Axiom of Choice
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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11 Hilberts First Problem
22 Some Concepts Related to the Axiom of Choice
32 Unnecessary Choice
Disasters without Choice
Function Spaces The Ascoli Theorem
The Baire Category Theorem
Disasters with Choice
Disasters either way
Beauty without Choice
72 Measurability The Axiom of Determinateness
42 Disasters in Cardinal Arithmetic
43 Disasters in Order Theory
The Reals and Continuity
Products The Tychonoff and the ČechStone Theorem