Walks on Ordinals and Their Characteristics
The analysis of the characteristics of walks on ordinals is a powerful new technique for building mathematical structures, developed by the author over the last twenty years. This is the first book-length exposition of this method. Particular emphasis is
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Series Editors H.Bass J. Oesterlé A. Weinstein
Stevo Todorcevic
Walks on Ordinals and Their Characteristics
Birkhäuser Basel · Boston · Berlin
Stevo Todorcevic Université Paris VII – C.N.R.S. UMR 7056 2, Place Jussieu – Case 7012 75251 Paris Cedex 05 France e-mail: [email protected]
Department of Mathematics University of Toronto Toronto M5S 2E4 Canada e-mail: [email protected]
and Mathematical Institute, SANU Kneza Mihaila 35 11000 Belgrad Serbia e-mail: [email protected]
2000 Mathematics Subject Classification 03E10, 03E75, 05D10, 06A07, 46B03, 54D65, 54A25 Library of Congress Control Number: 2007933914 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 http://dnb.ddb.de
ISBN 978-3-7643-8528-6 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. © 2007 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 from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8528-6
e-ISBN 978-3-7643-8529-3
987654321
www.birkhauser.ch
Contents 1 Introduction 1.1 Walks and the metric theory of ordinals 1.2 Summary of results . . . . . . . . . . . . 1.3 Prerequisites and notation . . . . . . . . 1.4 Acknowledgements . . . . . . . . . . . .
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2 Walks on Countable Ordinals 2.1 Walks on countable ordinals and their basic characteristics . 2.2 The coherence of maximal weights . . . . . . . . . . . . . . 2.3 Oscillations of traces . . . . . . . . . . . . . . . . . . . . . . 2.4 The number of steps and the last step functions . . . . . . .
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3 Metric Theory of Countable Ordinals 3.1 Triangle inequalities . . . . . . . . . . . . . . . 3.2 Constructing a Souslin tree using ρ . . . . . . . 3.3 A Hausdorff gap from ρ . . . . . . . . . . . . . 3.4 A general theory of subadditive functions on ω1 3.5 Conditional weakly null sequences based on subadditive functions . . . . . . . . . . . . . . . 4 Coherent Mappings and Trees 4.1 Coherent mappings . . . . . . . . . . . . . . 4.2 Lipschitz property of coherent trees . . . . . 4.3 The global structure of the class of coherent 4.4 Lexicographically ordered coherent trees . . 4.5 Stationary C-lines . . . . . . . . . . . . . . 5 The 5.1 5.2 5.3
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