The Legacy of Kurt Schütte.

By: Kahle, ReinhardContributor(s): Rathjen, MichaelMaterial type: TextTextSeries: eBooks on DemandPublisher: Cham : Springer International Publishing AG, 2020Copyright date: ©2020Description: 1 online resource (497 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783030494247Subject(s): Mathematicians-Germany-BiographyGenre/Form: Electronic books.Additional physical formats: Print version:: The Legacy of Kurt SchütteDDC classification: 510.922 LOC classification: QA1-939Online resources: Click here to view this ebook.
Contents:
Intro -- Preface -- Contents -- List of Contributors -- Part I History and Memories -- Chapter 1 "Sehr geehrter Herr Professor!" Proof Theory in 1949 in a Letter from Schütte to Bernays -- 1.1 Hilbert's Programme after Gödel and Gentzen -- 1.2 Schütte's Return to Logic -- 1.3 Schütte to Bernays, August 26th, 1949 -- 1.4 The ω-rule and Paul Lorenzen -- 1.5 The Legacy of Kurt Schütte -- References -- Chapter 2 Kurt Schütte's Way -- 2.1 Introduction -- 2.2 Beweistheorie -- 2.3 Predicativity -- 2.4 Breaching the Impredicative Barrier -- 2.5 Proof Theory, 2nd Edition -- 2.6 Evolution of the Munich School -- 2.7 Where DoWe Stand, and Where Do We Go from Here? -- Chapter 3 . . . and so on: Schütte on Naming Ordinals -- 3.1 Introduction -- 3.2 A Few Technicalities -- 3.3 The Klammersymbol Revelation -- 3.4 Natural Well-orderings -- References -- Kapitel 4 Kurt Schütte -- Chapter 4 Kurt Schütte -- Chapter 5 Memories of Kurt Schütte and the logic group in Munich: A personal report -- References -- Chapter 6 Reminiscences of Kurt Schütte -- 6.1 Introduction -- 6.2 Fragments of Schütte's Professional Career -- 6.3 The Beginnings of Schütte's Logic Research Group -- 6.4 Aspects of Schütte's Professional Work -- 6.5 Schütte's Role as My Supervisor -- References -- Kapitel 7 Mathematische Logik -- 7.1 Die Grundlegung der modernen mathematischen Logik -- 7.2 Der Logizismus -- 7.3 Die Grundlagenkrise der Mathematik -- 7.4 Die Hilbertsche Beweistheorie -- 7.5 Der Intuitionismus -- 7.6 Die Mengenlehre -- 7.7 Die Rekursionstheorie -- 7.8 Die Modelltheorie -- Kapitel 8 Bemerkungen zur Hilbertschen Beweistheorie -- Chapter 8 Remarks on Hilbert's Proof Theory -- Part II Proof Theory atWork -- Chapter 9 Having a Look Again at Some Theories of Proof-Theoretic Strengths around Γ_0 -- 9.1 Introduction -- 9.2 Subsystems of Second Order Arithmetic.
9.3 Subsystems of Set Theory -- References -- Chapter 10 The Limits of Predicativity Revisited -- 10.1 Introduction -- 10.2 The Vicious Circle Principle -- 10.3 Constructible Sets -- 10.4 Ramified Morse-Kelly Set Theory -- 10.5 The Boundedness Theorem -- 10.6 Attainability -- 10.7 The Attainability Proof -- 10.8 Conclusion -- 10.9 Erratum to "Semi-Formal Calculi and Their Applications" in [15] -- References -- Chapter 11 A Note on (Meta)predicative Wellordering Proofs -- 11.1 Introduction -- 11.2 Ordinal Theoretic Preliminaries -- 11.3 The Theories T^ν -- References -- Chapter 12 Well-ordering Principles, ω-models and Π^1_1-comprehension -- 12.1 Introduction -- 12.2 Relativizing the Ordinal for Π^1_1-comprehension -- 12.3 A Well-ordering Proof -- 12.4 Deduction Chains -- 12.5 Proof of the Main Theorem: the Hard Direction -- References -- Chapter 13 From Schütte's Formal Systems to Modern Automated Deduction -- 13.1 Introduction -- 13.2 Schütte's Influences on the History of Automated Deduction -- 13.3 Modern Connection Calculi -- 13.4 Conclusions -- References -- Chapter 14 Calculating Maximal Order Types for Finite Rooted Unstructured Labeled Trees -- 14.1 Introduction -- 14.2 Ordinal Background -- 14.3 Lower Bounds for the Maximal Order Types of Unstructured Trees -- 14.4 Applications -- References -- Chapter 15 Cut-Elimination for SBL -- 15.1 Introduction -- 15.2 Collapsing Functions ψσ -- 15.3 The Logic Calculus SBL -- 15.4 The Stratified Logic Calculus SBL' -- 15.5 Proof of Main Lemma 15.27 -- References -- Chapter 16 An Upper Bound for the Proof-Theoretic Strength of Martin-Löf Type Theory with W-type and One Universe -- 16.1 Introduction -- 16.2 Definition of the Formal System of Extensional Martin-Löf's Type Theory -- 16.3 Intensional Martin-Löf Type Theory and Its Embedding into Extensional Type Theory.
16.4 Embedding of the Russell Version of Martin-Löf Universes into the Tarski Version -- 16.5 Definition of KPI^+ -- 16.6 Interpretation of Terms and Types -- 16.7 Properties of the Interpretation -- 16.8 Main Lemma -- 16.9 Π^1_1-soundness of the Interpretation of Martin-Löf Type Theory into KPI^+ -- 16.10 Main Theorem -- References -- Chapter 17 Normalization Proof for Derivations in PA after P. Cohen -- 17.1 Introduction -- 17.2 Finite Trees as Ordinals. Termination of Reduction Sequence -- 17.3 Comparison with Gentzen's Second Consistency Proof -- References -- Part III Further Legacy -- Chapter 18 From Probability Measures to Each Lévy Triplet and Back -- 18.1 Introduction -- 18.2 Models of Type Logic -- 18.3 Some Properties of the Nonstandard Model -- 18.4 Finite-Dimensional Lévy Processes -- References -- Chapter 19 On the Strength of the Uniform Fixed Point Principle in Intuitionistic Explicit Mathematics -- 19.1 Introduction -- 19.2 Fixed Point Theories -- 19.3 Double-Negation Translation -- 19.4 Embedding into Intuitionistic Explicit Mathematics -- References -- Chapter 20 Foundations of Mathematics: an Optimistic Message -- References -- Chapter 21 A Glimpse of Σ_3-elementarity -- 21.1 Introduction -- 21.2 Digression: the Discovery of R_1 -- 21.3 A Journey from R_1 via R_2 toward R_3 -- 21.4 A Foretaste of R_3 -- References -- Part IV Kurt Schüttes Spätwerk -- Kapitel 22 Ein Wohlordnungsbeweis mit ∆^1_2-Komprehension und Bar-Induktion -- 22.1 Grundbegriffe -- 22.2 Herleitungen mit arithmetischer Komprehension -- 22.3 Herleitungen mit Π^1_1-Komprehension -- 22.4 Herleitungen mit ∆^1_2-Komprehension -- 22.5 Herleitungen mit ∆^1_2-Komprehension und Bar-Induktion -- Literatur -- Kapitel 23 Beziehungen des Ordinalzahlensystems OT(ϑ) zur Veblen-Hierarchie -- 23.1 Grundbegriffe -- 23.2 Das Ordinalzahlensystem OT(ϑ).
23.3 Der Ackermannsche Ordinalzahlenabschnitt -- 23.4 Die Veblen-Hierarchie der ε-Zahlen -- Literatur -- Kapitel 24 Zur Beweistheorie von KPM -- 24.1 Das mengentheoretische formale System KPM -- 24.2 Das Ordinalzahlensystem OT(M) -- 24.3 Das geschichtete halbformale System RS(M) -- Literatur -- Kapitel 25 Zur Beweistheorie von KP+ Π_3-Ref -- 25.1 Das formale System KP+ Π_3-Ref -- 25.2 Ordinalzahlentheoretische Grundbegriffe -- 25.3 Die Ordinalzahlenmengen M^α und Ordinalzahlen Ξ(α) -- 25.4 Die Ordinalzahlen ψ^µ_π(α) und ψΩ_γ+1^α -- 25.5 Die Ordinalzahlenmenge T(K) -- 25.6 Das geschichtete halbformale System RS(K) -- 25.7 H-kontrollierte Herleitungen -- 25.8 Einbettung von KP+ Π_3-Ref in RS(K) -- 25.9 Herleitungsreduktionen in RS(K) -- Literatur.
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Intro -- Preface -- Contents -- List of Contributors -- Part I History and Memories -- Chapter 1 "Sehr geehrter Herr Professor!" Proof Theory in 1949 in a Letter from Schütte to Bernays -- 1.1 Hilbert's Programme after Gödel and Gentzen -- 1.2 Schütte's Return to Logic -- 1.3 Schütte to Bernays, August 26th, 1949 -- 1.4 The ω-rule and Paul Lorenzen -- 1.5 The Legacy of Kurt Schütte -- References -- Chapter 2 Kurt Schütte's Way -- 2.1 Introduction -- 2.2 Beweistheorie -- 2.3 Predicativity -- 2.4 Breaching the Impredicative Barrier -- 2.5 Proof Theory, 2nd Edition -- 2.6 Evolution of the Munich School -- 2.7 Where DoWe Stand, and Where Do We Go from Here? -- Chapter 3 . . . and so on: Schütte on Naming Ordinals -- 3.1 Introduction -- 3.2 A Few Technicalities -- 3.3 The Klammersymbol Revelation -- 3.4 Natural Well-orderings -- References -- Kapitel 4 Kurt Schütte -- Chapter 4 Kurt Schütte -- Chapter 5 Memories of Kurt Schütte and the logic group in Munich: A personal report -- References -- Chapter 6 Reminiscences of Kurt Schütte -- 6.1 Introduction -- 6.2 Fragments of Schütte's Professional Career -- 6.3 The Beginnings of Schütte's Logic Research Group -- 6.4 Aspects of Schütte's Professional Work -- 6.5 Schütte's Role as My Supervisor -- References -- Kapitel 7 Mathematische Logik -- 7.1 Die Grundlegung der modernen mathematischen Logik -- 7.2 Der Logizismus -- 7.3 Die Grundlagenkrise der Mathematik -- 7.4 Die Hilbertsche Beweistheorie -- 7.5 Der Intuitionismus -- 7.6 Die Mengenlehre -- 7.7 Die Rekursionstheorie -- 7.8 Die Modelltheorie -- Kapitel 8 Bemerkungen zur Hilbertschen Beweistheorie -- Chapter 8 Remarks on Hilbert's Proof Theory -- Part II Proof Theory atWork -- Chapter 9 Having a Look Again at Some Theories of Proof-Theoretic Strengths around Γ_0 -- 9.1 Introduction -- 9.2 Subsystems of Second Order Arithmetic.

9.3 Subsystems of Set Theory -- References -- Chapter 10 The Limits of Predicativity Revisited -- 10.1 Introduction -- 10.2 The Vicious Circle Principle -- 10.3 Constructible Sets -- 10.4 Ramified Morse-Kelly Set Theory -- 10.5 The Boundedness Theorem -- 10.6 Attainability -- 10.7 The Attainability Proof -- 10.8 Conclusion -- 10.9 Erratum to "Semi-Formal Calculi and Their Applications" in [15] -- References -- Chapter 11 A Note on (Meta)predicative Wellordering Proofs -- 11.1 Introduction -- 11.2 Ordinal Theoretic Preliminaries -- 11.3 The Theories T^ν -- References -- Chapter 12 Well-ordering Principles, ω-models and Π^1_1-comprehension -- 12.1 Introduction -- 12.2 Relativizing the Ordinal for Π^1_1-comprehension -- 12.3 A Well-ordering Proof -- 12.4 Deduction Chains -- 12.5 Proof of the Main Theorem: the Hard Direction -- References -- Chapter 13 From Schütte's Formal Systems to Modern Automated Deduction -- 13.1 Introduction -- 13.2 Schütte's Influences on the History of Automated Deduction -- 13.3 Modern Connection Calculi -- 13.4 Conclusions -- References -- Chapter 14 Calculating Maximal Order Types for Finite Rooted Unstructured Labeled Trees -- 14.1 Introduction -- 14.2 Ordinal Background -- 14.3 Lower Bounds for the Maximal Order Types of Unstructured Trees -- 14.4 Applications -- References -- Chapter 15 Cut-Elimination for SBL -- 15.1 Introduction -- 15.2 Collapsing Functions ψσ -- 15.3 The Logic Calculus SBL -- 15.4 The Stratified Logic Calculus SBL' -- 15.5 Proof of Main Lemma 15.27 -- References -- Chapter 16 An Upper Bound for the Proof-Theoretic Strength of Martin-Löf Type Theory with W-type and One Universe -- 16.1 Introduction -- 16.2 Definition of the Formal System of Extensional Martin-Löf's Type Theory -- 16.3 Intensional Martin-Löf Type Theory and Its Embedding into Extensional Type Theory.

16.4 Embedding of the Russell Version of Martin-Löf Universes into the Tarski Version -- 16.5 Definition of KPI^+ -- 16.6 Interpretation of Terms and Types -- 16.7 Properties of the Interpretation -- 16.8 Main Lemma -- 16.9 Π^1_1-soundness of the Interpretation of Martin-Löf Type Theory into KPI^+ -- 16.10 Main Theorem -- References -- Chapter 17 Normalization Proof for Derivations in PA after P. Cohen -- 17.1 Introduction -- 17.2 Finite Trees as Ordinals. Termination of Reduction Sequence -- 17.3 Comparison with Gentzen's Second Consistency Proof -- References -- Part III Further Legacy -- Chapter 18 From Probability Measures to Each Lévy Triplet and Back -- 18.1 Introduction -- 18.2 Models of Type Logic -- 18.3 Some Properties of the Nonstandard Model -- 18.4 Finite-Dimensional Lévy Processes -- References -- Chapter 19 On the Strength of the Uniform Fixed Point Principle in Intuitionistic Explicit Mathematics -- 19.1 Introduction -- 19.2 Fixed Point Theories -- 19.3 Double-Negation Translation -- 19.4 Embedding into Intuitionistic Explicit Mathematics -- References -- Chapter 20 Foundations of Mathematics: an Optimistic Message -- References -- Chapter 21 A Glimpse of Σ_3-elementarity -- 21.1 Introduction -- 21.2 Digression: the Discovery of R_1 -- 21.3 A Journey from R_1 via R_2 toward R_3 -- 21.4 A Foretaste of R_3 -- References -- Part IV Kurt Schüttes Spätwerk -- Kapitel 22 Ein Wohlordnungsbeweis mit ∆^1_2-Komprehension und Bar-Induktion -- 22.1 Grundbegriffe -- 22.2 Herleitungen mit arithmetischer Komprehension -- 22.3 Herleitungen mit Π^1_1-Komprehension -- 22.4 Herleitungen mit ∆^1_2-Komprehension -- 22.5 Herleitungen mit ∆^1_2-Komprehension und Bar-Induktion -- Literatur -- Kapitel 23 Beziehungen des Ordinalzahlensystems OT(ϑ) zur Veblen-Hierarchie -- 23.1 Grundbegriffe -- 23.2 Das Ordinalzahlensystem OT(ϑ).

23.3 Der Ackermannsche Ordinalzahlenabschnitt -- 23.4 Die Veblen-Hierarchie der ε-Zahlen -- Literatur -- Kapitel 24 Zur Beweistheorie von KPM -- 24.1 Das mengentheoretische formale System KPM -- 24.2 Das Ordinalzahlensystem OT(M) -- 24.3 Das geschichtete halbformale System RS(M) -- Literatur -- Kapitel 25 Zur Beweistheorie von KP+ Π_3-Ref -- 25.1 Das formale System KP+ Π_3-Ref -- 25.2 Ordinalzahlentheoretische Grundbegriffe -- 25.3 Die Ordinalzahlenmengen M^α und Ordinalzahlen Ξ(α) -- 25.4 Die Ordinalzahlen ψ^µ_π(α) und ψΩ_γ+1^α -- 25.5 Die Ordinalzahlenmenge T(K) -- 25.6 Das geschichtete halbformale System RS(K) -- 25.7 H-kontrollierte Herleitungen -- 25.8 Einbettung von KP+ Π_3-Ref in RS(K) -- 25.9 Herleitungsreduktionen in RS(K) -- Literatur.

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