## Axiom of ChoiceAC, 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 |