The theory of Gödel / / Carlo Cellucci

• Record Nr.: UNISA996499868803316
• Autore: Cellucci Carlo
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031134173; 9783031134166
• Descrizione fisica: 1 online resource (164 pages)
• Collana: Synthese library ; ; Volume 470
• Disciplina: 511.3
• Soggetti: Gödel's theorem; Teorema de Gödel; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.; 8.9 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.