Infinity: New Research Frontiers

This page intentionally left blank Infinity New Research Frontiers This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: Can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity. Rev. Dr. Michael Heller is the Founder and Director of the Copernicus Center for Interdisciplinary Studies, Cracow. He is a Professor of Philosophy at the Pontifical University of John Paul II, Cracow, Poland; an Adjunct Member of the Vatican Observatory; and Ordinary Member of the Pontifical Academy of Sciences. Heller is the author of more than thirty books and nearly two hundred scientific papers. Dr. W. Hugh Woodin is a Professor of Mathematics and the Chair of the Mathematics Department at the University of California, Berkeley. Professor Woodin has published numerous articles and books and is the managing editor of the Journal of Mathematical Logic and editor of Mathematical Research Letters, Mathematical Logic Quarterly, and Electronic Research Announcements (American Mathematical Society). Infinity New Research Frontiers Edited by Michael Heller Pontifical University of John Paul II W. Hugh Woodin University of California, Berkeley cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107003873 c Cambridge University Press 2011 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 2011 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 Infinity : new research frontiers / edited by Michael Heller, W. Hugh Woodin. p. cm. Includes index. ISBN 978-1-107-00387-3 (hardback) 1. Infinite – History. 2. Mathematics – Philosophy. I. Heller, Michael. II. Woodin, W. H. (W. Hugh) III. Title. QA9.I46 2011 2010049374 111 .6–dc22 ISBN 978-1-107-00387-3 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. The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. . . . David Hilbert (1862–1943) Contents Contributors Preface Acknowledgments page ix xi xiii Introduction Rudy Rucker 1 I Perspectives on Infinity from History 1 Infinity as a Transformative Concept in Science and Theology