Book:Gaisi Takeuti/Proof Theory/Second Edition

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Gaisi Takeuti: Proof Theory (2nd Edition)

Published $\text {1987}$, Dover

ISBN 0-486-49073-4.


Subject Matter


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


Further Editions