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Large cardinals with few measures

Authors: Arthur W. Apter, James Cummings and Joel David Hamkins
Journal: Proc. Amer. Math. Soc. 135 (2007), 2291-2300
MSC (2000): Primary 03E35, 03E55
Published electronically: March 2, 2007
MathSciNet review: 2299507
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Abstract: We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $ \kappa^+$ many normal measures on the least measurable cardinal $ \kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $ \lambda$ strong compactness or $ \lambda$ supercompactness measures on $ P_\kappa(\lambda)$ can be exactly $ \lambda^+$ if $ \lambda > \kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

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Additional Information

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010

James Cummings
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Joel David Hamkins
Affiliation: Mathematics Program, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016 — Department of Mathematics, The College of Staten Island of CUNY, Staten Island, New York 10314

Keywords: Supercompact cardinal, strongly compact cardinal, measurable cardinal, normal measure.
Received by editor(s): March 14, 2006
Published electronically: March 2, 2007
Additional Notes: The research of the first and third authors was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive Grants. The second author’s research was partially supported by NSF Grant DMS-0400982.
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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