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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the strong equality between supercompactness and strong compactness

Authors: Arthur W. Apter and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 349 (1997), 103-128
MSC (1991): Primary 03E35; Secondary 03E55.
MathSciNet review: 1333385
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Abstract: We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models {}$ “$\kappa$ is $\lambda$ supercompact”, then $V[G] \models {}$ “$\kappa$ is $\lambda$ supercompact” and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models {}$ “$\kappa$ is $\lambda$ strongly compact” iff $V[{G}] \models {}$ “$\kappa$ is $\lambda$ supercompact”, except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.

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Additional Information

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010
MR Author ID: 26680

Saharon Shelah
Affiliation: Department of Mathematics, The Hebrew University, Jerusalem, Israel; Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08904
MR Author ID: 160185
ORCID: 0000-0003-0462-3152

Keywords: Strongly compact cardinal, supercompact cardinal.
Received by editor(s): May 2, 1994
Received by editor(s) in revised form: December 30, 1994
Additional Notes: The research of the first author was partially supported by PSC-CUNY Grant 662341 and a salary grant from Tel Aviv University. In addition, the first author wishes to thank the Mathematics Departments of The Hebrew University and Tel Aviv University for the hospitality shown him during his sabbatical in Israel. The second author wishes to thank the Basic Research Fund of the Israeli Academy of Sciences for partially supporting this research, which is Publication 495 of the second author.
Article copyright: © Copyright 1997 American Mathematical Society