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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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Cofinality and measurability of the first three uncountable cardinals
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by Arthur W. Apter, Stephen C. Jackson and Benedikt Löwe PDF
Trans. Amer. Math. Soc. 365 (2013), 59-98 Request permission


This paper discusses models of set theory without the Axiom of Choice. We investigate all possible patterns of the cofinality function and the distribution of measurability on the first three uncountable cardinals. The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under $\mathsf {AD}$) that there is a cardinal $\kappa$ such that the triple $(\kappa ,\kappa ^+,\kappa ^{++})$ satisfies the strong polarized partition property.
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Additional Information
  • Arthur W. Apter
  • Affiliation: Department of Mathematics, Baruch College, City University of New York, One Bernard Baruch Way, New York, New York 10010 – and – The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, New York 10016
  • MR Author ID: 26680
  • Email:
  • Stephen C. Jackson
  • Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
  • MR Author ID: 255886
  • ORCID: 0000-0002-2399-0129
  • Email:
  • Benedikt Löwe
  • Affiliation: Institute for Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands – and – Department Mathematik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany – and – Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Email:
  • Received by editor(s): May 4, 2009
  • Received by editor(s) in revised form: November 2, 2010
  • Published electronically: July 12, 2012
  • Additional Notes: The first author’s visits to Amsterdam and the third author’s visit to New York were partially supported by the DFG-NWO Bilateral Grant (DFG KO 1353/5-1; NWO 62-630). In addition, the first author wishes to acknowledge the support of various PSC-CUNY and CUNY Collaborative Incentive grants. The third author would like to thank the CUNY Graduate Center Mathematics Program for their hospitality and partial financial support during his stay in New York.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 59-98
  • MSC (2010): Primary 03E02, 03E35, 03E55, 03E60; Secondary 03E10, 03E15, 03E25
  • DOI:
  • MathSciNet review: 2984052