Provability, Computability and Reflection (eBook)

Front Cover 1
Mathematical Interpretation of Formal Systems 4
Copyright Page 5
Contents 7
Chapter I. Peano's Axioms and Models of Arithmetic 8
0. Introduction 8
1. Preliminary Remarks 9
2. An Arithmetical Lemma 12
3. The Proof of the Existence of N* 13
4. Some more Special Results 16
Chapter II. On Definability and Derivability 22
1. The Concept of Model 22
2. The Concept of Universal Consequence 23
3. Incompletability Theorems for the relation of Universal Consequence 25
4. The Concept of Universal K-Consequence 26
5. Special Cases of Non-derivability 27
6. Final remarks 31
Chapter III. Models, Translations and Interpretations 33
1. Notation 34
2. Models 35
3. Translations 37
4. Interpretations 40
5. Computable Functionals 46
6. .-sistency 54
Chapter IV. Ordered Structures and Related Concepts 58
1. Introduction 58
2. Model-Completeness 58
3. Applications 62
Chapter V. On Denumerable Bases of Formal Systems 64
1. Introduction 64
2. The Skolem Functions 64
3. Enumeration in a Formal System 66
4. Reduction to Finitely Many Axioms 68
5. Certain Special Methods of Enumeration 70
6. Principles of Choice and Hilbert's e-Operator 73
7. Unintended Interpretations of Formal Systems 76
8.Completeness of Quantification Theory 79
9. Arbitrary Functions 84
Chapter VI. The Representation Theorem for Cylindrical Algebras 92
1. Basic Concepts 92
2. Some Remarks on a Certain Feature of the Proof of the Representation Theorem 99
3. The Representation Theorem for Cylindrical Algebras 102
Chapter V. Quelques Remarques, Théorèmes et Problèmes sur lee Classes Définissables d'algèbres 105