Gödel's Theorem: An Incomplete Guide to Its Use and Abuse

184 Pages·2005·0.81 MB·English

by  Torkel 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
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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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