The Continuum A Constructive Approach to Basic Concepts of Real Anal
In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the
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The Continuum
Rudolf Taschner
The Continuum A Constructive Approach to Basic Concepts of Real Analysis
aI
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Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the Internet at .
Prof. Dr. Rudolf Taschner
Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstr. 8 A-I040 WIEN E-Mail: [email protected]
First edition, September 2005 All rights reserved © Friedr. Vieweg & Sohn VerlagjGWV Fachverlage GmbH, Wiesbaden 2005 Softcover reprint of the hardcover 1st edition 2005
Editorial office: Ulrike Schmickler-Hirzebruch / Petra RuBkamp Vieweg is a company in the specialist publishing group Springer Science+Business Media. www.vieweg.de
No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder.
Cover design: Ulrike Weigel, www.CorporateDesignGroup.de Printed on acid-free paper ISBN-I3: 978-3-322-82038-9 DOl: 10.1 007/978-3-322-82036-5
e-ISBN-13: 978-3-322-82036-5
Preface
"Few mathematical structures have undergone as many revlSlons or have been presented in as many guises as the real numbers. Every generation re-examines the reals in the light of its values and mathematical objectives." This citation is said to be due to Gian-Carlo Rota, and in this book its correctness again is affirmed. Here I propose to investigate the structure of the mathematical continuum by undertaking a rather unconventional access to the real numbers: the intuitionistic one. The traces can be tracked back at least to L.E.J. Brouwer and to H. Weyl. Largely unknown photographies of Weyl in Switzerland after World War II provided by Peter Bettschart enliven the abstract text full of subtle definitions and sophisticated estimations. The book can be read by students who have undertaken the usual analysis courses and want to know more about the intrinsic details of the underlying concepts, and it can also be used by university teachers in lectures for advanced undergraduates and in seminaries for graduate students. I wish to thank Walter Lummerding and Gottfried Oehl who helped me with their impressive expert knowledge of the English language. I also take the opportunity to express my gratitude to Ulrike Schmickler-Hirzebruch and to the staff of Vieweg-Verlag for editing my manuscript just now, exactly 50 years after the death of Hermann Weyl, in their renowned publishing house.
Vienna, 2005
Rudolf Taschner
Contents
1 Introduction and historical remarks 1.1 F AREY fractions. . 1.2 The pentagram ... 1.3 Continued fractions . 1.4 Special square roots . 1.5 DEDEKIND cuts . . . 1.6 WEYL'S alternative 1.7 BROUWER's alternative. 1.8 Integration in traditional and in intuitionistic framework . 1.9 The wager . . . . . . . . . . . . 1.10 How to read the following pages
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Real numbers Definition of real numbers ..
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