Provability, Computability and Reflection

Lev D. Beklemishev

• Publisher: Elsevier Science
• Year: 2000
• Print ISBN: 9780444533852
• eBook ISBN: 9780080957401
• Topic: Informatique
• Subtopic: Sciences générales de l'informatique

Studies in Logic publishes monographs and occasionally edited volumes in the area of mathematical logic and its applications.

Table of contents

• Front Cover
• Recursive Number Theory
• Copyright Page
• CONTENTS
• Preface
• Introduction
• Chapter I. The arithmetical operations. Definition by iteration and recursion. Contracted notation. Single and multiple recursions
• Chapter II. Definition of proof. Commutative and associative properties of addition. The equations x + (y–x) = y + (x–y) and (1– Ix,yI)f(x) = (1– Ix,yI)f(y). The functionsΣt, and μt Inequalities. Verifiability and freedom from contradiction. EXA⤀
• Chapter III. The propositional calculus. Propositional functions. The limited universal, existential and minimal operators Anx, Enx, and Lnx, Mathematical induction. The counting operator Nnx
• Chapter IV. The fundamental theorems of arithmetic. Prime numbers. Uniqueness of the resolution into prime factors. The greatest common factor
• Chapter V. Formalisations of recursive arithmetic. The Deduction Theorem. Reduction of the uniqueness schema
• Chapter VI. Reductions to primitive recursion. Course-of-values recursion. Recursion with parameter substitution. Simultaneous recursions. Generalised induction schemata. Permutation.
• Chapter VII. Elimination of parameters. The initial functions and the generation of all primitive recursive functions by the single schema. Fx = Ax0. The enumerating function. A doubly recursive function which cannot be defined by primitive recursion and substitut
• Chapter VIII. Godel numbering and the incompleteness of arithmetic. The arithmetisation of syntax. The construction of a verifiable unprovable equation. Skolem's non-standard model for arithmetic.
• Chapter VIII. Gödel numbering and the incompleteness of arithmetic. The arithmetisation of syntax. The construction of a verifiable unprovable equation. Skolem's non-standard model for arithmetic
• Solutions to Examples
• Bibliographical Notes
• Bibliography
• Index