Studies in Logic and the Foundations of Mathematics (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
292 Seiten
Elsevier Science (Verlag)
978-0-08-095762-3 (ISBN)

Infos

Provability, Computability and Reflection

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Studies in Logic and the Foundations of Mathematics

Inhalt

Front Cover 1
Non-Standard Analysis 4
Copyright Page 5
LIST OF CONTENTS 10
CHAPTER I. GENERAL INTRODUCTION 14
1.1 Purpose of this book 14
1.2 Summary of contents 15
CHAPTER II. TOOLS FROM LOGIC 19
2.1 The Lower Predicate Calculus 19
2.2 Interpretation 21
2.3 Ultraproducts 23
2.4 Prenex normal form 23
2.5 The finiteness principle 26
2.6 Higher order structures and corresponding languages 32
2.7 Type symbols 36
2.8 Finiteness principle for higher order theories 39
2.9 Enlargements 43
2.10 Examples of enlargements 47
2.11 General properties of enlargements 54
2.12 Remarks and references 61
CHAPTER III. DIFFERENTIAL AND INTEGRAL CALCULUS 62
3.1 Non-standard Arithmetic 62
3.2 Non-standard Analysis 68
3.3 Convergence 71
3.4 Continuity and differentiation 78
3.5 Integration 84
3.6 Differentials 92
3.7 Total differentials 94
3.8 Elementary Differential Geometry 96
3.9 Remarks and references 101
CHAPTER IV. GENERAL TOPOLOGY 102
4.1 Topological spaces 102
4.2 Sequences, nets, mappings 108
4.3 Metric spaces 113
4.4 Topologies in *T 119
4.5 Functions, limits, continuity in metric spaces 123
4.6 Sequences of functions, Compact mappings 129
4.7 Euclidean space 133
4.8 Remarks and references 135
CHAPTER V. FUNCTIONS OF A REAL VARlABLE 136
5.1 Measure and integration 136
5.2 Sequences of functions 143
5.3 Distributions 146
5.4 Remarks and references 158
CHAPTER VI. FUNCTIONS OF A COMPLEX VARIABLE 160
6.1 Analytic theory of polynomials 160
6.2 Analytic functions 168
6.3 Picard’s theorems and Julia’s directions 173
6.4 Compactness arguments in classical Function Theory 186
6.5 Remarks and references 189
CHAPTER VII. LINEAR SPACES 190
7.1 Normed spaces 190
7.2 Hilbert space 194
7.3 Spectral theory of compact operators 198
7.4 An invariant subspace problem 208
7.5 Remarks and references 213
CHAPTER VIIl. TOPOLOGICAL GROUPS AND LIE GROUPS 215
8.1 Topological groups 215
8.2 Metric groups 220
8.3 One-parametric subgroups 228
8.4 The Lie algebra of a group 238
8.5 Remarks and references 242
CHAPTER IX. SELECTED TOPICS 243
9.1 Variations 243
9.2 Riemann’s mapping theorem 245
9.3 Dirichlet’s principle 246
9.4 Sources and doublets 251
9.5 Local perturbations 254
9.6 Boundary layer theory 260
9.7 Saint-Venant’s principle 266
9.8 Remarks and references 271
CHAPTER X. CONCERNING THE HISTORY OF THE CALCULUS 273
10.1 Introduction 273
10.2 Leibniz 274
10.3 De I’Hospital 277
10.4 Lagrange and d’Alembert 280
10.5 Cauchy 282
10.6 Bolzano, Weierstrass. and after 289
10.7 Infinitely small and large numbers and the infinite 292
BIBLIOGRAPHY 296
INDEX OF AUTHORS 303
SUBJECT INDEX 305

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 ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095762-5 / 0080957625
• ISBN-13: 978-0-08-095762-3 / 9780080957623