Level by level equivalence and the number of normal measures over $P_{\kappa} (\lambda )$

Volume 194 / 2007

Arthur W. Apter Fundamenta Mathematicae 194 (2007), 253-265 MSC: 03E35, 03E55. DOI: 10.4064/fm194-3-3


We construct two models for the level by level equivalence between strong compactness and supercompactness in which if $\kappa $ is $\lambda $ supercompact and $\lambda \ge \kappa $ is regular, we are able to determine exactly the number of normal measures $P_\kappa (\lambda )$ carries. In the first of these models, $P_\kappa (\lambda )$ carries $2^{2^{[\lambda ]^{< \kappa }}}$ many normal measures, the maximal number. In the second of these models, $P_\kappa (\lambda )$ carries $2^{2^{[\lambda ]^{< \kappa }}}$ many normal measures, except if $\kappa $ is a measurable cardinal which is not a limit of measurable cardinals. In this case, $\kappa $ (and hence also $P_\kappa (\kappa )$) carries only $\kappa ^+$ many normal measures. In both of these models, there are no restrictions on the structure of the class of supercompact cardinals.


  • Arthur W. ApterDepartment of Mathematics
    Baruch College of CUNY
    New York, NY 10010, U.S.A.
    The CUNY Graduate Center, Mathematics
    365 Fifth Avenue
    New York, NY 10016, U.S.A.

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