The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

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Wikipedia contributors. (2024, February 2). Axiom of choice. In Wikipedia, The Free Encyclopedia. Retrieved February 5, 2024, from https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1202259837

Well-ordering theorem (Wikipedia) In mathematics, the well-ordering theorem, also known as Zermelo’s theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn’s lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents). Ernst Zermelo introduced the axiom of choice as an “unobjectionable logical principle” to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox.

Axiom of choice (Wikipedia) In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite.