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The Axiom of Choice (AC) is an [[axiom]] of [[Zermelo-Fraenkel]] set theory that states:

The Axiom of Choice (AC)<ref>There are numerous equivalent mathematical descriptions of the Axiom of Choice.  Here are two:

::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is an [[axiom]] of [[Zermelo-Fraenkel]] set theory holding that:<ref>Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

Or more compactly:

:given any collection of sets, however large, we can pick one element from each set in the collection.

::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math>

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."  More precisely, the Axiom of Choice states that:

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."  More precisely, the Axiom of Choice states that:

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the [[Well-Ordering Theorem]], the existence of [[cardinal numbers]], the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]].

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the [[Well-Ordering Theorem]], the existence of [[cardinal numbers]], the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]].

Despite its usefulness, many mathematicians reject the Axiom of Choice. This rejection is based on the belief that all [[mathematical proofs]] should be constructive. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Suppose we consider a set consisting of many pairs of shoes. The Axiom of Choice says that we can create a new set by selecting just the left shoe from each pair. It is obvious that we can do this for shoes, since left and right shoes are distinguishable. What if the pairs of objects in question are <i>not</i> distinguishable and it is not acceptable to say 'just pick any one of them'?

Despite its usefulness, many mathematicians reject the Axiom of Choice. "It bothers some people because it asserts the existence of a set ... without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the ''only'' formal set-theoretical axiom which does this.  For this reason it is customary to mention the axiom of choice whenever it is used.  It need not be used if the number of sets is finite."<ref>Willard, ''supra'', 1.17, p. 9.</ref>

 

The rejection of the Axiom of Choice reflects the preference that all [[mathematical proofs]] should be constructive. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Suppose we consider a set consisting of many pairs of shoes. The Axiom of Choice says that we can create a new set by selecting just the left shoe from each pair. It is obvious that we can do this for shoes, since left and right shoes are distinguishable. What if the pairs of objects in question are <i>not</i> distinguishable and it is not acceptable to say 'just pick any one of them'?

Put another way, it may be objectionable to insist that a function exists based on the Axiom of Choice, and then to derive results based on the existence of that function, when in fact there may be no known way to construct such a function. This objection to use of the Axiom of Choice is analogous to the objection to the use of non-[[falsifiable]] claims in science.  Since the Axiom of Choice is independent of the other axioms, results provable with it may be verified with the other axioms.

Put another way, it may be objectionable to insist that a function exists based on the Axiom of Choice, and then to derive results based on the existence of that function, when in fact there may be no known way to construct such a function. This objection to use of the Axiom of Choice is analogous to the objection to the use of non-[[falsifiable]] claims in science.  Since the Axiom of Choice is independent of the other axioms, results provable with it may be verified with the other axioms.