A new Easton theorem for supercompactness 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, USA <a href="http://faculty.baruch.cuny.edu/aapter">http://faculty.baruch.cuny.edu/aapter</a>

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 1, pp. 1-10

Résumé

We establish a new Easton theorem for the least supercompact cardinal $\kappa $ that is consistent with the level by level equivalence between strong compactness and supercompactness. This theorem is true in any model of ZFC containing at least one supercompact cardinal, regardless if level by level equivalence holds. Unlike previous Easton theorems for supercompactness, there are no limits on the Easton functions $F$ used, other than the usual constraints given by Easton’s theorem and the fact that if $\delta \lt \kappa $ is regular, then $F(\delta ) \lt \kappa $. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.

Détail

DOI : 10.4064/ba8080-6-2017

Keywords: establish easton theorem least supercompact cardinal kappa consistent level level equivalence between strong compactness supercompactness theorem model zfc containing least supercompact cardinal regardless level level equivalence holds unlike previous easton theorems supercompactness there limits easton functions other usual constraints given easton theorem delta kappa regular delta kappa ground model model witnessing conclusions theorem there restrictions structure class supercompact cardinals