Large cardinals with few measures

Apter, Arthur; Cummings, James; Hamkins, Joel

doi:10.1090/S0002-9939-07-08786-2

Large cardinals with few measures
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by Arthur W. Apter, James Cummings and Joel David Hamkins PDF

Proc. Amer. Math. Soc. 135 (2007), 2291-2300 Request permission

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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