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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cofinality and measurability of the first three uncountable cardinals


Authors: Arthur W. Apter, Stephen C. Jackson and Benedikt Löwe
Journal: Trans. Amer. Math. Soc. 365 (2013), 59-98
MSC (2010): Primary 03E02, 03E35, 03E55, 03E60; Secondary 03E10, 03E15, 03E25
Published electronically: July 12, 2012
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Abstract: 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 The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under $ \mathsf {AD}$) that there is a cardinal $ \kappa $ such that the triple $ (\kappa ,\kappa ^+,\kappa ^{++})$ satisfies the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
Email: awapter@alum.mit.edu

Stephen C. Jackson
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
Email: jackson@unt.edu

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: bloewe@science.uva.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05497-3
PII: S 0002-9947(2012)05497-3
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.