The Axiom

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, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every set I {\displaystyle I} and every I {\displaystyle I} -indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an I {\displaystyle I} -indexed set ( x i ) i ∈ I {\displaystyle (x_{i})_{i\in I}} of elements of ∪ i ∈ I S i {\displaystyle \cup _{i\in I}S_{i}} such that x i ∈ S i {\displaystyle x_{i}\in S_{i}} for every i ∈ I {\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. In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite (in which induction can be applied), or if a canonical rule on how to choose the elements is available – some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to form a choice function from choosing the smallest element from each set to produce a set; the axiom of choice is not needed here. On the other hand, for the collection of all non-empty subsets of the real numbers, there is no known canonical rule by which one can choose one element from each of these subsets. In that case, the axiom of choice must be invoked to construct the desired…