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There's Something About Gödel
The Complete Guide to the Incompleteness Theorem
"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
Francesco Berto (Author)
9781405197663, Wiley
Hardback, published 6 November 2009
256 pages
23.9 x 16 x 2.5 cm, 0.526 kg
"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)
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.
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
Subject Areas: Philosophy [HP]
