Infinity and truth

245 Pages·2014·1.52 MB·English

by  Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin, Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin (eds)

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
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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
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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
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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