Stationary reflection and level by level equivalence

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.

Colloquium Mathematicum, Tome 115 (2009) no. 1, pp. 113-128.

Résumé

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness and supercompactness in which on a stationary subset of the least supercompact cardinal $\kappa $, there are non-weakly compact Mahlo cardinals which reflect stationary sets. We also examine some extensions and limitations on what is possible in our theorems. Finally, we indicate how to ensure in our models that $\diamondsuit _\delta $ holds for every successor and Mahlo cardinal $\delta $, and below the least supercompact cardinal $\kappa $, $\square _\delta $ holds on a stationary subset of $\kappa $. There are no restrictions in our main models on the structure of the class of supercompact cardinals.

Détail

DOI : 10.4064/cm115-1-10

Keywords: force construct model which level level equivalence between strong compactness supercompactness holds along certain additional inner model properties particular model class mahlo cardinals reflecting stationary sets the class weakly compact cardinals every regular nsson cardinal weakly compact other force construct model level level equivalence between strong compactness supercompactness which stationary subset least supercompact cardinal kappa there non weakly compact mahlo cardinals which reflect stationary sets examine extensions limitations what possible theorems finally indicate ensure models diamondsuit delta holds every successor mahlo cardinal delta below least supercompact cardinal kappa square delta holds stationary subset kappa there restrictions main models structure class supercompact cardinals