axiom of choice: a standalone construct

Ernst Zermelo introduced the axiom of choice in 1904 in Beweis, daß jede Menge wohlgeordnet werden kann (Proof that every set can be well-ordered), stating that for any collection of nonempty sets there exists a choice function, a rule that selects exactly one element from each set, even when the collection is infinite and no explicit recipe for choosing is available. The axiom is logically equivalent to two other cornerstone statements, Zorn's lemma and the well-ordering theorem, so a proof may invoke whichever form is most convenient. Its standing within standard set theory was settled in two stages: Kurt Gödel showed in 1938 that the axiom cannot be disproved from the other Zermelo-Fraenkel axioms, and Paul Cohen showed in 1963 that it cannot be proved from them either, so the axiom is independent of that system. What makes the axiom of choice a landmark is that it lets a mathematician assert that a selection across infinitely many sets exists without exhibiting any procedure that builds it, a nonconstructive step that underlies large parts of analysis, algebra, and topology while remaining something one may assume or do without.[1][2][3]

Unit

A foundational axiom of set theory, asserting the existence of a choice function over any collection of nonempty sets

Recovery means

Not applicable

Measured by

Identified in a proof by its reliance on a choice function, Zorn's lemma, or the well-ordering theorem, forms shown equivalent within Zermelo-Fraenkel set theory