Book:Gaisi Takeuti/Proof Theory/Second Edition

Subject Matter

 * Logic

Contents

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


 * PART I. FIRST ORDER SYSTEMS


 * $\S 1$. 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
 * $\S 8$. The completeness theorem


 * $\S 9$. 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
 * $\S 14$. An additional topic


 * PART II. SECOND ORDER AND FINITE ORDER SYSTEMS


 * $\S 15$. 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
 * $\S 21$. The cut-elimination theorem for simple type theory


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


 * PART III. CONSISTENCY PROBLEMS


 * $\S 25$. 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
 * $\S 28$. A consistency proof for a system with inductive definitions


 * $\S 29$. Provable well-orderings
 * $\S 30$. The $\Pi_1^1$-comprehension axiom and the $\omega$-rule
 * $\S 31$. Reflection principles
 * $\S 31$. Reflection principles




 * APPENDIX
 * by Georg Kreisel
 * by Wolfram Pohlers
 * by Stephen G. Simpson
 * by Solomon Feferman


 * Index