Provability, Computability and Reflection (eBook)

Front Cover 1
Introduction to Model Theory and to The Metamathematics of Algebra 4
Copyright Page 5
List of Contents 8
CHAPTER I. THE LOWER PREDICATE CALCULUS 12
1.1. General introduction 12
1.2. Rules of formation 14
1.3. Rules of deduction 17
1.4. Semantic interpretation 20
1.5. The relation between deductive and descriptive concepts 22
1.6. Sets of sentences and their varieties 31
1.7. Problems 33
CHAPTER II. ALGEBRAIC THEORIES 34
2.1. Equality 34
2.2. Specification of axiomatic systems 36
2.3. Related sets of sentences 42
2.4. Embedding theorems and transfer principles 45
2.5. Malcev’s theory of normal chains 56
2.6. Problems 60
CHAPTER III. SOME CONCEPTS AND METHODS OF MODEL THEORY 62
3.1. Skolem functions relativization
3.2. The extension of models 65
3.3. The prefix problem 76
3.4. Obstructions to arithmetical extension 86
3.5. Convex systems 91
3.6. Model-consistency 95
3.7. Problems 97
CHAPTER IV. COMPLETENESS 99
4.1. A test for completeness 99
4.2. Model-completeness 102
4.3. Relative model-completeness 116
4.4. Problems 121
CHAPTER V. DEFINABILITY 123
5.1. The consistency lemma 123
5.2. Beth's theorem 128
5.3. Relative definitions 130
5.4. Application to the Nullstellensatz 136
5.5. Model-completion 139
5.6. Problems 148
CHAPTER VI. GENERALIZATION OF ALGEBRAIC CONCEPTS 150
6.1. Polynomials in general axiomatic systems 150
6.2. Bounded predicates 160
6.3. Algebraic predicates 165
6.4. Algebraic predicates and convex systems 172
6.5. Separability 178
6.6. Problems 182
CHAPTER VII. METAMATHEMATICAL THEORY OF IDEALS 183
7.1. Introduction 183
7.2. Metamathematical ideals 184
7.3. Connection between ideals in different domains 186
7.4. Disjunctive ideals 190
7.5. Ideals and homomorphisms 196
7.6. Problems 202
CHAPTER VIII. METAMATHEMATICAL THEORY OF VARIETIES 203
8.1. Varieties of structures 203
8.2. Pre-ideals and their varieties 211
8.3. Metamathematical varieties and algebraic varieties 215
8.4. Differential ideals 220
8.5. Hilbert's seventeenth problem 225
8.6. Problems 235
CHAPTER IX. SELECTED TOPICS 237
9.1. Introduction of function symbols 237
9.2. The elimination of quantifiers 242
9.3. Direct products and ultraproducts 249
9.4. Non-standard analysis 255
9.5. Non-standard theory of functions of a real variable 263
9.6. Non-standard analysis of functions of several variables 272
9.7. Problems 281
BIBLIOGRAPHY 283
INDEX OF AUTHORS 292
SUBJECT INDEX 294