Problems and Theorems in Classical Set Theory
This is the first comprehensive collection of problems in set theory. Most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come from the period betwe
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Edited by P. Winkler
Péter Komjáth and Vilmos Totik
Problems and Theorems in Classical Set Theory
Péter Komjáth Department of Computer Science Eotvos Lorand University, Budapest Budapest 1117 Hungary Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755-3551 [email protected]
Vilmos Totik Department of Mathematics University of South Florida Tampa, FL 33620 USA and Bolyai Institute University of Szeged Szeged Hungary 6720 [email protected]
Mathematics Subject Classification (2000): 03Exx, 05-xx, 11Bxx Library of Congress Control Number: 2005938489 ISBN-10: 0-387-30293-X ISBN-13: 978-0387-30293-5 Printed on acid-free paper. © (2006) Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springer.com
(MVY)
Dedicated to Andr´ as Hajnal and to the memory of Paul Erd˝ os and G´eza Fodor
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Part I Problems 1
Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3
Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4
Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5
Sets of reals and real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6
Ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7
Order types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8
Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9
Ordinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
10 Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 11 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 12 Transfinite enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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