LEADER 04047nam 2200481 450 001 996499868803316 005 20230508170301.0 010 $a9783031134173$b(electronic bk.) 010 $z9783031134166 035 $a(MiAaPQ)EBC7144570 035 $a(Au-PeEL)EBL7144570 035 $a(CKB)25456759600041 035 $a(PPN)266349021 035 $a(EXLCZ)9925456759600041 100 $a20230408d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe theory of Go?del /$fCarlo Cellucci 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (164 pages) 225 1 $aSynthese library ;$vVolume 470 311 08$aPrint version: Cellucci, Carlo The Theory of Gödel Cham : Springer International Publishing AG,c2023 9783031134166 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- References -- Contents -- 1 First-Order Logic -- 1.1 First-Order Languages -- 1.2 Free Variables -- 1.3 Substitution -- 1.4 A Deductive Calculus -- 1.5 Connective Rules -- 1.6 Quantifier Rules -- 1.7 Substitutivity of Equivalence -- 1.8 Equality -- 2 Completeness -- 2.1 Interpretations -- 2.2 Substitution Properties -- 2.3 Soundness -- 2.4 Consistency -- 2.5 Rich Sets -- 2.6 Completeness Theorem -- 2.7 Isomorphisms of Interpretations -- 2.8 Elementary Equivalence -- 2.9 Definability in an Interpretation -- 3 First-Order Theories -- 3.1 Generalities on First-Order Theory -- 3.2 Extensions of Theories -- 3.3 Definitional Extensions -- 4 Primitive Recursive Arithmetic -- 4.1 Primitive Recursive Functions -- 4.2 The Theories PRA and PA -- 4.3 Elementary Properties of PRA -- 4.4 Developing Arithmetic in PRA -- 4.5 Bounded Formulas -- 4.6 Non-Standard Models of PA -- 5 Encoding -- 5.1 Encoding of Finite Sequences -- 5.2 Encoding of Syntax -- 5.3 RE-Theories -- 6 Incompleteness -- 6.1 Traditional Gödel's First Incompleteness Theorem -- 6.2 Gödel's First Incompleteness Theorem -- 6.3 Corollaries of Gödel's First Incompleteness Theorem -- 6.4 Gödel's Second Incompleteness Theorem -- 6.5 Expressing Consistency -- 6.6 Rosser's Incompleteness Theorem -- 6.7 Gödel's Third Incompleteness Theorem -- 6.8 Reflection Principle -- 6.9 Löb's Theorem -- 6.10 Extension to Other First-Order Theories -- 7 Other Limitative Results -- 7.1 Tarski's Undefinability Theorems -- 7.2 Undecidability Theorem -- 7.3 Church's Theorem -- 7.4 Extension to Other First-Order Theories -- 7.5 Decidability of Monadic First-Order Logic -- 8 Second-Order Logic -- 8.1 Second-Order Languages -- 8.2 Free Variables -- 8.3 Substitution -- 8.4 A Deductive Calculus -- 8.5 Quantifier Rules -- 8.6 Substitutivity of Equivalence -- 8.7 Equality -- 8.8 Interpretations. 327 $a8.9 Substitution Properties -- 8.10 Soundness -- 8.11 Consistency -- 8.12 Negative Results -- 8.13 Isomorphism of Interpretations -- 9 Second-Order Arithmetic -- 9.1 Second-Order Theories -- 9.2 The Theory PA2 -- 9.3 Primitive Recursive Functions in PA2 -- 9.4 Limitative Results for PA2 -- 9.5 Categoricity of PA2 -- 9.6 Strong Incompleteness Theorem for Second-Order Logic -- 9.7 Non-recursive Enumerability of Consequences of PA2 -- Appendix -- A.1 Hilbert's Approach -- A.2 The Conservation Program -- A.3 The Consistency Program -- A.4 Equivalence of the Two Programs -- A.5 Fall of the Consistency Program -- A.6 Fall of the Conservation Program -- A.7 Other Shortcomings of Hilbert's Approach -- References -- Index. 410 0$aSynthese library ;$vVolume 470. 606 $aGo?del's theorem 606 $aTeorema de Gödel$2thub 608 $aLlibres electrňnics$2thub 615 0$aGo?del's theorem. 615 7$aTeorema de Gödel 676 $a511.3 700 $aCellucci$b Carlo$045650 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996499868803316 996 $aThe Theory of Gödel$92982662 997 $aUNISA