Axiom of Choice (eBook)

Preface 7
Contents 12
1 Origins 14
1.1 Hilbert’s First Problem 15
2 Choice Principles 22
2.1 Some Equivalents to the Axiom of Choice 22
2.2 Some Concepts Related to the Axiom of Choice 26
3 Elementary Observations 34
3.1 Hidden Choice 34
3.2 Unnecessary Choice 40
3.3 Concepts Split Up: Compactness 45
4 Disasters without Choice 56
4.1 Finiteness 56
4.2 Disasters in Cardinal Arithmetic 64
4.3 Disasters in Order Theory 69
4.4 Disasters in Algebra I: Vector Spaces 79
4.5 Disasters in Algebra II: Categories 84
4.6 Disasters in Elementary Analysis: The Reals and Continuity 85
4.7 Disasters in Topology I: Countable Sums 92
4.8 Disasters in Topology II: Products (The Tychono. and the . Cech–Stone Theorem) 98
4.9 Disasters in Topology III: Function Spaces (The Ascoli Theorem) 108
4.10 Disasters in Topology IV: The Baire Category Theorem 115
4.11 Disasters in Graph Theory: Coloring Problems 122
5 Disasters with Choice 130
5.1 Disasters in Elementary Analysis 130
5.2 Disasters in Geometry: Paradoxical Decompositions 139
6 Disasters either way 150
6.1 Disasters in Game Theory 150
7 Beauty without Choice 155
7.1 Lindel¨ of = Compact 155
7.2 Measurability (The Axiom of Determinateness) 162
Appendix: Models 170
References 180
Selected Books and Longer Articles 191
List of Symbols 193
List of Axioms 195
Index 198