We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures carries. In the first of these models, carries many normal measures, the maximal number. In the second of these models, carries many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and hence also ) carries only κ⁺ many normal measures. In both of these models, there are no restrictions on the structure of the class of supercompact cardinals.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-3,
author = {Arthur W. Apter},
title = {Level by level equivalence and the number of normal measures over $P\_{$\kappa$}($\lambda$)$
},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {253-265},
zbl = {1121.03067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-3}
}
Arthur W. Apter. Level by level equivalence and the number of normal measures over $P_{κ}(λ)$
. Fundamenta Mathematicae, Tome 193 (2007) pp. 253-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-3/