Supercompactness and partial level by level equivalence between strong compactness and strongness
Volume 182 / 2004
Fundamenta Mathematicae 182 (2004), 123-136
MSC: 03E35, 03E55.
DOI: 10.4064/fm182-2-3
Abstract
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.
