.org/10.1007/s00153-012-0300-0
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DOI: https://doi.org/10.1007/s00153-012-0300-0