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M. Holz K. Steffens E. Weitz
Introduction to Cardinal Arithmetic
Reprint of the 1999 Edition Birkhäuser Verlag Basel · Boston · Berlin
Authors: Michael Holz Unter den Bäumchen 17 30926 Seelze Germany e-mail:
[email protected] Edmund Weitz Bernadottestr. 38 22763 Hamburg Germany
Karsten Steffens Institut für Algebra, Zahlentheorie und Diskrete Mathematik Universität Hannover Welfengarten 1 30167 Hannover Germany e-mail:
[email protected] Originally published under the same title in the Birkhäuser Advanced Texts – Basler Lehrbücher series by Birkhäuser Verlag, Switzerland, ISBN 978-3-7643-6124-7 © 1999 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
1991 Mathematics Subject Classification 04-01, 04A10, 03E10 Library of Congress Control Number: 2009937809 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-0346-0327-0 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2010 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞
ISBN 978-3-0346-0327-0 987654321
e-ISBN 978-3-0346-0330-0 www.birkhauser.ch
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1 Foundations 1.1 The Axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . 1.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Transfinite Induction and Recursion . . . . . . . . . . . . . 1.4 Arithmetic of Ordinals . . . . . . . . . . . . . . . . . . . . . 1.5 Cardinal Numbers and their Elementary Properties . . . . . 1.6 Infinite Sums and Products . . . . . . . . . . . . . . . . . . 1.7 Further Properties of κλ – the Singular Cardinal Hypothesis 1.8 Clubs and Stationary Sets . . . . . . . . . . . . . . . . . . . 1.9 The Erd¨ os-Rado Partition Theorem . . . . . . . . . . . . . 2 The 2.1 2.2 2.3
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5 15 20 30 40 58 70 79 96
Galvin-Hajnal Theorem Ideals and the Reduction of Relations . . . . . . . . . . . . . . 103 The Galvin-Hajnal Formula . . . . . . . . . . . . . . . . . . . . 108 Applications of the Galvin-Hajnal Formula . . . . . . . . . . . 121
3 Ordinal Functions 3.1 Suprema and Cofinalities . . . . . . . . . . . . . . . . 3.2 κ-rapid Sequences and the Main Lemma of pcf-Theory 3.3 The Definition and Simple Properties of pcf(a) . . . . 3.4 The Ideal J
