There's Something About Gödel: The Complete Guide to the Incompleteness Theorem

257 Pages·2009·1.15 MB·English

by  Francesco Berto

THERE’S SOMETHING
ABOUT GÖDEL
There’s Something about G ö del: The Complete Guide to the Incompleteness Theorem Francesco Berto
© 2009 Francesco Berto. ISBN: 978-1-405-19766-3
99778811440055119977666633__11__pprreettoocc..iinndddd ii 77//2222//22000099 1111::3355::1166 AAMM

O U T
· M· E T H I N G A B
O
T H E R E ’ S S
L
E
D
O
G
T H E C OMMP LPE LTEETNEE SGSU TI DHEE OT OR ETMH E
I N C O
F R A N C E S C O B E R T O
A John Wiley & Sons, Ltd., Publication
99778811440055119977666633__11__pprreettoocc..iinndddd iiiiii 77//2222//22000099 1111::3355::1166 AAMM
This edition first published in English 2009
English translation © 2009 Francesco Berto
Original Italian text (Tutti pazzi per Gödel!) © 2008, Gius. Laterza & Figli, All rights reserved
Published by agreement with Marco Vigevani Agenzia Letteraria
Edition history: Gius. Laterza & Figli (1e in Italian, 2008); Blackwell Publishing Ltd (1e in
English, 2009)
Blackwell Publishing was acquired by John Wiley & Sons in February 2007. Blackwell’s
publishing program has been merged with Wiley’s global Scientific, Technical, and Medical
business to form Wiley-Blackwell.
Registered Office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United
Kingdom
Editorial Offices
350 Main Street, Malden, MA 02148-5020, USA
9600 Garsington Road, Oxford, OX4 2DQ, UK
The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
For details of our global editorial offices, for customer services, and for information about
how to apply for permission to reuse the copyright material in this book please see our
website at www.wiley.com/wiley-blackwell.
The right of Francesco Berto to be identified as the author of this work has been asserted in
accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without
the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears
in print may not be available in electronic books.
Designations used by companies to distinguish their products are often claimed as t rademarks.
All brand names and product names used in this book are trade names, service marks,
trademarks or registered trademarks of their respective owners. The publisher is not
associated with any product or vendor mentioned in this book. This publication is designed
to provide accurate and authoritative information in regard to the subject matter covered. It
is sold on the understanding that the publisher is not engaged in rendering professional
services. If professional advice or other expert assistance is required, the services of a
competent professional should be sought.
Library of Congress Cataloging-in-Publication Data
Berto, Francesco.
[Tutti pazzi per Gödel! English]
There’s something about Gödel! : the complete guide to the incompleteness theorem /
Francesco Berto.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4051-9766-3 (hardcover : alk. paper) — ISBN 978-1-4051-9767-0 (pbk. : alk. paper)
1. Incompleteness theorem. 2. Gödel’s theorem. 3. Mathematics--Philosophy. 4. Gödel, Kurt.
I. Title.
QA9.54B4713 2009
511.3–dc22
2009020156
A catalogue record for this book is available from the British Library.
Set in 10.5/13pt Garamond by SPi Publisher Services, Pondicherry, India
Printed in Singapore
1 2009
99778811440055119977666633__11__pprreettoocc..iinndddd iivv 77//2222//22000099 1111::3355::1166 AAMM
Contents
Prologue xi
Acknowledgments xix
Part I: The Gödelian Symphony 1
1 Foundations and Paradoxes 3
1 “This sentence is false” 6
2 The Liar and Gödel 8
3 Language and metalanguage 10
4 The axiomatic method, or how to get the non-obvious
out of the obvious 13
5 Peano’s axioms … 14
6 … and the unsatisfied logicists, Frege and Russell 15
7 Bits of set theory 17
8 The Abstraction Principle 20
9 Bytes of set theory 21
10 Properties, relations, functions, that is, sets again 22
11 Calculating, computing, enumerating, that is, the notion
of algorithm 25
12 Taking numbers as sets of sets 29
13 It’s raining paradoxes 30
14 Cantor’s diagonal argument 32
15 Self-reference and paradoxes 36
2 Hilbert 39
1 Strings of symbols 39
2 “… in mathematics there is no ignorabimus” 42
99778811440055119977666633__22__ttoocc..iinndddd vv 77//2200//22000099 88::2299::3399 PPMM
vi Contents
3 Gödel on stage 46
4 Our first encounter with the Incompleteness
Theorem … 47
5 … and some provisos 51
3 Gödelization, or Say It with Numbers! 54
1 TNT 55
2 The arithmetical axioms of TNT and the “standard
model” N 57
3 The Fundamental Property of formal systems 61
4 The Gödel numbering … 65
5 … and the arithmetization of syntax 69
4 Bits of Recursive Arithmetic … 71
1 Making algorithms precise 71
2 Bits of recursion theory 72
3 Church’s Thesis 76
4 The recursiveness of predicates, sets,
properties, and relations 77
5 … And How It Is Represented in Typographical
Number Theory 79
1 Introspection and representation 79
2 The representability of properties, relations,
and functions … 81
3 … and the Gödelian loop 84
6 “I Am Not Provable” 86
1 Proof pairs 86
2 The property of being a theorem
of TNT (is not recursive!) 87
3 Arithmetizing substitution 89
4 How can a TNT sentence refer to itself? 90
5 γ 93
6 Fixed point 95
99778811440055119977666633__22__ttoocc..iinndddd vvii 77//2200//22000099 88::2299::3399 PPMM
Contents vii
7 Consistency and omega-consistency 97
8 Proving G1 98
9 Rosser’s proof 100
7 The Unprovability of Consistency and the “Immediate
Consequences” of G1 and G2 102
1 G2 102
2 Technical interlude 105
3 “Immediate consequences” of G1 and G2 106
4 Undecidable and undecidable 107
1 2
5 Essential incompleteness, or the syndicate
of mathematicians 109
6 Robinson Arithmetic 111
7 How general are Gödel’s results? 112
8 Bits of Turing machine 113
9 G1 and G2 in general 116
10 Unexpected fish in the formal net 118
11 Supernatural numbers 121
12 The culpability of the induction scheme 123
13 Bits of truth (not too much of it, though) 125
Part II: The World after Gödel 129
8 Bourgeois Mathematicians! The Postmodern
Interpretations 131
1 What is postmodernism? 132
2 From Gödel to Lenin 133
3 Is “Biblical proof” decidable? 135
4 Speaking of the totality 137
5 Bourgeois teachers! 139
6 (Un)interesting bifurcations 141
9 A Footnote to Plato 146
1 Explorers in the realm of numbers 146
2 The essence of a life 148
3 “The philosophical prejudices of our times” 151
99778811440055119977666633__22__ttoocc..iinndddd vviiii 77//2200//22000099 88::2299::3399 PPMM
viii Contents
4 From Gödel to Tarski 153
5 Human, too human 157
10 Mathematical Faith 162
1 “I’m not crazy!” 163
2 Qualified doubts 166
3 From Gentzen to the Dialectica interpretation 168
4 Mathematicians are people of faith 170
11 Mind versus Computer: Gödel and Artificial Intelligence 174
1 Is mind (just) a program? 174
2 “Seeing the truth” and “going outside the system” 176
3 The basic mistake 179
4 In the haze of the transfinite 181
5 “Know thyself”: Socrates and the inexhaustibility
of mathematics 185
12 Gödel versus Wittgenstein and the Paraconsistent
Interpretation 189
1 When geniuses meet … 190
2 The implausible Wittgenstein 191
3 “There is no metamathematics” 194
4 Proof and prose 196
5 The single argument 201
6 But how can arithmetic be inconsistent? 206
7 The costs and benefits of making Wittgenstein plausible 213
Epilogue 214
References 217
Index 225
99778811440055119977666633__22__ttoocc..iinndddd vviiiiii 77//2200//22000099 88::2299::3399 PPMM