For any set, a mapping exists that chooses one element of each subset {axiom of choice} {Zermelo's axiom}. Elements are not in any other non-empty set, even if number of non-empty subsets is infinite. Axiom of choice is independent of set theory. Zermelo set theory has no paradoxes but is not consistent. If Zermelo-Fraenkel set theory is consistent without axiom of choice, then ZF set theory is consistent with axiom of choice.
Axiom of choice leads to unexpected consequences if applied to sets with uncountably infinite members. In such set, object can divide into five pieces that can rotate, translate, and invert to make much greater volume {Banach-Tarski theorem}.
In partially ordered sets in which subsets have upper bounds, sets have greatest member {Zorn's lemma} {Zorn lemma}. Zorn's lemma is equivalent to axiom of choice.
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Date Modified: 2022.0225