Provability, Computability and Reflection (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
194 Seiten
Elsevier Science (Verlag)
978-0-08-095724-1 (ISBN)

Infos

Provability, Computability and Reflection

Inhalt

Front Cover 1
On the Metamathematics of Algebra 4
Copyright Page 5
LIST OF CONTENTS 8
CHAPTER I. GENERAL INTRODUCTION 12
1.1 Purpose of treatise 12
1.2 Fundamental logical outlook 13
1.3 Metamathematical theorems in Algebra 13
1.4 Generalised concepts of Algebra 14
1.5 Definition of Equality 15
1.6 Example of a generalised concept in Algebra 15
1.7 Extension of models 17
1.8 Summary 17
1.9 Conclusion 18
CHAPTER II. CONSTRUCTION OF A FORMAL LANGUAGE 21
2.1 Introduction 21
2.2 Symbols 22
2.3 Formulae 22
2.4 Statements and predicates 23
2.5 Calculus of deduction 25
2.6 Calculus of continued 27
2.7 Descriptive interpretation 28
2.8 Descriptive interpretation of continued 30
2.9 Infinite conjunctions and disjunctions 32
CHAPTER III. RELATIONS BETWEEN DEDUCTIVE AND DESCRIPTIVE CONCEPTS 34
3.1 Compatibility of deductive and descriptive concepts 34
3.2 Completeness of the calculus of deduction 35
3.3 Proof of the completeness theorem for special cases 36
3.4 Proof of the completeness theorem for special cases, continued 37
3.5 Proof of the completeness theorem for special cases, continued 40
3.6 Proof of the completeness theorem for the general case 42
3.7 Proof of the completeness theorem for the general case, continued 43
3.8 Systems of statements 47
3.9 Lindenbaum’s theorem 50
CHAPTER IV. SPECIFICATION OF AXIOMATIC SYSTEMS 51
4.1. Axiom for groups, rings and fields 51
4.2 Characteristic of a field and Archimedes axiom 53
CHAPTER V. METAMATHEMATICAL THEOREMS ON ALGEBRAIC FIELDS 56
5.1 Chains of statements 56
5.2 Metamathematical theorem on fields with prime characteristic 57
5.3 Applications 58
5.4 Properties of infinite disjunctions 62
5.5 Properties of infinite disjunctions, continued 64
5.6 Archimedean and non-Archimedean fields 67
5.7 Examples 68
5.8 Algebraically closed fields 70
CHAPTER VI. ALGEBRAIC SYSTEMS 73
6.1 Algebras of axioms 73
6.2 Algebraic structures 75
6.3 Isomorphisms and homomorphisms 76
6.4 Similarity of algebras 79
6.5 Extension of general structures 82
6.6 Extension of algebraic structures 84
6.7 Applications 88
CHAPTER VII. POLYNOMIALS IN GENERAL ALGEBRAS 90
7.1 Introduction 90
7.2 Prepolynomials and polynomials 91
7.3 Properties of prepolynomial structures 93
7.4 Augmented systems of axioms 96
7.5 Applications to ring extensions 97
7.6 Applications to ring extensions, continued 102
7.7 Homomorphisms of prepolynomial structures 103
CHAPTER VIII. ALGEBRAIC PREDICATES 106
8.1 Introduction 106
8.2 Bounded predicates 108
8.3 Decomposit ion of bounded predicates 113
8.4 Algebraic predicates 116
8.5 Properties of algebraic predicates 119
8.6 Applications to algebraic fields 122
8.7 Applications to algebraic fields, continued 124
CHAPTER IX. CONVEX SYSTEMS 127
9.1 Definition and properties of convex systems 127
9.2 Convex systems and algebraic predicates 133
9.3 Definite algebras 138
9.4 Properties of definite algebras 141
9.5 Separable and perfect systems 144
9.6 Symmetrical predicates 147
CHAPTER X. IDEALS 150
10.1 Introduction 150
10.2 Metamathematical ideals 151
10.3 Connection between ideals in different domains 153
10.4 Maximum condition for disjunctive domains 155
10.5 Properties of disjunctive ideals 159
10.6 Ideals and homomorphisms 163
10.7 Disjunctive ideals in rings 166
CHAPTER XI. PRE-IDEALS 171
11.1 Properties of varieties 171
11.2 Definition of pre-ideals 173
11.3 Pre-ideals and their varieties 176
11.4 Pseudo-ideals and pre-ideals 182
11.5 Pseudo-ideals and pre-ideals, continued 185
11.6 Application to the theory of polynomial ideals 188
11.7 Disjunctive pseudo-ideals and pre-ideals 191
11.8 Application to primary rings 197
11.9 Ideals in the theory of algebraic numbers 201
Bibliography 203

Produkt-Details

• Erscheint lt. Verlag: 1.4.2000
• Sprache: englisch
• Themenwelt: Informatik ► Software Entwicklung ► User Interfaces (HCI)
• Themenwelt: Informatik ► Theorie / Studium ► Algorithmen
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095724-2 / 0080957242
• ISBN-13: 978-0-08-095724-1 / 9780080957241