LEADER 05116nam 22005893u 450 001 9910792101703321 005 20230120031134.0 010 $a1-4443-1502-1 035 $a(CKB)2580000000004811 035 $a(EBL)694276 035 $a(OCoLC)819641609 035 $a(SSID)ssj0000506364 035 $a(PQKBManifestationID)11332879 035 $a(PQKBTitleCode)TC0000506364 035 $a(PQKBWorkID)10515346 035 $a(PQKB)11195842 035 $a(MiAaPQ)EBC694276 035 $a(MiAaPQ)EBC7104594 035 $a(EXLCZ)992580000000004811 100 $a20130418d2009|||| u|| | 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aThere's Something About Gödel$b[electronic resource] $eThe Complete Guide to the Incompleteness Theorem 210 $aHoboken $cJohn Wiley & Sons$d2009 215 $a1 online resource (255 p.) 300 $aDescription based upon print version of record. 311 $a1-4051-9766-8 327 $aGODEL; Contents; Prologue; Acknowledgments; Part I: The Go?delian Symphony; 1 Foundations and Paradoxes; 1 "This sentence is false"; 2 The Liar and Go?del; 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 327 $a14 Cantor's diagonal argument 15 Self-reference and paradoxes; 2 Hilbert; 1 Strings of symbols; 2 "... in mathematics there is no ignorabimus"; 3 Go?del on stage; 4 Our first encounter with the Incompleteness Theorem ...; 5 ... and some provisos; 3 Go?delization, 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 Go?del 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 327 $a4 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 Go?delian 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 ?; 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 327 $a1 G22 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 Go?del'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 Go?del; 8 Bourgeois Mathematicians! The Postmodern Interpretations; 1 What is postmodernism?; 2 From Go?del to Lenin 327 $a3 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 Go?del 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: Go?del and Artificial Intelligence; 1 Is mind (just) a program?; 2 "Seeing the truth" and "going outside the system" 327 $a3 The basic mistake 330 $aBerto's highly readable and lucid guide introduces students and the interested reader to Go?del's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Go?del's arguments.Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters. Discusses interpretations of the Theorem made by celebrated contemporary thinkers. Sheds light on the wider extra-mathematical and philosophical implications of Go?del's theories. 606 $aIncompleteness theorems$xPhilosophy 606 $aGo?del's theorem 606 $aMathematics 615 0$aIncompleteness theorems$xPhilosophy 615 0$aGo?del's theorem. 615 0$aMathematics 676 $a511.3 700 $aBerto$b Francesco$0286940 801 0$bAU-PeEL 801 1$bAU-PeEL 801 2$bAU-PeEL 906 $aBOOK 912 $a9910792101703321 996 $aThere's Something About Gödel$93682440 997 $aUNINA