axiom of choice  axiom of choice, sometimes called Zermelo’s
axiom of choice,  statement in the language of set theory that
makes it possible to form sets by choosing an element
simultaneously from each member of an infinite collection of
sets even when no algorithm exists for the selection. The axiom
of choice has many mathematically equivalent formulations, some
of which were not immediately realized to be equivalent. One
version states that, given any collection of disjoint sets (sets
having no common elements), there exists at least one set
consisting of one element from each of the nonempty sets in the
collection; collectively, these chosen elements make up the
“choice set.” Another common formulation is to say that for any
set S there exists a function f (called a “choice function”)
such that, for any nonempty subset s of S, f(s) is an element of
s.  The axiom of choice was first formulated in 1904 by the
German mathematician Ernst Zermelo in order to prove the “well-
ordering theorem” (every set can be