Axiom of Choice

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Springer Science & Business Media, May 11, 2006 - Mathematics - 194 pages
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AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that:

- Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).

- Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).

- Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness).

Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

 

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Contents

Origins
1
11 Hilberts First Problem
2
Choice Principles
9
22 Some Concepts Related to the Axiom of Choice
13
Elementary Observations
21
32 Unnecessary Choice
27
Compactness
32
Disasters without Choice
43
Function Spaces The Ascoli Theorem
95
The Baire Category Theorem
102
Coloring Problems
109
Disasters with Choice
117
Paradoxical Decompositions
126
Disasters either way
137
Beauty without Choice
143
72 Measurability The Axiom of Determinateness
150

42 Disasters in Cardinal Arithmetic
51
43 Disasters in Order Theory
56
Vector Spaces
66
Categories
71
The Reals and Continuity
72
Countable Sums
79
Products The Tychonoff and the ČechStone Theorem
85
Models
158
References
169
Selected Books and Longer Articles
181
List of Symbols
183
List of Axioms
185
Index
188
Copyright

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