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.
Mathematical Sciences>Set Theory>Axioms>Axiom Of Choice
3-Set Theory-Axioms-Axiom Of Choice
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Date Modified: 2022.0224