There's Something About Gödel

Berto's highly readable and lucid guide introduces students and the interested reader to Gödel's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.

• Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
• Discusses interpretations of the Theorem made by celebrated contemporary thinkers
• Sheds light on the wider extra-mathematical and philosophical implications of Gödel's theories
• Written in an accessible, non-technical style

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

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

7 Consistency and omega-consistency 97

8 Proving G 1 98

9 Rosser’s proof 100

7 The Unprovability of Consistency and the “Immediate Consequences” of G1 and G2 102

1 G 2 102

2 Technical interlude 105

3 “Immediate consequences” of G1 and G 2 106

4 Undecidable 1 and undecidable 2 107

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

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

"There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011)

“There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)

Francesco Berto teaches logic, ontology, and philosophy of mathematics at the universities of Aberdeen in Scotland, and Venice and Milan-San Raffaele in Italy. He holds a Chaire d'Excellence fellowship at CNRS in Paris, where he has taught ontology at the École Normale Supérieure, and he is a visiting professor at the Institut Wiener Kreis of the University of Vienna. He has written papers for American Philosophical Quarterly, Dialectica, The Philosophical Quarterly, the Australasian Journal of Philosophy, the European Journal of Philosophy, Philosophia Mathematica, Logique et Analyse, and Metaphysica, and runs the entries “Dialetheism” and “Impossible Worlds” in the Stanford Encyclopedia of Philosophy. His book How to Sell a Contradiction has won the 2007 Castiglioncello prize for the best philosophical book by a young philosopher. There’s Something About Gödel is a lucid and accessible guide to Gödel’s revolutionary Incompleteness Theorem, considered one of the most astounding argumentative sequences in the history of human thought. It is also an exploration of the most controversial alleged philosophical outcomes of the Theorem.

Divided into two parts, the first section introduces the reader to the Incompleteness Theorem – the argument that all mathematical systems contain statements which are true, yet which cannot be proved within the system. Berto describes the historical context surrounding Gödel's accomplishment, explains step-by-step the key aspects of the Theorem, and explores the technical issues of incompleteness in formal logical systems. The second half, The World After Gödel, considers some of the most famous – and infamous – claims arising from Gödel's theorem in the areas of the philosophy of mathematics, metaphysics, the philosophy of mind, Artificial Intelligence, and even sociology and politics.

This book requires only minimal knowledge of aspects of elementary logic, and is written in a user-friendly style that enables it to be read by those outside the academic field, as well as students of philosophy, logic, and computing.

"Berto's book will tell you everything you wanted to know about Gödel's theorem, but were too afraid to ask. Read it if you want your biggest organ pleasurably stimulated."
—Graham Priest, University of Melbourne