: Mathematics: axiom of choice

set theory > axiom of choice

axiom of choice

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 arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ${\\displaystyle (S_{i})_{i\\in I}}$ of nonempty sets, there exists an indexed set ${\\displaystyle (x_{i})_{i\\in I}}$ such that ${\\displaystyle x_{i}\\in S_{i}}$ for every ${\\displaystyle i\\in I}$ . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Axiom_of_choice)