Exactly controlling the non-supercompact strongly compact cardinals

Journal of Symbolic Logic 68 (2):669-688 (2003)

Authors
Joel David Hamkins
Oxford University
Abstract
We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [5], due to the first author
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DOI 10.2178/jsl/1052669070
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References found in this work BETA

The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.

View all 10 references / Add more references

Citations of this work BETA

Tall Cardinals.Joel D. Hamkins - 2009 - Mathematical Logic Quarterly 55 (1):68-86.
On the Consistency Strength of Level by Level Inequivalence.Arthur W. Apter - 2017 - Archive for Mathematical Logic 56 (7-8):715-723.
Tallness and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (1):4-12.
More on Regular and Decomposable Ultrafilters in ZFC.Paolo Lipparini - 2010 - Mathematical Logic Quarterly 56 (4):340-374.

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