Axiom of Choice

The Axiom of Choice (AC) is a powerful axiom of Zermelo-Fraenkel set theory that is useful for proving otherwise unprovable problems, but which has also been used to prove statements that appear absurd. The Axiom of Choice holds that:

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

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:

For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S.

Mathworld explains the Axiom of Choice as follows:

Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.

Yet another helpful explanation of the Axiom of Choice is this:

If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from each other.

Use of the Axiom of Choice

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. In algebra it is common to use Zorn's Lemma (also equivalent to the Axiom of Choice) to study ideals in infinite Noetherian rings.

Use of the Axiom of Choice has led to some seemingly absurd results. In the Banach-Tarski Paradox, the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1. This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom. One attempt to resolve this apparent contradiction is to show that physical spheres are not Lebesgue measurable.

The highly publicized proof of Fermat's Last Theorem relies on the Axiom of Choice, which was one of the points of criticism by Marilyn vos Savant in her column and book.

Controversy

Despite its usefulness, many mathematicians reject the Axiom of Choice. Stephen Willard, author of the textbook "General Topolog"y, wrote the following after describing the axiom of choice:

It is left to the reader to decide that these two statements both say the same thing. What they say is: given any collection of sets, however large, we can pick one element from each set in the collection. 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-theoretic 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.

The rejection of the Axiom of Choice reflects a preference for constructive mathematical proofs. 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. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.

Sources

• http://www.math.vanderbilt.edu/~schectex/ccc/choice.html