Axiom of Choice (eBook)

Front Cover 1
The Axiom of Choice 4
Copyright Page 5
Preface 8
Contents 10
Preface 8
Chapter 1. Introduction 14
1.1. The Axiom of Choice 14
1.2. A nonmeasurable set of real numbers 15
1.3. A paradoxical decomposition of the sphere 16
1.4. Problems 19
1.5. Historical remarks 21
Chapter 2. Use of the Axiom of Choice 22
2.1. Equivalents of the Axiom of Choice 22
2.2. Some applications of the Axiom of Choice in mathematics 24
2.3. The Prime Ideal Theorem 27
2.4. The Countable Axiom of Choice 33
2.5. Cardinal numbers 36
2.6. Problems 38
2.7. Historical remarks 42
Chapter 3. Consistency of the Axiom of Choice 44
3.1. Axiomatic systems and consistency 44
3.2. Axiomatic set theory 44
3.3. Transitive models of ZF 46
3.4. The constructible universe 51
3.5. Problems 53
3.6. Historical remarks 56
Chapter 4. Permutation models 57
4.1. Set theory with atoms 57
4.2. Permutation models 58
4.3. The basic Fraenkel model 61
4.4. The second Fraenkel model 61
4.5. The ordered Mostowski model 62
4.6. Problems 64
4.7. Historical remarks 67
Chapter 5. Independence of the Axiom of Choice 68
5.1. Generic models 68
5.2. Symmetric submodels of generic models 77
5.3. The basic Cohen model 79
5.4. The second Cohen model 81
5.5. Independence of the Axiom of Choice from the Ordering Principle 84
5.6. Problems 91
5.7. Historical remarks 97
Chapter 6. Embedding Theorems 98
6.1. The First Embedding Theorem 98
6.2. Refinements of the First Embedding Theorem 102
6.3. Problems 107
6.4. Historical remarks 109
Chapter 7. Models with finite supports 110
7.1. Independence of the Axiom of Choice from the Prime Ideal Theorem 110
7.2. Independence of the Prime Ideal Theorem from the Ordering Principle 113
7.3. Independence of the Ordering Principle from the Axiom of Choice for Finite Sets 117
7.4. The Axiom of Choice for Finite Sets 120
7.5. Problems 125
7.6. Historical remarks 130
Chapter 8. Some weaker versions of the Axiom of Choice 132
8.1. The Principle of Dependent Choices and its generalization 132
8.2. Independence results concerning the Principle of Dependent Choices 135
8.3. Problems 144
8.4. Historical remarks 145
Chapter 9. Nontransferable statements 146
9.1. Statements which imply AC in ZF but are weaker than AC in ZFA 146
9.2. Independence results in ZFA 147
9.3. Problems 151
9.4. Historical remarks 153
Chapter 10. Mathematics without choice 154
10.1. Properties of the real line 154
10.2. Algebra without choice 158
10.3. Problems 161
10.4. Historical remarks 163
Chapter 11. Cardinal numbers in set theory without choice 164
11.1. Ordering of cardinal numbers 164
11.2. Definability of cardinal numbers 165
11.3. Arithmetic of cardinal numbers 169
11.4. Problems 174
11.5. Historical remarks 178
Chapter 12. Some properties contradicting the Axiom of Choice 180
12.1. Measurability of N1 180
12.2. Closed unbounded sets and partition properties 182
12.3. The Axiom of Determinateness 189
12.4. Problems 194
12.5. Historical remarks 195
Appendix 196
A.1. Equivalents of the Axiom of Choice 196
A.2. Equivalents of the Prime Ideal Theorem 196
A.3. Various independence results 196
A.4. Miscellaneous examples 198
References 199
Author index 208
Subject index 210
List of symbols 213