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Foreword; Contents; 1 Introduction: The Internal Logic of Arithmetic; 2 Arithmetization of Analysis and Algebra; 2.1 Cauchy and Weierstrass; 2.2 Dedekind and Cantor; 2.3 Frege; 2.4 Russell, Peano and Zermelo; 2.5 Kronecker and the Arithmetization of Algebra; 3 Arithmetization of Logic; 3.1 Hilbert after Kronecker; 3.2 Hilbert's Arithmetization of Logic and the Epsilon Calculus; 3.3 Herbrand's Theorem; 3.4 Tarski's Quantifier Elimination; 3.5 Gödel's Functional Interpretation; 3.6 Skolem and Brouwer; 3.7 Gödel and Turing; 3.8 Arithmetic; 3.9 Constructive Arithmetic and Analysis
3.10 Complexity4 Kronecker's Foundational Programme in Contemporary Mathematics; 4.1 Introduction; 4.2 Grothendieck's Programme; 4.3 Descent; 4.4 Langlands' Programme; 4.5 Kronecker's and Hilbert's Programmes in Contemporary Mathematical Logic; 4.6 Conclusion: Finitism and Arithmetism; 5 Arithmetical Foundations for Physical Theories; 5.1 Introduction: The Notion of Analytical Apparatus; 5.2 Analytical and Empirical Apparatuses; 5.3 Models; 5.4 The Consistency of Physical Theories; 5.5 Quantum Mechanics; 5.5.1 Hilbert Space; 5.5.2 Probabilities; 5.5.3 Logics; 5.5.4 Local Complementation
5.5.5 The Total Hilbert Space5.5.6 Finite Derivation of the Local Complement; 5.6 Riemannian Geometry; 5.7 Riemann's ``Hypotheses''; 5.8 Physical Geometry; 5.9 Minkowski's Spacetime; 5.10 Geometry of Numbers; 5.11 Spacetime Diagrams; 5.12 Physical Axiomatics; 5.13 Hermann Weyl and the Free-Will Theorem; 5.14 The Conway-Kochen Free-Will Theorem; 5.15 A General No-Cloning Theorem in the Multiversal Cosmology; 5.15.1 The No-Cloning Theorem in QM; 5.15.2 A No-Cloning Theorem in the Multiverse Cosmology; 5.16 Conclusion; 5.17 Appendix to Chapter 5
5.17.1 Principles for a Theory of Measurement in QM6 The Internal Logic of Constructive Mathematics; 6.1 Transcendental Versus Elementary: The Gel'fond-Schneider Theorem; 6.2 Transcendental Number Theory; 6.3 The Internal Logic; 6.4 Descent or Descending Induction; 6.5 Induction Principles; 6.6 Intuitionistic Logic and Transfinite Induction; 6.7 Transfinite Induction; 6.8 Conclusion: A Finitist Logic for Constructive Mathematics; 7 The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof; 7.1 Preamble; 7.2 Introduction; 7.3 Arithmetic; 7.4 Arithmetization of Syntax
7.5 Reducibility and Divisibility7.6 Elimination of Logical Constants; 7.7 The Elimination of Implication; 7.8 The Elimination of the Effinite Quantifier Through Infinite Descent; 7.9 Conclusion: The Polynomial Extension from a Finitist Point of View; 8 Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege; 8.1 Introduction: Arithmetical Philosophy; 8.2 Kronecker Today; 8.3 Arithmetization of Geometry: From Algebraic Geometry to Arithmetic Geometry; 8.4 From Geometry of Numbers to Physical Geometry and Physics; 8.5 Arithmetization of Logic; References
3.10 Complexity4 Kronecker's Foundational Programme in Contemporary Mathematics; 4.1 Introduction; 4.2 Grothendieck's Programme; 4.3 Descent; 4.4 Langlands' Programme; 4.5 Kronecker's and Hilbert's Programmes in Contemporary Mathematical Logic; 4.6 Conclusion: Finitism and Arithmetism; 5 Arithmetical Foundations for Physical Theories; 5.1 Introduction: The Notion of Analytical Apparatus; 5.2 Analytical and Empirical Apparatuses; 5.3 Models; 5.4 The Consistency of Physical Theories; 5.5 Quantum Mechanics; 5.5.1 Hilbert Space; 5.5.2 Probabilities; 5.5.3 Logics; 5.5.4 Local Complementation
5.5.5 The Total Hilbert Space5.5.6 Finite Derivation of the Local Complement; 5.6 Riemannian Geometry; 5.7 Riemann's ``Hypotheses''; 5.8 Physical Geometry; 5.9 Minkowski's Spacetime; 5.10 Geometry of Numbers; 5.11 Spacetime Diagrams; 5.12 Physical Axiomatics; 5.13 Hermann Weyl and the Free-Will Theorem; 5.14 The Conway-Kochen Free-Will Theorem; 5.15 A General No-Cloning Theorem in the Multiversal Cosmology; 5.15.1 The No-Cloning Theorem in QM; 5.15.2 A No-Cloning Theorem in the Multiverse Cosmology; 5.16 Conclusion; 5.17 Appendix to Chapter 5
5.17.1 Principles for a Theory of Measurement in QM6 The Internal Logic of Constructive Mathematics; 6.1 Transcendental Versus Elementary: The Gel'fond-Schneider Theorem; 6.2 Transcendental Number Theory; 6.3 The Internal Logic; 6.4 Descent or Descending Induction; 6.5 Induction Principles; 6.6 Intuitionistic Logic and Transfinite Induction; 6.7 Transfinite Induction; 6.8 Conclusion: A Finitist Logic for Constructive Mathematics; 7 The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof; 7.1 Preamble; 7.2 Introduction; 7.3 Arithmetic; 7.4 Arithmetization of Syntax
7.5 Reducibility and Divisibility7.6 Elimination of Logical Constants; 7.7 The Elimination of Implication; 7.8 The Elimination of the Effinite Quantifier Through Infinite Descent; 7.9 Conclusion: The Polynomial Extension from a Finitist Point of View; 8 Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege; 8.1 Introduction: Arithmetical Philosophy; 8.2 Kronecker Today; 8.3 Arithmetization of Geometry: From Algebraic Geometry to Arithmetic Geometry; 8.4 From Geometry of Numbers to Physical Geometry and Physics; 8.5 Arithmetization of Logic; References