There's Something About Gödel

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.

• Prologue xiAcknowledgments xixPart I: The Gödelian Symphony 11 Foundations and Paradoxes 31 “This sentence is false” 62 The Liar and Gödel 83 Language and metalanguage 104 The axiomatic method, or how to get the non-obvious out of the obvious 135 Peano’s axioms … 146 … and the unsatisfied logicists, Frege and Russell 157 Bits of set theory 178 The Abstraction Principle 209 Bytes of set theory 2110 Properties, relations, functions, that is, sets again 2211 Calculating, computing, enumerating, that is, the notion of algorithm 2512 Taking numbers as sets of sets 2913 It’s raining paradoxes 3014 Cantor’s diagonal argument 3215 Self-reference and paradoxes 362 Hilbert 391 Strings of symbols 392 “… in mathematics there is no ignorabimus” 423 Gödel on stage 464 Our first encounter with the Incompleteness Theorem … 475 … and some provisos 513 Gödelization, or Say It with Numbers! 541 TNT 552 The arithmetical axioms of TNT and the “standard model” N 573 The Fundamental Property of formal systems 614 The Gödel numbering … 655 … and the arithmetization of syntax 694 Bits of Recursive Arithmetic … 711 Making algorithms precise 712 Bits of recursion theory 723 Church’s Thesis 764 The recursiveness of predicates, sets, properties, and relations 775 … And How It Is Represented in Typographical Number Theory 791 Introspection and representation 792 The representability of properties, relations, and functions … 813 … and the Gödelian loop 846 “I Am Not Provable” 861 Proof pairs 862 The property of being a theorem of TNT (is not recursive!) 873 Arithmetizing substitution 894 How can a TNT sentence refer to itself? 905 γ 936 Fixed point 957 Consistency and omega-consistency 978 Proving G 1 989 Rosser’s proof 1007 The Unprovability of Consistency and the “Immediate Consequences” of G1 and G2 1021 G 2 1022 Technical interlude 1053 “Immediate consequences” of G1 and G 2 1064 Undecidable 1 and undecidable 2 1075 Essential incompleteness, or the syndicate of mathematicians 1096 Robinson Arithmetic 1117 How general are Gödel’s results? 1128 Bits of Turing machine 1139 G1 and G2 in general 11610 Unexpected fish in the formal net 11811 Supernatural numbers 12112 The culpability of the induction scheme 12313 Bits of truth (not too much of it, though) 125Part II: The World after Gödel 1298 Bourgeois Mathematicians! The Postmodern Interpretations 1311 What is postmodernism? 1322 From Gödel to Lenin 1333 Is “Biblical proof” decidable? 1354 Speaking of the totality 1375 Bourgeois teachers! 1396 (Un)interesting bifurcations 1419 A Footnote to Plato 1461 Explorers in the realm of numbers 1462 The essence of a life 1483 “The philosophical prejudices of our times” 1514 From Gödel to Tarski 1535 Human, too human 15710 Mathematical Faith 1621 “I’m not crazy!” 1632 Qualified doubts 1663 From Gentzen to the Dialectica interpretation 1684 Mathematicians are people of faith 17011 Mind versus Computer: Gödel and Artificial Intelligence 1741 Is mind (just) a program? 1742 “Seeing the truth” and “going outside the system” 1763 The basic mistake 1794 In the haze of the transfinite 1815 “Know thyself”: Socrates and the inexhaustibility of mathematics 18512 Gödel versus Wittgenstein and the Paraconsistent Interpretation 1891 When geniuses meet … 1902 The implausible Wittgenstein 1913 “There is no metamathematics” 1944 Proof and prose 1965 The single argument 2016 But how can arithmetic be inconsistent? 2067 The costs and benefits of making Wittgenstein plausible 213Epilogue 214References 217Index 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)