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INFINITY AND TRUTH LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Chitat Chong and Wing Keung To Institute for Mathematical Sciences National University of Singapore ISSN: 1793-0758 Published Vol. 14 Computational Prospects of Infinity — Part I: Tutorials edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 15 Computational Prospects of Infinity — Part II: Presented Talks edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 16 Mathematical Understanding of Infectious Disease Dynamics edited by Stefan Ma & Yingcun Xia Vol. 17 Interface Problems and Methods in Biological and Physical Flows edited by Boo Cheong Khoo, Zhilin Li & Ping Lin Vol. 18 Random Matrix Theory and Its Applications: Multivariate Statistics and Wireless Communications edited by Zhidong Bai, Yang Chen & Ying-Chang Liang Vol. 19 Braids: Introductory Lectures on Braids, Configurations and Their Applications edited by A Jon Berrick, Frederick R Cohen, Elizabeth Hanbury, Yan-Loi Wong & Jie Wu Vol. 20 Mathematical Horizons for Quantum Physics edited by H Araki, B-G Englert, L-C Kwek & J Suzuki Vol. 21 Environmental Hazards: The Fluid Dynamics and Geophysics of Extreme Events edited by H K Moffatt & E Shuckburgh Vol. 22 Multiscale Modeling and Analysis for Materials Simulation edited by Weizhu Bao & Qiang Du Vol. 23 Geometry, Topology and Dynamics of Character Varieties edited by W Goldman, C Series & S P Tan Vol. 24 Complex Quantum Systems: Analysis of Large Coulomb Systems edited by Heinz Siedentop Vol. 25 Infinity and Truth edited by Chitat Chong, Qi Feng, Theodore A Slaman & W Hugh Woodin *For the complete list of titles in this series, please go to http://www.worldscientific.com/series/LNIMSNUS Lecture Notes Series, Institute for Mathematical Sciences, Vol. 25 National University of Singapore INFINITY AND TRUTH Editors Chitat Chong Qi Feng National University of Singapore, Singapore Theodore A. Slaman W. Hugh Woodin University of California, Berkeley, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Infinity and truth / edited by Chitat Chong (National University of Singapore, Singapore), Qi Feng (National University of Singapore, Singapore), Theodore A. Slaman (University of California, Berkeley, USA), and W. Hugh Woodin (University of California, Berkeley, USA). pages cm. -- (Lecture notes series (Institute for Mathematical Sciences, National University of Singapore) ; volume 25) “Based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011.” Includes bibliographical references. ISBN 978-9814571036 (hardcover : alk. paper) 1. Logic, Symbolic and mathematical--Congresses. 2. Mathematics--Philosophy--Congresses. 3. Set theory--Congresses. 4. Axiomatic set theory--Congresses. I. Chong, C.-T. (Chi-Tat), 1949– editor of compilation. II. Feng, Qi, 1955– editor of compilation. III. Slaman, T. A. (Theodore Allen), 1954– editor of compilation. IV. Woodin, W. H. (W. Hugh), editor of compilation. QA9.A1I54 2013 510.1--dc23 2013041952 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore October18,2013 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in 00a˙toc CONTENTS Foreword vii Preface ix Section I. Invited Lectures Absoluteness, Truth, and Quotients 1 Ilijas Farah A Multiverse Perspective on the Axiom of Constructibility 25 Joel David Hamkins Hilbert, Bourbaki and the Scorning of Logic 47 A. R. D. Mathias Toward Objectivity in Mathematics 157 Stephen G. Simpson Sort Logic and Foundations of Mathematics 171 Jouko V¨aa¨n¨anen Reasoning about Constructive Concepts 187 Nik Weaver Perfect Infinities and Finite Approximation 199 Boris Zilber Section II. Special Session An Objective Justification for Actual Infinity? 225 Stephen G. Simpson Oracle Questions 229 Theodore Slaman and W. Hugh Woodin v May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk October3,2013 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in 00b˙foreword FOREWORD TheInstituteforMathematicalSciences(IMS)attheNationalUniversityof Singaporewasestablishedon1July2000.Itsmissionistofostermathemat- icalresearch,bothfundamentalandmultidisciplinary,particularlyresearch that links mathematics to other efforts of human endeavor, and to nurture the growth of mathematical talent and expertise in research scientists, as well as to serve as a platform for research interaction between scientists in Singapore and the international scientific community. TheInstituteorganizesthematicprogramsoflongerdurationandmath- ematicalactivitiesincludingworkshopsandpubliclectures.Theprogramor workshopthemes are selected from among areasat the forefrontof current researchin the mathematical sciences and their applications. EachvolumeoftheIMSLectureNotesSeriesisacompendiumofpapers basedonlecturesortutorialsdeliveredataprogram/workshop.Itbringsto the internationalresearchcommunity originalresults or expositoryarticles on a subject of current interest. These volumes also serve as a record of activities that took place at the IMS. WehopethatthroughtheregularpublicationoftheseLectureNotesthe Institutewillachieve,inpart,itsobjectiveofreachingouttothecommunity of scholars in the promotion of researchin the mathematical sciences. August 2013 Chitat Chong Wing Keung To Series Editors vii May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk October21,2013 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in 00c˙preface PREFACE AWorkshop on Infinity and TruthwasheldattheInstituteforMathemat- ical Sciences from 25 to 29 July 2011. The theme of the Workshop was on basic foundational questions such as: (i) What is the nature of mathemati- cal truth and how does one resolve questions that are formally unsolvable within ZFC, an example being the Continuum Hypothesis? And (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view, such as Platonism or Formalism, over others? Leading experts repre- senting areas across the mathematical and philosophical logic participated in this Workshop. Altogether there were ten lectures delivered by Ilijas Farah, Joel Hamkins, LeoA. Harrington, Kai Hauser, MenachemMagidor, StephenG.Simpson,JoukoVa¨¨ana¨nen,NikWeaver,W.HughWoodin,and Boris Zilber. The Workshop concluded with a special session on problems that would drive progress in foundational research (“Oracle Questions”). This volume consists of two sections: (I) written articles of talks pre- sented at the Workshop received by the editors, and (II) a contribution by Stephen G. Simpson as well as a compendium of foundational questions discussed during the concluding session prepared by Theodore A. Slaman and W. Hugh Woodin. The volume also includes an article submitted by A. R. D. Mathias, who was invited but not able to attend the Workshop. The meeting was supported by a grant from the John Templeton Foun- dation and formed part of the larger program Computational Prospects of Infinity II hosted and funded by the Institute for Mathematical Sciences. We wish to record here our gratitude for the support. August 2013 Chitat Chong National University of Singapore, Singapore Qi Feng∗ National University of Singapore, Singapore Theodore A. Slaman University of California at Berkeley, USA W. Hugh Woodin University of California at Berkeley, USA Editors ∗Currentaddress:ChineseAcademyofSciences,China. ix
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