Grzegorz Tomkowicz - The BanachTarski Paradox

Thisneweditionofaclassicbookunifiescontemporaryresearchontheparadox.Ithasbeenupdatedwithmanynewproofsandresultsanddiscussionsofthemanyproblemsthatremainunsolved.Amongthenewresultspresentedareseveralunusualparadoxesinthehyperbolicplane,oneofwhichinvolvestheshapesofEschersfamousAngelandDevilswoodcut.Anewchapterisdevotedtoacompleteproofoftheremarkableresultthatthecirclecanbesquaredusingsettheory,aproblemthathadbeenopenforoversixtyyears.

GrzegorzTomkowicz isaself-educatedPolishmathematicianwhohasmadeseveralimportantcontributionstothetheoryofparadoxicaldecompositionsandinvariantmeasures.

StanWagon isaProfessorofMathematicsatMacalesterCollege.HeisawinneroftheWolframResearchInnovatorAward,aswellasnumerouswritingawardsincludingtheFord,Evans,andtheAllendoerferAwards.Hispreviousworkincludes ACourseinComputationalNumberTheory (2000), TheSIAM100-DigitChallenge (2004)and MathematicainAction (3rdEd.2010).

EncyclopediaofMathematicsanditsApplications

Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationinmathematicsormathematicalscienceandforwhichadetaileddevelopmentoftheabstracttheoryislessimportantthanathoroughandconcreteexplorationoftheimplicationsandapplications.

Booksinthe EncyclopediaofMathematicsanditsApplications covertheirsubjectscomprehensively.Lessimportantresultsmaybesummarizedasexercisesattheendsofchapters.Fortechnicalities,readerscanbereferredtothebibliography,whichisexpectedtobecomprehensive.Asaresult,volumesareencyclopedicreferencesormanageableguidestomajorsubjects.

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The BanachTarski Paradox

Second Edition

Grzegorz Tomkowicz

Centrum Edukacji G2, Bytom, Poland

Stan Wagon

Macalester College, St. Paul, Minnesota

One Liberty Plaza, New York, NY 10006

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Information on this title: www.cambridge.org/9781107042599

Cambridge University Press 2016

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2016

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Names: Tomkowicz, Grzegorz. | Wagon, S.

Title: The Banach-Tarski paradox.

Description: Second edition / Grzegorz Tomkowicz, Centrum Edukacji G2, Bytom,

Poland, Stan Wagon, Macalester College, St. Paul, Minnesota. | New York,

NY : Cambridge University Press, [2016] | Series: Encyclopedia of

mathematics and its applications ; 163 | Previous edition: The

Banach-Tarski paradox / Stan Wagon (Cambridge : Cambridge University

Press, 1985). | Includes bibliographical references and index.

Identifiers: LCCN 2015046410 | ISBN 9781107042599 (hardback : alk. paper)

Subjects: LCSH: Banach-Tarski paradox. | Measure theory. | Decomposition

(Mathematics)

Classification: LCC QA248 .W22 2016 | DDC 511.3/22dc23

LC record available at http://lccn.loc.gov/2015046410

ISBN 978-1-107-04259-9 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

DELIANS: How can we be rid of the plague?
DELPHIC ORACLE: Construct a cubic altar having double the size of the existing one.
BANACH AND TARSKI: Can we use the Axiom of Choice?

This book is motivated by the following theorem of Hausdorff, Banach, andTarski: Given any two bounded sets and in three-dimensional space, each having nonempty interior, one can partition into finitely manydisjoint parts and rearrange them by rigid motions to form . This, I believe,is the most surprising result of theoretical mathematics. It shows the imaginarycharacter of the unrestricted idea of a set in . It precludes the existence offinitely additive, congruence-invariant measures over all bounded subsets of, and it shows the necessity of more restricted constructions, such asLebesgue measure.

In the 1950s, the years of my mathematical education in Poland, this resultwas often discussed. J. F. Adams, T. J. Dekker, J. von Neumann, R. M.Robinson, and W. Sierpiski wrote about it; my PhD thesis was motivated byit. (All this is referenced in this book.) Thus it is a great pleasure to introduceyou to this book, where this striking theorem and many related results ingeometry and measure theory, and the underlying tools of group theory, arepresented with care and enthusiasm. The reader will also find someapplications of the most recent advances of group theory to measuretheory: the work of Gromov, Margulis, Rosenblatt, Sullivan, Tits, andothers.