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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the strong equality between supercompactness and strong compactness
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by Arthur W. Apter and Saharon Shelah PDF
Trans. Amer. Math. Soc. 349 (1997), 103-128 Request permission


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
  • Email:
  • 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
  • Email:,
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 103-128
  • MSC (1991): Primary 03E35; Secondary 03E55
  • DOI:
  • MathSciNet review: 1333385