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Exactly controlling the non-supercompact strongly compact cardinals

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter  and
Joel David Hamkins
Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA, E-mail: [email protected], URL: http://math.baruch.cuny.edu/~apter
Joel David Hamkins
Affiliation:
Department of Mathematics, The College of Staten Island of Cuny Department of Mathematics, The Graduate Center of Cuny, 365 Fifth Avenue, New York, New York 10016, USA, E-mail: [email protected], URL: http://jdh.hamkins.org
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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.

Keywords

Supercompact cardinalstrongly compact cardinalstrong cardinalindestructibilitynon-reflecting stationary set of ordinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 669 - 688
Copyright
Copyright © Association for Symbolic Logic 2003

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