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

Arthur W. Apter¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.

Fundamenta Mathematicae, Tome 194 (2007) no. 3, pp. 253-265.

Résumé

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.

Détail

DOI : 10.4064/fm194-3-3

Keywords: construct models level level equivalence between strong compactness supercompactness which kappa lambda supercompact lambda kappa regular able determine exactly number normal measures kappa lambda carries first these models kappa lambda carries lambda kappa many normal measures maximal number second these models kappa lambda carries lambda kappa many normal measures except kappa measurable cardinal which limit measurable cardinals kappa hence kappa kappa carries only kappa many normal measures these models there restrictions structure class supercompact cardinals