de Gruyter Series in Logic and Its Applications 6 Editors: W.A. Hodges (London) · R. Jensen (Berlin) S. Lempp (Madison) · M. Magidor (Jerusalem) One Hundred Years of Russell’s Paradox Mathematics, Logic, Philosophy Editor Godehard Link Walter de Gruyter ≥ Berlin · New York Editor Godehard Link Philosophie-Department Ludwig-Maximilians-Universität München Ludwigstr. 31/I 80539 München Germany Series Editors Wilfrid A. Hodges Ronald Jensen School of Mathematical Sciences Institut für Mathematik Queen Mary and Westfield College Humboldt-Universität University of London Unter den Linden 6 Mile End Road 10099 Berlin London E1 4NS Germany United Kingdom Steffen Lempp Menachem Magidor Department of Mathematics Institute of Mathematics University of Wisconsin The Hebrew University 480 Lincoln Drive Givat Ram Madison, WI 53706-1388 91904 Jerusalem USA Israel Mathematics Subject Classification 2000: 03A05; 00A30, 03-06 Țȍ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability Library of Congress Ϫ Cataloging-in-Publication Data One hundred years of Russell’s paradox : mathematics, logic, phi- losophy / edited by Godehard Link. p. cm. Ϫ (De Gruyter series in logic and its applications; 6) Includes bibliographical references and indexes. ISBN 3-11-017438-3 (acid-free paper) 1. Paradox. 2. Liar paradox. I. Title: 100 years of Russell’s paradox. II. Link, Godehard. III. Series. BC199.P2O54 2004 165Ϫdc22 2004006962 ISBN 3-11-017438-3 ISSN 1438-1893 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 Ͻhttp://dnb.ddb.deϾ. Ą Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting using the Authors’ TEXfiles: I. Zimmermann, Freiburg Ϫ Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Ϫ Cover design: Rainer Engel, Berlin. Preface In June 2001 an international conference was held at the University of Munich, Ger- many, to commemorate the centenary of the discovery of Russell’s paradox. It brought together leading scholars from the following fields: • Russell Studies • Mathematical Logic • Set Theory • Philosophy of Mathematics The aims of the event were twofold, historical and systematic. One focus was on Russell’s logic and logical philosophy, which were shaped so much by his own and related paradoxes, in particular since new material has become available in recent years from the rich sources of the Russell Archives through the publication of the Collected Papers of Bertrand Russell. But the second aim was of equal importance: to present original research in the broad range of foundational studies that draws on both current conceptions and recent technical advances in the above-mentioned fields. It was hoped to contribute this way to the well-established body of mathematical philosophy initiated to a large extent by Russell a hundred years ago. The conference featured plenary sessions with distinguished invited speakers, sec- tion meetings with both invited and contributed papers, and as special events two panel discussions, “Russell in Context” and “The Meaning of Set Theory”, as well as two symposia, one on “Propositional Functions” and the other on the “Finite Mathematics of Set Theory”. The Russell panel, chaired by A. Irvine, was composed of N. Griffin, P. Hylton, D. Kaplan, and A. Urquhart. Y. Moschovakis chaired the panel on Set The- ory, with S. Feferman, H. Friedman, D. C. McCarty, and W. H. Woodin as panelists. The symposiasts of the first symposium, chaired by R. Wahl, were G. Landini and B. Linsky, those of the second S. Lavine and K.-G. Niebergall, chaired by the editor. The papers collected in this volume represent the main body of research emanating from the proceedings of the conference. All authors delivered a talk at the conference except J. Mycielski, who was unable to attend but sent his invited paper, and H. Field, who attended the conference as a discussant and was invited by the editor to contribute his present paper. G. Jäger and H. Schwichtenberg were each joined by a co-author for their paper included here. A number of invited speakers delivered a talk, but were unable to contribute a paper to the volume because of other commitments; these are W. Buchholz, P. Hylton, H. Kamp, P. Martin-Löf, Y. Moschovakis, C. Parsons, G. Priest, and A. Urquhart. vi Preface The papers were all originally written for this volume, with two exceptions. The original place of publication of H. Field’s paper “The Consistency of the Naive Theory of Properties” is The Philosophical Quarterly,volume 54, No. 214, January 2004. It is reprinted here by kind permission of Blackwell Publishing and the author. A condensed version of Hazen’s paper appeared as A. P. Hazen and J. M. Davoren, Russell’s 1925 logic, in: Australasian Journal of Philosophy,volume 78, No. 4, pp. 534–556, December 2000. Reprint of this material as part of the present paper by kind permission of Oxford University Press and the co-author. The occasion to publish this volume would never have arisen without the essen- tial financial support from a number of institutions that made the conference possible in the first place. It is therefore fitting to express my deep appreciation here to the Deutsche Forschungsgemeinschaft (DFG ) for covering the main bulk of the confer- ence costs through funds made available to the interdisciplinary DFG graduate program Graduiertenkolleg “Sprache, Information, Logik” (SIL ) at Munich University, and special conference funds; furthermore to the following institutions and private sponsors for filling the remaining financial gaps: the Bayerisches Staatsministerium für Wis- senschaft, Forschung und Kunst, the Philosophy Department, University of Munich, the Gesellschaft von Freunden und Förderern der Universität München (Münchener Universitätsgesellschaft e.V. ), the University of the German Federal Armed Forces Munich, Apple Computer Germany, and Alpina Burkard Bovensiepen GmbH & Co. Many people helped in various ways to organize the conference and to prepare this volume. I wish to thank all of them, in particular Solomon Feferman for his encouragement, and Andrew Irvine for generous advice, in the early stages of the conference project. Ulrich Albert carried the main organizational burden of the con- ference as “front liner” on the internet and at the conference site; Sebastian Paasch was responsible for the technical part of the preparation of the volume. My thanks to both of them for their efficiency and their commitment far beyond the call of duty. Ulrich and Sebastian were assisted at different stages by Marie-Nicole Ehlers, Martin Fischer, Roland Kastler, Uwe Lück, Michaela Paivarinta, Hannes Petermair, Marek Polanski, Christian Tapp, Mai-Lan Thai, Julia Zink, and Mauricio Zuluaga. Daniel Mook helped with checking the English of non-native speakers. Iowe a special debt to Karl-Georg Niebergall, who was always close at hand for any advise on matters of content pertaining to the contributions of the volume, and whose philosophical seriousness, enthusiasm and unfailing technical judgment I have had the privilege to enjoy in many years of joint seminars and private discussions. Finally, I wish to extend my thanks to Dr. Manfred Karbe of Walter de Gruyter Publishers for supporting the decision to publish the volume, and for his sustained encouragement, advice, and patience during the phase of its preparation. May 2004 G. L. Table of Contents Preface v Godehard Link Introduction. Bertrand Russell—The Invention of Mathematical Philosophy 1 W. Hugh Woodin Set Theory after Russell: The Journey Back to Eden 29 Harvey M. Friedman AWayOut 49 Sy D. Friedman Completeness and Iteration in Modern Set Theory 85 Kai Hauser Was sind und was sollen (neue) Axiome? 93 Gerhard Jäger and Dieter Probst Iterating Operations in Admissible Set Theory without Foundation: A Further Aspect of Metapredicative Mahlo 119 Solomon Feferman Typical Ambiguity: Trying to Have Your Cake and Eat It Too 135 Karl-Georg Niebergall Is ZF Finitistically Reducible? 153 Tobias Hürter Inconsistency in the Real World 181 Michael Rathjen Predicativity, Circularity, and Anti-Foundation 191 John L. Bell Russell’s Paradox and Diagonalization in a Constructive Context 221 viii Table of Contents Peter Schuster and Helmut Schwichtenberg Constructive Solutions of Continuous Equations 227 Kai F. Wehmeier Russell’s Paradox in Consistent Fragments of Frege’s Grundgesetze der Arithmetik 247 Andrea Cantini On a Russellian Paradox about Propositions and Truth 259 Hartry Field The Consistency of the Naive Theory of Properties 285 Ulrich Blau The Significance of the Largest and Smallest Numbers for the Oldest Paradoxes 311 Nicholas Griffin The Prehistory of Russell’s Paradox 349 Gregory Landini Logicism’s ‘Insolubilia’ and Their Solution by Russell’s Substitutional Theory 373 Philippe de Rouilhan Substitution and Types: Russell’s Intermediate Theory 401 Francisco Rodríguez-Consuegra Propositional Ontology and Logical Atomism 417 Bernard Linsky Classes of Classes and Classes of Functions in Principia Mathematica 435 Allen P. Hazen A “Constructive” Proper