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Large cardinals with few measures

Authors:	Arthur W Apter James Cummings Joel David Hamkins

Institution:	Department of Mathematics, Baruch College of CUNY, New York, New York 10010 ; Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 ; 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

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.

Keywords:	Supercompact cardinal strongly compact cardinal measurable cardinal normal measure

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