Gödel's theorem : an incomplete guide to its use and abuse

182 Pages·2005·0.85 MB·English

by  Franzén

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Gödel’s Theorem
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Gödel’s Theorem
An Incomplete Guide to
Its Use and Abuse
Torkel Franzén
Luleå University of Technology, Sweden
AKPeters
Wellesley,Massachusetts
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Editorial, Sales, and Customer Service Office
A K Peters, Ltd.
888 Worcester Street, Suite 230
Wellesley, MA 02482
www.akpeters.com
Copyright © 2005 by A K Peters, Ltd.
All rights reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form, electronic or mechanical, including
photocopying, recording, or by any information storage and retrieval system,
without written permission from the copyright owner.
Library of Congress Cataloging-in-Publication Data
Franz´en, Torkel.
G¨odel’s theorem : an incomplete guide to its use and abuse / Torkel Franz´en.
p. cm.
Includes bibliographical references and index.
ISBN 1-56881-238-8
1. G¨odel’s theorem. 2. Incompleteness theorems. I. Title.
QA9.65.F73 2005
511.3–dc22
2005045868
About the cover: Sampled on the cover are various arguments and reflections
invoking G¨odels theorem. The aim of the book is to allow a reader with no
knowledge of formal logic to form a sober and soundly based opinion of these
uses and abuses.
Printed in the United States of America
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For Marcia,
still bright and beautiful, in her joy, in her despair
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Contents
Preface ix
1 Introduction 1
1.1 The Incompleteness Theorem . . . . . . . . . . . . . . . . . 1
1.2 Go¨del’s Life and Work . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Rest of the Book . . . . . . . . . . . . . . . . . . . . . 7
2 The Incompleteness Theorem: An Overview 9
2.1 Arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The First Incompleteness Theorem . . . . . . . . . . . . . . 15
2.3 Some Limitations of the First Incompleteness Theorem . . . 24
2.4 The First Incompleteness Theorem and Mathematical Truth 28
2.5 The First Incompleteness Theorem and Hilbert’s
Non Ignorabimus . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 The Second Incompleteness Theorem . . . . . . . . . . . . . 34
2.7 Proving the Incompleteness Theorem . . . . . . . . . . . . . 39
2.8 A “Postmodern Condition”? . . . . . . . . . . . . . . . . . . 50
2.9 Mind vs. Computer . . . . . . . . . . . . . . . . . . . . . . 55
2.10 Some Later Developments . . . . . . . . . . . . . . . . . . . 56
3 Computability, Formal Systems, and Incompleteness 59
3.1 Strings of Symbols . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Computable Enumerability and Decidability . . . . . . . . . 61
3.3 Undecidable Sets . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Computability and the First Incompleteness Theorem . . . 72
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viii Contents
4 Incompleteness Everywhere 77
4.1 The Incompleteness Theorem Outside Mathematics . . . . . 77
4.2 “Human Thought” and the Incompleteness Theorem . . . . 80
4.3 Generalized Go¨del Sentences . . . . . . . . . . . . . . . . . 83
4.4 Incompleteness and the TOE . . . . . . . . . . . . . . . . . 87
4.5 Theological Applications . . . . . . . . . . . . . . . . . . . . 90
5 Skepticism and Confidence 97
5.1 The Second Incompleteness Theorem . . . . . . . . . . . . . 97
5.2 Skepticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Consistency Proofs . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Inexhaustibility . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Gödel, Minds, and Computers 115
6.1 Go¨del and the UTM . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Penrose’s “Second Argument” . . . . . . . . . . . . . . . . . 119
6.3 Inexhaustibility Revisited . . . . . . . . . . . . . . . . . . . 121
6.4 Understanding One’s Own Mind . . . . . . . . . . . . . . . 124
7 Gödel’s Completeness Theorem 127
7.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 PA as a First-Order Theory . . . . . . . . . . . . . . . . . . 131
7.3 Incompleteness and Nonstandard Models. . . . . . . . . . . 134
8 Incompleteness, Complexity, and Infinity 137
8.1 Incompleteness and Complexity . . . . . . . . . . . . . . . . 137
8.2 Incompleteness and Randomness . . . . . . . . . . . . . . . 145
8.3 Incompleteness and Infinity . . . . . . . . . . . . . . . . . . 149
A Appendix 155
A.1 The Language of Elementary Arithmetic . . . . . . . . . . . 155
A.2 The First Incompleteness Theorem . . . . . . . . . . . . . . 157
A.3 Goldbach-Like Statements . . . . . . . . . . . . . . . . . . . 160
References 165
Index 169
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Preface
My excuse for presenting yet another book on Go¨del’s incompleteness the-
orem written for a general audience is that no existing book both explains
the theorem from a mathematical point of view, including that of com-
putability theory, and comments on a fairly wide selection of the many
invocations of the incompleteness theorem outside mathematics.
Toaconsiderableextent,thebookreflectsmyexperiencesovertheyears
ofreadingandcommentingonreferencestotheincompletenesstheoremon
the Internet. Quotations from named sources that as far as I know exist
onlyinelectronicformarenotaccompaniedbyanyURLs,sincesuchoften
become obsolete. However, using a search engine, the reader can easily
locate any quoted text that is still extant somewhere on the Internet.
In quite a few cases, comments that I have encountered on the Internet
ininformalcontextsarequoted(sometimesinslightlyeditedform)without
attribution. They are used to represent commonly occurring ideas and
arguments.
In thanking those who have helped me write this book, I must begin
withthemanypeoplediscussingGo¨del’stheoremontheInternet, whether
named in the book or not, without whose contributions it is unlikely that
thebookwouldeverhaveappeared. IalsothankAndrewBoucher,Damjan
Bojadziev,AlexBlum,JeffDalton,SolomonFeferman,JohnHarrison,Jef-
frey Ketland, Panu Raatikainen, and Charles Silver for helpful comments.
In writing the book, I have drawn on the resources of Lule˚a University of
Technology in several essential ways, for which I am grateful.
For any remaining instances of incompleteness or inconsistency in the
book,Iconsidermyselfentirelyblameless,sinceafterall,Go¨delprovedthat
any book on the incompleteness theorem must be incomplete or inconsis-
tent. Well, maybe not. Although the book will perhaps in part be heavy
going for readers not used to mathematical proofs and definitions, I hope
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x Preface
itwillgiveevencasualreadersabasisforjudgingforthemselvesthemerits
of such nonmathematical appeals to the incompleteness theorem and an
appreciation of some of the philosophical and mathematical perspectives
opened up by the theorem.
Torkel Franz´en
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