Book:Gaisi Takeuti/Proof Theory/Second Edition

Subject Matter

 * Logic

Contents

 * Preface
 * Preface to the second edition
 * Contents
 * Introduction


 * PART I. FIRST ORDER SYSTEMS


 * Chapter 1. First Order Predicate Calculus
 * $\S 1$. Formalization of statements
 * $\S 2$. Formal proofs and related concepts
 * $\S 3$. A formulation of intuitionistic predicate calculus
 * $\S 4$. Axiom systems
 * $\S 5$. The cut-elimination theorem
 * $\S 6$. Some consequences of the cut-elimination theorem
 * $\S 7$. The predicate calculus with equality
 * $\S 8$. The completeness theorem


 * Chapter 2. Peano Arithmetic
 * $\S 9$. A formulation of Peano arithmetic
 * $\S 10$. The incompleteness theorem
 * $\S 11$. A discussion of ordinals from a finitist standpoint
 * $\S 12$. A consistency proof of $\mathbf{PA}$
 * $\S 13$. Provable well-orderings
 * $\S 14$. An additional topic


 * PART II. SECOND ORDER AND FINITE ORDER SYSTEMS


 * Chapter 3. Second Order Systems and Simple Type Theory
 * $\S 15$. Second order predicate calculus
 * $\S 16$. Some systems of second order predicate calculus
 * $\S 17$. The theory of relativization
 * $\S 18$. Truth definition for first order arithmetic
 * $\S 19$. The interpretation of a system of second order arithmetic
 * $\S 20$. Simple type theory
 * $\S 21$. The cut-elimination theorem for simple type theory


 * Chapter 4. Infinitary Logic
 * $\S 22$. Infinitary logic with homogeneous quantifiers
 * $\S 23$. Determinate logic
 * $\S 24$. A general theory of heterogeneous quantifiers


 * PART III. CONSISTENCY PROBLEMS


 * Chapter 5. Consistently Proofs
 * $\S 25$. Introduction
 * $\S 26$. Ordinal diagrams
 * $\S 27$. A consistency proof of second order arithmetic with the $\Pi_1^1$-comprehension axiom
 * $\S 28$. A consistency proof for a system with inductive definitions


 * Chapter 6 Some Applications of Consistency Proofs
 * $\S 29$. Provable well-orderings
 * $\S 30$. The $\Pi_1^1$-comprehension axiom and the $\omega$-rule
 * $\S 31$. Reflection principles


 * Postscript


 * APPENDIX
 * Proof Theory by Georg Kreisel
 * Contributions of the Schütte School by Wolfram Pohlers
 * Subsystems of $Z_2$ and Reverse Mathematics by Stephen G. Simpson
 * Proof Theory: A Personal Report by Solomon Feferman


 * Index