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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...