Provability, Computability and Reflection (eBook)

Front Cover 1
Recursive Number Theory 4
Copyright Page 5
CONTENTS 10
Preface 6
Introduction 12
The nature of numbers. ArIthmetic and the game of chess. Definition of counting. Evolution of the concept of a formal system 12
Chapter I. The arithmetical operations. Definition by iteration and recursion. Contracted notation. Single and multiple recursions 24
Examples I 36
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 functionsSt, and µt Inequalities. Verifiability and freedom from contradiction. EXA 38
Examples II 62
Chapter III. The propositional calculus. Propositional functions. The limited universal, existential and minimal operators Anx, Enx, and Lnx, Mathematical induction. The counting operator Nnx 67
Examples III 91
Chapter IV. The fundamental theorems of arithmetic. Prime numbers. Uniqueness of the resolution into prime factors. The greatest common factor 97
Examples IV 112
Chapter V. Formalisations of recursive arithmetic. The Deduction Theorem. Reduction of the uniqueness schema 115
Chapter VI. Reductions to primitive recursion. Course-of-values recursion. Recursion with parameter substitution. Simultaneous recursions. Generalised induction schemata. Permutation. 130
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 145
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. 154
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 154
Solutions to Examples 171
Bibliographical Notes 198
Bibliography 199
Index 200