Combinatorial Set Theory With a Gentle Introduction to Forcing
This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and
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Lorenz J. Halbeisen
Combinatorial Set Theory With a Gentle Introduction to Forcing
Lorenz J. Halbeisen Institut für Mathematik Universität Zürich Zürich Switzerland [email protected]
ISSN 1439-7382 Springer Monographs in Mathematics ISBN 978-1-4471-2172-5 e-ISBN 978-1-4471-2173-2 DOI 10.1007/978-1-4471-2173-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011942598 Mathematics Subject Classification: 03E35, 03E17, 03E25, 05D10, 03E30, 03E50, 03E05, 03E10, 03E75 © Springer-Verlag London Limited 2012 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Joringel, Meredith, Andrin, and Salome
Preface
By the campanologist, the playing of tunes is considered to be a childish game; the proper use of bells is to work out mathematical permutations and combinations. His passion finds its satisfaction in mathematical completeness and mechanical perfection.
D OROTHY L. S AYERS The Nine Tailors, 1934 This book provides a self-contained introduction to Axiomatic Set Theory with main focus on Infinitary Combinatorics and the Forcing Technique. The book is intended to be used as a textbook in undergraduate and graduate courses of various levels, as well as for self-study. To make the book valuable for experienced researchers also, some historical background and the sources of the main results have been provided in the N OTES, and some topics for further studies are given in the section R ELATED R ESULTS—where those containing open problems are marked with an asterisk. The axioms of Set Theory ZFC, consisting of the axioms of Zermelo–Fraenkel Set Theory (denoted ZF) and the Axiom of Choice, are the foundation of Mathematics in the sense that essentially all Mathematics can be formalised within ZFC. On the other hand, Set Theory can also be considered as a mat
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