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 and every -indexed family of nonempty sets, there exists an -indexed set of elements of such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. The axiom of choice is equivalent to the statement that every partition has a transversal.
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 â 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 <nowiki></nowiki>, 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 choice function.
Bertrand Russell coined an analogy: for any (even infinite) collection of unordered pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function without using the axiom of choice. However, for an infinite collection of unordered pairs of socks (assumed to have no distinguishing features such as being a left sock rather than a right sock), there is no natural (i.e., canonical) way of choosing one sock from each pair, so one must appeal to the axiom of choice to construct the desired choice function.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and is included in the standard form of axiomatic set theory, ZermeloâÂÂFraenkel set theory with the axiom of choice (ZFC). One motivation for this is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. While some varieties of constructive mathematics avoid the axiom of choice, others embrace it.
A choice function (also called selector or selection) is a function , defined on a collection of nonempty sets, such that for every set in , is an element of . With this concept, the axiom can be stated:
Formally, this may be expressed as follows:
Each choice function on a family of nonempty sets is an element of the Cartesian product of the sets in , and vice versa. Therefore an equivalent form of the axiom of choice is:
This form implies a more general form where the Cartesian product is of a general indexed family of sets (which may contain duplicates), since one can always select the same element from duplicate factors.
In this article and other discussions of the Axiom of Choice the following abbreviations are common:
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.
One variation avoids the use of choice functions by, in effect, replacing each choice function with its range:
This can be formalized in first-order logic as:
Note that is logically equivalent to .<br> In English, this first-order sentence reads:
This guarantees for any partition of a set the existence of a subset of containing exactly one element from each part of the partition.
Another equivalent axiom only considers collections that are essentially powersets of other sets:
Authors who use this formulation often speak of the choice function on , but this is a slightly different notion of choice function. Its domain is the power set of (with the empty set removed), and so makes sense for any set , whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as
which is equivalent to
The negation of the axiom can thus be expressed as:
Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.
The existence of a choice function for a finite collection of nonempty sets can be proved by the principle of finite induction, without appealing to the axiom of choice. The proof uses the fact that, given a single nonempty set , first-order logic allows choosing some concrete . However, since a proof in first-order logic must be finite, one cannot make an infinite number of choices with first-order logic alone.
Another case where the axiom of choice is not needed is when there exists an explicit rule that gives a canonical choice function. For example, if each member of the collection is a nonempty subset of the natural numbers, then one such explicit rule is to choose the smallest element of each . The canonical choice function that maps each to its smallest element can again be constructed in ZF without the axiom of choice.
In general, if the union of all sets in can be well-ordered, then a choice function for can be constructed without using the axiom of choice. Note that it does not suffice that each can be well-ordered, since the axiom of choice may be needed to choose a canonical well-ordering for each anyway.
As an example where the axiom of choice is required, let be set of all non-empty subsets of the real numbers. Choosing the least element from each set no longer works, because some subsets of the real numbers do not have least elements. For example, the open interval does not have a least element: if is in , then so is , and is always strictly smaller than . This strategy fails here because the natural order of real numbers is not a well-order.
If there exists a different ordering of the real numbers which is a well-ordering, then applying the least-element strategy with respect to that ordering would give a choice function for . Conversely, if there exists a choice function for , then the proof of the well-ordering theorem would show that a well-ordering of the real numbers does exist.
Let be the unit circle, and be the group consisting of all rational rotations (i.e., rotations by angles which are rational multiples of ). Since is countable while is uncountable, must break up into uncountably many orbits under the action of .
Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset of with the property that all of its translates by are disjoint from . The set of those translates partitions the circle into a countable collection of pairwise disjoint sets, which are all pairwise congruent. The set will be non-measurable for any rotation-invariant countably additive measure on : if has zero measure, countable additivity would imply that the whole circle has zero measure. If has positive measure, countable additivity would show that the circle has infinite measure.
Applying a similar construction to the three-dimensional ball can result in a set that is non-measurable even for any rotation-invariant finitely additive measure, as shown by the BanachâÂÂTarski paradox.
A proof requiring the axiom of choice may establish the existence of an object without canonically defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proved to exist using the axiom of choice, it is consistent that no such set is definable.
The axiom of choice asserts the existence of these intangibles (objects that are proved to exist, but which cannot be constructed in any canonical way), which may conflict with some philosophical principles. Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in category theory). This has been used as an argument against the use of the axiom of choice.
Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive. One example is the BanachâÂÂTarski paradox, which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.
Despite these seemingly paradoxical results, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. But the debate is interesting enough that it is considered notable when a theorem in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type that requires the axiom of choice to be true.
Theorems of ZF hold true in any model of that theory, regardless of the truth or falsity of the axiom of choice in that particular model. The implications of choice below, including weaker versions of the axiom itself, are listed because they are not theorems of ZF. The BanachâÂÂTarski paradox, for example, is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition. Such statements can be rephrased as conditional statementsâÂÂfor example, "If AC holds, then the decomposition in the BanachâÂÂTarski paradox exists." Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice.
As discussed above, in the classical theory of ZFC, the axiom of choice enables nonconstructive proofs in which the existence of a type of object is proved without an explicit canonical construction of an instance of this type. In fact, in set theory and topos theory, Diaconescu's theorem shows that the axiom of choice implies the law of excluded middle. The principle is thus not available in constructive set theory, where non-classical logic is employed.
The situation is different when the principle is formulated in Martin-Löf type theory. There and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem. A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does. The type theoretical context is discussed further below.
Different choice principles have been thoroughly studied in the constructive contexts and the principles' status varies between different school and varieties of the constructive mathematics. Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle. Errett Bishop, who is notable for developing a framework for constructive analysis, argued that an axiom of choice was constructively acceptable, saying
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
It has been known since as early as 1922 that the axiom of choice may fail in a variant of ZF with urelements, through the technique of permutation models introduced by Abraham Fraenkel and developed further by Andrzej Mostowski. The basic technique can be illustrated as follows: Let x<sub>n</sub> and y<sub>n</sub> be distinct urelements for , and build a model where each set is symmetric under the interchange x<sub>n</sub> â y<sub>n</sub> for all but a finite number of n. Then the set