Axiom of Choice

The Axiom of Choice is an axiom of Zermelo-Fraenkel set theory that states:

<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>

Or more compactly:

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

The English translation of the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We have the right to 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."

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 not distinguishable and it is not acceptable to say 'just pick any one of them'?

The formulation of constructive axiom of choice is one of three major problems which challenge 21st century logicians.