The concept of number: from quaternions to monads and topological fields

Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Translator's Foreword to the English Edition......Page 8
Foreword to the German Edition......Page 11
CHAPTER I : THE COMPLETE ORDERED FIELD R......Page 14
1A The Concept of a Field: Algebraic Preliminaries......Page 15
1B Order Relations, Completeness......Page 16
1C Ordered Groups and Fields......Page 21
1D Recognition of R......Page 30
1E On Properties Equivalent to Completion......Page 37
1F Cantor's Characterisation of (E, ')......Page 46
CHAPTER 2 : CONSTRUCTIONS OF P......Page 52
2A Decimal Representations of the Real Numbers......Page 53
2B Constructions of P with Decimal Sequences......Page 56
2C Construction of R Following Dedekind......Page 57
2D Cantor's Construction of R......Page 60
2E Continued Fractions......Page 62
2F Closing Remarks on Chapter 2......Page 67
3A Decimal Expansions......Page 72
3B Algebraic Numbers......Page 75
3C Quadratic Irrational Numbers......Page 77
3D Transcendental Numbers......Page 80
3E Multiples of Irrational Numbers Modulo 1......Page 85
4A Constructions of C......Page 88
4B Some Structural Properties of C......Page 93
4C The Fundamental Properties of C......Page 101
4D The Fundamental Theorem as an Assertion About Extension Fields of R......Page 109
5A Preliminary Remarks......Page 112
5B Embedding P and C in H......Page 117
5C Quaternions and Vector Calculations......Page 120
5D The Multiplicative Group of Quaternions......Page 124
5E Quaternions and Orthogonal Mappings in R^3......Page 126
5F The Theorems of Frobenius......Page 136
5G The Cayley Numbers (Octaves)......Page 141
5H Relations with Geometry 1: Vector Fields on Spheres......Page 143
5I Relations with Geometry 2: Affine Planes......Page 145
CHAPTER 6 : SETS AND NUMBERS......Page 150
6A Equipotent Sets......Page 151
6B The Number Systems as Unstructured Sets (Comparison of Cardinals)......Page 155
6C Cardinal Numbers......Page 163
6D Zorn's Lemma as a Proof Principle......Page 170
6E The Arithmetic of Cardinal Numbers......Page 174
6F Vector Spaces of Infinite Dimension, and the Cauchy Functional Equation......Page 178
APPENDIX : Hamel's Existence Proof for a Basis......Page 181
CHAPTER 7 : NON- STANDARD NUMBERS......Page 186
7A Preparation: The Non- Archimedean Ordered Field R(x) of Rational Functions......Page 187
7B The Ring OR of Schmieden and Laugwitz......Page 190
7C Filters and Ultrafilters......Page 192
7D The Fields *R(I, U) as Ultraproducts......Page 196
7E An Axiomatic Approach to Non- Standard Analysis......Page 205
CHAPTER 8 : PONTRJAGIN'S TOPOLOGICAL CHARACTERIZATION OF R, C AND H......Page 214
8A Topological Groups......Page 215
8B Topological Fields......Page 223
8C Pontrjagin's Theorem......Page 228
APPENDIX : NOTATION AND TERMINOLOGY......Page 242
COMMENTS ON THE LITERATURE......Page 249
REFERENCES......Page 250
INDEX......Page 255
LIST OF SYMBOLS......Page 264