{"product_id":"theres-something-about-godel-isbn-9781405197663","title":"There's Something About Gödel","description":"\u003cp\u003e\u003cb\u003eBerto's highly readable and lucid guide introduces students and the interested reader to Gödel's celebrated \u003ci\u003eIncompleteness Theorem\u003c\/i\u003e, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.\u003c\/b\u003e\u003c\/p\u003e \u003cul\u003e \u003cli\u003eOffers a clear understanding of this difficult subject by presenting each of the key steps of the \u003ci\u003eTheorem\u003c\/i\u003e in separate chapters\u003c\/li\u003e \u003cli\u003eDiscusses interpretations of the \u003ci\u003eTheorem\u003c\/i\u003e made by celebrated contemporary thinkers\u003c\/li\u003e \u003cli\u003eSheds light on the wider extra-mathematical and philosophical implications of Gödel's theories\u003c\/li\u003e \u003cli\u003eWritten in an accessible, non-technical style\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003ePrologue xi\u003c\/p\u003e \u003cp\u003eAcknowledgments xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I: The Gödelian Symphony 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Foundations and Paradoxes 3\u003c\/p\u003e \u003cp\u003e1 “This sentence is false” 6\u003c\/p\u003e \u003cp\u003e2 The Liar and Gödel 8\u003c\/p\u003e \u003cp\u003e3 Language and metalanguage 10\u003c\/p\u003e \u003cp\u003e4 The axiomatic method, or how to get the non-obvious out of the obvious 13\u003c\/p\u003e \u003cp\u003e5 Peano’s axioms … 14\u003c\/p\u003e \u003cp\u003e6 … and the unsatisfied logicists, Frege and Russell 15\u003c\/p\u003e \u003cp\u003e7 Bits of set theory 17\u003c\/p\u003e \u003cp\u003e8 The Abstraction Principle 20\u003c\/p\u003e \u003cp\u003e9 Bytes of set theory 21\u003c\/p\u003e \u003cp\u003e10 Properties, relations, functions, that is, sets again 22\u003c\/p\u003e \u003cp\u003e11 Calculating, computing, enumerating, that is, the notion of algorithm 25\u003c\/p\u003e \u003cp\u003e12 Taking numbers as sets of sets 29\u003c\/p\u003e \u003cp\u003e13 It’s raining paradoxes 30\u003c\/p\u003e \u003cp\u003e14 Cantor’s diagonal argument 32\u003c\/p\u003e \u003cp\u003e15 Self-reference and paradoxes 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Hilbert 39\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Strings of symbols 39\u003c\/p\u003e \u003cp\u003e2 “… in mathematics there is no \u003ci\u003eignorabimus\u003c\/i\u003e” 42\u003c\/p\u003e \u003cp\u003e3 Gödel on stage 46\u003c\/p\u003e \u003cp\u003e4 Our first encounter with the Incompleteness Theorem … 47\u003c\/p\u003e \u003cp\u003e5 … and some provisos 51\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Gödelization, or Say It with Numbers! 54\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 \u003cb\u003eTNT\u003c\/b\u003e 55\u003c\/p\u003e \u003cp\u003e2 The arithmetical axioms of \u003cb\u003eTNT\u003c\/b\u003e and the “standard model” N 57\u003c\/p\u003e \u003cp\u003e3 The Fundamental Property of formal systems 61\u003c\/p\u003e \u003cp\u003e4 The Gödel numbering … 65\u003c\/p\u003e \u003cp\u003e5 … and the arithmetization of syntax 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Bits of Recursive Arithmetic … 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Making algorithms precise 71\u003c\/p\u003e \u003cp\u003e2 Bits of recursion theory 72\u003c\/p\u003e \u003cp\u003e3 Church’s Thesis 76\u003c\/p\u003e \u003cp\u003e4 The recursiveness of predicates, sets, properties, and relations 77\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 … And How It Is Represented in Typographical Number Theory 79\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introspection and representation 79\u003c\/p\u003e \u003cp\u003e2 The representability of properties, relations, and functions … 81\u003c\/p\u003e \u003cp\u003e3 … and the Gödelian loop 84\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 “I Am Not Provable” 86\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Proof pairs 86\u003c\/p\u003e \u003cp\u003e2 The property of being a theorem of \u003cb\u003eTNT\u003c\/b\u003e (is not recursive!) 87\u003c\/p\u003e \u003cp\u003e3 Arithmetizing substitution 89\u003c\/p\u003e \u003cp\u003e4 How can a \u003cb\u003eTNT\u003c\/b\u003e sentence refer to itself? 90\u003c\/p\u003e \u003cp\u003e5 γ 93\u003c\/p\u003e \u003cp\u003e6 Fixed point 95\u003c\/p\u003e \u003cp\u003e7 Consistency and omega-consistency 97\u003c\/p\u003e \u003cp\u003e8 Proving G 1 98\u003c\/p\u003e \u003cp\u003e9 Rosser’s proof 100\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 The Unprovability of Consistency and the “Immediate Consequences” of G1 and G2 102\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 G 2 102\u003c\/p\u003e \u003cp\u003e2 Technical interlude 105\u003c\/p\u003e \u003cp\u003e3 “Immediate consequences” of G1 and G 2 106\u003c\/p\u003e \u003cp\u003e4 Undecidable 1 and undecidable 2 107\u003c\/p\u003e \u003cp\u003e5 Essential incompleteness, or the syndicate of mathematicians 109\u003c\/p\u003e \u003cp\u003e6 Robinson Arithmetic 111\u003c\/p\u003e \u003cp\u003e7 How general are Gödel’s results? 112\u003c\/p\u003e \u003cp\u003e8 Bits of Turing machine 113\u003c\/p\u003e \u003cp\u003e9 G1 and G2 in general 116\u003c\/p\u003e \u003cp\u003e10 Unexpected fish in the formal net 118\u003c\/p\u003e \u003cp\u003e11 Supernatural numbers 121\u003c\/p\u003e \u003cp\u003e12 The culpability of the induction scheme 123\u003c\/p\u003e \u003cp\u003e13 Bits of truth (not too much of it, though) 125\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II: The World after Gödel 129\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Bourgeois Mathematicians! The Postmodern Interpretations 131\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 What is postmodernism? 132\u003c\/p\u003e \u003cp\u003e2 From Gödel to Lenin 133\u003c\/p\u003e \u003cp\u003e3 Is “Biblical proof” decidable? 135\u003c\/p\u003e \u003cp\u003e4 Speaking of the totality 137\u003c\/p\u003e \u003cp\u003e5 Bourgeois teachers! 139\u003c\/p\u003e \u003cp\u003e6 (Un)interesting bifurcations 141\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 A Footnote to Plato 146\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Explorers in the realm of numbers 146\u003c\/p\u003e \u003cp\u003e2 The essence of a life 148\u003c\/p\u003e \u003cp\u003e3 “The philosophical prejudices of our times” 151\u003c\/p\u003e \u003cp\u003e4 From Gödel to Tarski 153\u003c\/p\u003e \u003cp\u003e5 Human, too human 157\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Mathematical Faith 162\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 “I’m not crazy!” 163\u003c\/p\u003e \u003cp\u003e2 Qualified doubts 166\u003c\/p\u003e \u003cp\u003e3 From Gentzen to the Dialectica interpretation 168\u003c\/p\u003e \u003cp\u003e4 Mathematicians are people of faith 170\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Mind versus Computer: Gödel and Artificial Intelligence 174\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Is mind (just) a program? 174\u003c\/p\u003e \u003cp\u003e2 “Seeing the truth” and “going outside the system” 176\u003c\/p\u003e \u003cp\u003e3 The basic mistake 179\u003c\/p\u003e \u003cp\u003e4 In the haze of the transfinite 181\u003c\/p\u003e \u003cp\u003e5 “Know thyself”: Socrates and the inexhaustibility of mathematics 185\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Gödel versus Wittgenstein and the Paraconsistent Interpretation 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 When geniuses meet … 190\u003c\/p\u003e \u003cp\u003e2 The implausible Wittgenstein 191\u003c\/p\u003e \u003cp\u003e3 “There is no metamathematics” 194\u003c\/p\u003e \u003cp\u003e4 Proof and prose 196\u003c\/p\u003e \u003cp\u003e5 The single argument 201\u003c\/p\u003e \u003cp\u003e6 But how can arithmetic be inconsistent? 206\u003c\/p\u003e \u003cp\u003e7 The costs and benefits of making Wittgenstein plausible 213\u003c\/p\u003e \u003cp\u003eEpilogue 214\u003c\/p\u003e \u003cp\u003eReferences 217\u003c\/p\u003e \u003cp\u003eIndex 225\u003c\/p\u003e  \"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)\u003cbr\u003e \u003cbr\u003e   \u003cp\u003e“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.” (\u003ci\u003eTimes Higher Education Supplement\u003c\/i\u003e, February 2010)\u003c\/p\u003e \u003cb\u003eFrancesco Berto\u003c\/b\u003e 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 \u003ci\u003eChaire d'Excellence\u003c\/i\u003e 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 \u003ci\u003eAmerican Philosophical Quarterly\u003c\/i\u003e, \u003ci\u003eDialectica\u003c\/i\u003e, \u003ci\u003eThe Philosophical Quarterly\u003c\/i\u003e, the \u003ci\u003eAustralasian Journal of Philosophy\u003c\/i\u003e, the \u003ci\u003eEuropean Journal of Philosophy\u003c\/i\u003e, \u003ci\u003ePhilosophia Mathematica\u003c\/i\u003e, \u003ci\u003eLogique et Analyse\u003c\/i\u003e, and \u003ci\u003eMetaphysica\u003c\/i\u003e, and runs the entries “Dialetheism” and “Impossible Worlds” in the \u003ci\u003eStanford Encyclopedia of Philosophy\u003c\/i\u003e. His book \u003ci\u003eHow to Sell a Contradiction\u003c\/i\u003e has won the 2007 Castiglioncello prize for the best philosophical book by a young philosopher.  \u003ci\u003eThere’s Something About Gödel\u003c\/i\u003e is a lucid and accessible guide to Gödel’s revolutionary \u003ci\u003eIncompleteness Theorem\u003c\/i\u003e, 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 \u003ci\u003eTheorem\u003c\/i\u003e.  \u003cp\u003eDivided into two parts, the first section introduces the reader to the \u003ci\u003eIncompleteness Theorem\u003c\/i\u003e – 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 \u003ci\u003eTheorem\u003c\/i\u003e, and explores the technical issues of incompleteness in formal logical systems. The second half, \u003ci\u003eThe World After Gödel\u003c\/i\u003e, 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.\u003c\/p\u003e \u003cp\u003eThis 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.\u003c\/p\u003e  \"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.\"\u003cbr\u003e —\u003cb\u003eGraham Priest\u003c\/b\u003e, \u003ci\u003eUniversity of Melbourne\u003c\/i\u003e","brand":"Wiley-Blackwell","offers":[{"title":"Default Title","offer_id":47990382592229,"sku":"NP9781405197663","price":86.5,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781405197663.jpg?v=1761787601","url":"https:\/\/k12savings.com\/products\/theres-something-about-godel-isbn-9781405197663","provider":"K12savings","version":"1.0","type":"link"}