There's Something about Godel: The Complete Guide to the Incompleteness Theorem

247 Pages·2009·2.138 MB·English

by  Francesco Berto(auth.)

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
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A John Wiley & Sons, Ltd., Publication
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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)
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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
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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
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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
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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
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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
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For Marta Rossi
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Prologue
In 1930, a youngster of about 23 proved a theorem in mathematical
logic. His result was published the following year in an Austrian scien-
tific review. The title of the paper (written in German) containing the
proof, translated, was: “On Formally Undecidable Propositions of
Principia mathematica and Related Systems I.” Principia mathemat-
ica is a big three-volume book, written by the famous philosopher
Bertrand Russell and by the mathematician Alfred North Whitehead,
and including a system of logical-mathematical axioms within which all
mathematics was believed to be expressible and provable. The theorem
proved by the youngster referred to (a modification of ) that system. It
is known to the world as the Incompleteness Theorem, and its proof is
one of the most astonishing argumentations in the history of human
thought. The unknown youngster’s name was Kurt Gödel, and the book
you are now holding in your hands is a guide to his Theorem.
In fact, in his paper Gödel presented a sequence of theorems, but
the most important among them are Theorem VI, and the last of the
series, Theorem XI. These are nowadays called, respectively, Gödel’s
First and Second Incompleteness Theorems. When scholars simply
talk of Gödel’s Incompleteness Theorem, they usually refer to the con-
junction of the two.
Gödel’s Theorem is a technical result. Its original proof included
such innovative techniques that in 1931 (and for years to follow)
many logicians, philosophers, and mathematicians of the time – from
Ernst Zermelo to Rudolf Carnap and Russell himself – had a hard time
understanding exactly what had been accomplished. Nowadays, (the
proof of ) the Theorem is not considered too complex, and all logi-
cians have met it, in some version or other, in some textbook of inter-
mediate logic. Nevertheless, it remains a technical fact, inaccessible to
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xii Prologue
amateurs. It is therefore surprising how much this proof has changed
our u nderstanding of logic, perhaps of mathematics and, according to
some, even of ourselves and our world.
Everyone agrees, to begin with, that Gödel’s result is a terrific
achievement. Gödel’s official biographer John Dawson has noted that
it seems customary to invoke geological metaphors in this context.
Here is Karl Popper:
The work on formally undecidable propositions was felt as an earth-
quake, particularly also by Carnap.1
And here is John von Neumann, Princeton’s “human calculator,” in a
speech he gave in 1951 when Gödel was given the Einstein Award:
Kurt Gödel’s achievement in modern logic is singular and monumental –
indeed it is more than a monument, it is a landmark which will remain
visible far in space and time.2
As for the legendary friendship between Gödel and Einstein, the latter
once confessed to the economist Oskar Morgenstern that he had gone
to Princeton’s Institute for Advanced Study just “um das Privileg zu
haben, mit Gödel zu Fuss nach Hause gehen zu dürfen” – to have the
privilege of walking home with Gödel.
But this is not enough. Other technical results in contemporary
mathematics have received attention from popular books and news-
papers. Recently, this happened with Andrew Wiles’ proof of Fermat’s
Last Theorem (a 130 page demonstration – in fact, a proof of the
Taniyama–Shimura conjecture on elliptic curves, which in its turn
entails Fermat’s Theorem) that has inspired a nice book by Simon
Singh.3 However, no mathematical result has ever had extra-mathematical
1 Quoted in Dawson (1984), p. 74.
2 The New York Times, March 15, 1951, p. 31.
3 Fermat’s Last Theorem (which before Wiles’ proof should rather have been called
Fermat’s conjecture) says that no equation of the simple form xn + yn = zn has solu-
tions in positive integers for n greater than 2. Pierre de Fermat became famous
because he claimed he had a “marvelous proof” of this fact, which unfortunately the
page margin of the book on Diophantine equations he was reading was too narrow
to contain.
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