Book:Gaisi Takeuti/Proof Theory/Second Edition

Gaisi Takeuti: Proof Theory (2nd Edition)

Published $\text {1987}$, Dover

ISBN 0-486-49073-4

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. Consistency 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

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Further Editions

• 1975: Gaisi Takeuti: Proof Theory

Source work progress

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