Level by level equivalence and the number of normal measures over $P_{\kappa} (\lambda )$
Volume 194 / 2007
Abstract
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.
