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Also by Michael Huemer ETHICAL INTUITIONISM THE PROBLEM OF POLITICAL AUTHORITY SKEPTICISM AND THE VEIL OF PERCEPTION

Approaching Infinity Michael Huemer Professor of Philosophy, University of Colorado at Boulder, USA

© Michael Huemer 2016 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2016 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries.

ISBN 978-1-137-56086-5 ISBN 978-1-137-56087-2 (eBook) DOI 10.1057/9781137560872 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Huemer, Michael, 1969– author. Title: Approaching infinity / Michael Huemer. Description: New York : Palgrave Macmillan, 2016. | Includes index. Identifiers: LCCN 2015040546| ISBN 9781137560865 (pbk.) Subjects: LCSH: Infinite. Classification: LCC BD411 .H84 2016 | DDC 111/.6—dc23 LC record available at http://lccn.loc.gov/2015040546

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Contents List of Figures

xi

Preface

xii Part I

The Need for a Theory of Infinity

1

The Prevalence of the Infinite 1.1 The concept of infinity and the infinite 1.2 The infinite in mathematics 1.3 The infinite in philosophy 1.4 The infinite in the physical world 1.5 The infinite in modern physics 1.6 Controversies

2

Six Infinite Regresses 2.1 The regress of causes 2.2 The regress of reasons 2.3 The regress of forms 2.4 The regress of resemblances 2.5 The regress of temporal series 2.6 The regress of truths 2.7 Conclusion

9 9 10 11 12 14 15 16

3

Seventeen Paradoxes of the Infinite 3.1 A word about paradoxes 3.2 The arithmetic of infinity 3.3 The paradox of g