The Online Encyclopedia and Dictionary






Axiom of choice

In mathematics, the axiom of choice is an axiom of set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following:

Let X be a collection of non-empty sets. Then we can choose a member from each set in that collection.

Stated more formally:

There exists a function f defined on X such that for each set S in X, f(S) is an element of S.

Another formulation of the axiom of choice (AC) states:

Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

For many years, the axiom of choice was used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let F(X) be one of the members of X for all X in S." Here, the existence of the function F depends on the axiom of choice.

The principle seems obvious: if there are several boxes, each containing at least one item, the axiom simply states that one can choose exactly one item from each box. Although the statement sounds straightforward, there is controversy over what it means to choose something from these sets. To see the issue, let us look at some sample sets.

1. Let X be any finite collection of non-empty sets.
Then f can be stated explicitly (out of set A choose a, ...), since the number of sets is finite.
Here the axiom of choice is not needed; the existence of the choice set follows from the other axioms of set theory.
2. Let X be the collection of all non-empty subsets of the natural numbers {0, 1, 2, 3, ... }.
Then f can be the function that chooses the smallest element in each set.
Again the axiom of choice is not needed, since we have a rule for doing the choosing.
3. Let X be the set of all sub-intervals of (0,1) with a length greater than 0.
Then f can be the function that chooses the midpoint of each interval.
Again the axiom of choice is not needed.
4. Let X be the collection of all non-empty subsets of the reals.
Now the existence of f is not so straightforward. There is no obvious definition of f that will guarantee success. We cannot simply have f pick the smallest element as we did in example 2 because a set of real numbers need not have a smallest element; there is not, for example, a smallest rational number or a smallest positive real number. Perhaps under some ordering of the reals other than the usual one there would always be a smallest element. However, the other axioms of ZF set theory do not guarantee the existence of a well-ordering of the real numbers (or of any other uncountable set). In fact the statement that every set can be well-ordered is equivalent to the axiom of choice.

The axiom of choice asserts that there is some function f that will choose an element out of each set in the collection. It gives no indication of how the function would be defined, it simply mandates its existence. Theorems whose proofs involve the axiom of choice are always nonconstructive: they demonstrate the existence of something without telling us how to get it.

The axiom of choice has been proven to be logically independent of the remaining axioms of set theory; that is, it can be neither proven nor disproven from them (unless those remaining axioms contain an unknown contradiction). This is the result of work by Kurt Gödel and Paul Cohen. Thus no contradictions arise if the axiom of choice is rejected. However, most mathematicians accept either it, or a weakened variant of it, because it makes their jobs easier. Despite this, there is some study of systems in which the axiom of choice is either not true or at least not assumed (see also axiom of regularity). It is important to be aware of which proofs in mathematics use the axiom of choice and which do not.

One reason that some mathematicians dislike the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach-Tarski paradox which says in effect that it is possible to "carve up" the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell us how to carve up the unit sphere to make this happen, it simply tells us that it can be done.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up. There are also a remarkable number of important statements that are equivalent to the axiom of choice, most important among them Zorn's lemma and the well-ordering theorem: every set can be well-ordered. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle.

Several central theorems in different branches of mathematics require the axiom of choice (or a weak version of it, such as the Boolean prime ideal theorem, the axiom of countable choice, or the axiom of dependent choice):


The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's Lemma?
— Jerry Bona

(The joke here is that, in truth, all three of these are mathematically equivalent, but the statement underscores the fact that most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex to form any intuitive feeling about).

The axiom gets its name not because mathematicians prefer it to other axioms.
— A. K. Dewdney

From the famous April Fool's Day article in the computer recreations column of the Scientific American, April 1989.

External links

  • A leisurely introduction to the axiom, popular consequences, and further links are found at Eric Schechter's homepage .
  • There are many people still doing work on the axiom of choice and its consequences. If you are interested in more, look up Paul Howard at EMU .

Last updated: 02-08-2005 12:31:52
Last updated: 05-03-2005 02:30:17