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• W574013885 abstract "Prologue. Acknowledgments. Part I: Godelian Symphony. 1 Foundations and Paradoxes. 1 This sentence is false. 2 Liar and Godel. 3 Language and metalanguage. 4 axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms 6 and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 ... in mathematics there is no ignorabimus . 3 Godel on stage. 4 Our first encounter with the Incompleteness Theorem 5 and some provisos. 3 Godelization, or Say It with Numbers! 1 TNT. 2 arithmetical axioms of TNT and the standard model N. 3 Fundamental Property of formal systems. 4 Godel numbering 5 and the arithmetization of syntax. 4 Bits of Recursive Arithmetic 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 recursiveness of predicates, sets, properties, and relations. 5 And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 representability of properties, relations, and functions 3 and the Godelian loop. 6 I Am Not Provable. 1 Proof pairs. 2 property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 gamma 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 Unprovability of Consistency and the Immediate Consequences of G1 and G2. 1 G2. 2 Technical interlude. 3 Immediate consequences of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Godel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 culpability of the induction scheme. 13 Bits of (not too much of it, though). Part II: World after Godel. 8 Bourgeois Mathematicians! Postmodern Interpretations. 1 What is postmodernism? 2 From Godel to Lenin. 3 Is Biblical proof decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 essence of a life. 3 The philosophical prejudices of our times. 4 From Godel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 I'm not crazy! 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Godel and Artificial Intelligence. 1 Is mind (just) a program? 2 Seeing the truth and going outside the system. 3 basic mistake. 4 In the haze of the transfinite. 5 Know thyself: Socrates and the inexhaustibility of mathematics. 12 Godel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet 2 implausible Wittgenstein. 3 There is no metamathematics. 4 Proof and prose. 5 single argument. 6 But how can arithmetic be inconsistent? 7 costs and benefits of making Wittgenstein plausible. Epilogue. References. Index." @default.
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• W574013885 date "2009-11-09" @default.
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• W574013885 title "There's Something About Gödel: The Complete Guide to the Incompleteness Theorem" @default.
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