Supercompactness and partial level by level equivalence between strong compactness and strongness

Arthur W. Apter¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010 U.S.A.

Fundamenta Mathematicae, Tome 182 (2004) no. 2, pp. 123-136

Résumé

We force and construct a model containing supercompact cardinals in which, for any measurable cardinal $\delta $ and any ordinal $\alpha $ below the least beth fixed point above $\delta $, if $\delta ^{+ \alpha }$ is regular, $\delta $ is $\delta ^{+ \alpha }$ strongly compact iff $\delta $ is $\delta + \alpha + 1$ strong, except possibly if $\delta $ is a limit of cardinals $\gamma $ which are $\delta ^{+ \alpha }$ strongly compact. The choice of the least beth fixed point above $\delta $ as our bound on $\alpha $ is arbitrary, and other bounds are possible.

Détail

DOI : 10.4064/fm182-2-3

Keywords: force construct model containing supercompact cardinals which measurable cardinal delta ordinal alpha below least beth fixed point above delta delta alpha regular delta delta alpha strongly compact delta delta alpha strong except possibly delta limit cardinals gamma which delta alpha strongly compact choice least beth fixed point above delta bound alpha arbitrary other bounds possible