Berto, Francesco.
There's Something about G�odel : The Complete Guide to the Incompleteness Theorem. - 1 online resource (255 pages) - New York Academy of Sciences Ser. ; v.36 . - New York Academy of Sciences Ser. .
Intro -- GODEL -- Contents -- Prologue -- Acknowledgments -- Part I: The G�odelian Symphony -- 1 Foundations and Paradoxes -- 1 "This sentence is false" -- 2 The Liar and G�odel -- 3 Language and metalanguage -- 4 The 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 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 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 G�odel on stage -- 4 Our first encounter with the Incompleteness Theorem … -- 5 … and some provisos -- 3 G�odelization, 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 G�odel 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 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 G�odelian 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? -- 5 (Sd(B -- 6 Fixed point -- 7 Consistency and omega-consistency -- 8 Proving G1 -- 9 Rosser's 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 G�odel's 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 G�odel -- 8 Bourgeois Mathematicians! The Postmodern Interpretations -- 1 What is postmodernism? -- 2 From G�odel 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 G�odel 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: G�odel 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 G�odel 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.
9781444315011
Electronic books.
There's Something about G�odel : The Complete Guide to the Incompleteness Theorem. - 1 online resource (255 pages) - New York Academy of Sciences Ser. ; v.36 . - New York Academy of Sciences Ser. .
Intro -- GODEL -- Contents -- Prologue -- Acknowledgments -- Part I: The G�odelian Symphony -- 1 Foundations and Paradoxes -- 1 "This sentence is false" -- 2 The Liar and G�odel -- 3 Language and metalanguage -- 4 The 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 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 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 G�odel on stage -- 4 Our first encounter with the Incompleteness Theorem … -- 5 … and some provisos -- 3 G�odelization, 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 G�odel 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 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 G�odelian 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? -- 5 (Sd(B -- 6 Fixed point -- 7 Consistency and omega-consistency -- 8 Proving G1 -- 9 Rosser's 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 G�odel's 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 G�odel -- 8 Bourgeois Mathematicians! The Postmodern Interpretations -- 1 What is postmodernism? -- 2 From G�odel 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 G�odel 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: G�odel 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 G�odel 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.
9781444315011
Electronic books.