Prologue Acknowledgments Part I: The Gödelian Symphony: 1. Foundations and Paradoxes 1 "This sentence is false" 2 The Liar and Gödel 3 Language and metalanguage 4 The axiomatic method, or how to get the non-obvious out of the obvious 5 Peanos axioms 6 and the unsatisfied logicists, Frege and Russell 7 Bits of set theory 8 The 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 Its raining paradoxes 14 Cantors diagonal argument 15 Self-reference and paradoxes 2. Hilbert 1 Strings of symbols 2 " in mathematics there is no ignorabimus" 3 Gödel on stage 4 Our first encounter with the Incompleteness Theorem 5 and some provisos 3. Gödelization, or Say It with Numbers! 1 TNT 2 The arithmetical axioms of TNT and the "standard model" N 3 The Fundamental Property of formal systems 4 The Gödel numbering 5 and the arithmetization of syntax 4. Bits of Recursive Arithmetic 1 Making algorithms precise 2 Bits of recursion theory 3 Churchs Thesis 4 The recursiveness of predicates, sets, properties, and relations 5. And How It Is Represented in Typographical Number Theory 1 Introspection and representation 2 The representability of properties, relations, and functions 3 and the Gödelian loop 6. "I Am Not Provable" 1 Proof pairs 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution 4 How can a TNT sentence refer to itself? 5g 6 Fixed point 7 Consistency and omega-consistency 8 Proving G1 9 Rossers proof 7. The 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 Gödels results? 8 Bits of Turing machine 9 G1 and G2 in general 10 Unexpected fish in the formal net 11 Supernatural numbers 12 The culpability of the induction scheme 13 Bits of truth (not too much of it, though) Part II: The World after Gödel: 8. Bourgeois Mathematicians! The Postmodern Interpretations 1 What is postmodernism? 2 From Gödel 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 The essence of a life 3 "The philosophical prejudices of our times" 4 From Gödel to Tarski 5 Human, too human 10. Mathematical Faith 1 "Im not crazy!" 2 Qualified doubts 3 From Gentzen to the Dialectica interpretation 4 Mathematicians are people of faith 11. Mind versus Computer: Gödel and Artificial Intelligence 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system" 3 The basic mistake 4 In the haze of the transfinite 5 "Know thyself": Socrates and the inexhaustibility of mathematics 12. Gödel versus Wittgenstein and the Paraconsistent Interpretation 1 When geniuses meet 2 The implausible Wittgenstein 3 "There is no metamathematics" 4 Proof and prose 5 The single argument 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible Epilogue References Index
