axiom of choice

Illustration of the axiom of choice, with each set Si represented as a jar and its elements represented as marbles. Each element xi is represented as a marble on the right. Colors are used to suggest a functional association of marbles after adopting the choice axiom. The existence of such a choice function is in general independent of ZF for collections of infinite cardinality, even if all Si are finite. (Si) is an infinite indexed family of sets indexed over the real numbers R; that is, there is a set Si for each real number i, with a small sample shown above. Each set contains at least one, and possibly infinitely many, elements. The axiom of choice allows us to select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. In general, the collections may be indexed over any set I, (called index set whose elements are used as indices for elements in a set) not just R.

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from each set, even if the collection is infinite. Formally, the axiom establishes existence rather than a construction; it states that for every set