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In mathematics , the axiom of choice , or AC , is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite.
The axiom of choice was formulated in by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of sets is finite, or if a selection rule is available — 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. In this case, "select the smallest number" is a choice function. Even if infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers if there are non-constructible reals. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any even infinite collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function.
For an infinite collection of pairs of socks assumed to have no distinguishing features , there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,  and it is included in the standard form of axiomatic set theory , Zermelo—Fraenkel set theory with the axiom of choice ZFC.
One motivation for this use 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. The axiom of choice is avoided in some varieties of constructive mathematics , although there are varieties of constructive mathematics in which the axiom of choice is embraced.
A choice function is a function f , defined on a collection X of nonempty sets, such that for every set A in X , f A is an element of A. With this concept, the axiom can be stated:.
Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function. Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.
The axiom of choice asserts the existence of such elements; it is therefore equivalent to:. 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 guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition. Another equivalent axiom only considers collections X that are essentially powersets of other sets:.
Authors who use this formulation often speak of the choice function on A , but be advised that this is a slightly different notion of choice function. Its domain is the powerset of A with the empty set removed , and so makes sense for any set A , 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.
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo—Fraenkel set theory without the axiom of choice ZF ; it is easily proved by mathematical induction. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.
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.
Not every situation requires the axiom of choice. For finite sets X , the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several a finite number of boxes, each containing at least one item, then we can choose exactly one item from each box.
Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. The number of boxes is finite, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. A formal proof for all finite sets would use the principle of mathematical induction to prove "for every natural number k , every family of k nonempty sets has a choice function.
The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections. For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.
Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set, and makes it unnecessary to apply the axiom of choice. The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists?
For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we shall never be able to produce a choice function for all of X.
Next we might try specifying the least element from each set. But some subsets of the real numbers do not have least elements. So this attempt also fails. Additionally, consider for instance the unit circle S , and the action on S by a group G consisting of all rational rotations. Here G is countable while S is uncountable. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.
Since X is not measurable for any rotation-invariant countably additive finite measure on S , finding an algorithm to select a point in each orbit requires the axiom of choice. See non-measurable set for more details. The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are well-ordered : every nonempty subset of the natural numbers has a unique least element under the natural ordering.
One might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering. A proof requiring the axiom of choice may establish the existence of an object without explicitly 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 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 proves the existence of these intangibles objects that are proved to exist, but which cannot be explicitly constructed , which may conflict with some philosophical principles. 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. The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.
Despite these seemingly paradoxical facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. The debate is interesting enough, however, that it is considered of note 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 which requires the axiom of choice to be true.
It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of ZF, regardless of the truth or falsity of the axiom of choice in that particular model. The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable. For example, the Banach—Tarski paradox 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.
Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true. Statements such as the Banach—Tarski paradox can be rephrased as conditional statements, for example, "If AC holds, then the decomposition in the Banach—Tarski paradox exists. As discussed above, in ZFC, the axiom of choice is able to provide " nonconstructive proofs " in which the existence of an object is proved although no explicit example is constructed.
ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed. The status of the axiom of choice varies between different varieties of constructive mathematics. A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence. Thus the axiom of choice is not generally available in constructive set theory.
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. 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 in constructive set theory.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned. In , Paul Cohen employed the technique of forcing , developed for this purpose, to show that, assuming ZF is consistent, the axiom of choice itself is not a theorem of ZF.
Together these results establish that the axiom of choice is logically independent of ZF. The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse. Because of independence, the decision whether to use the axiom of choice or its negation in a proof cannot be made by appeal to other axioms of set theory.
The decision must be made on other grounds. One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal , every vector space has a basis , and every product of compact spaces is compact.
Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality. The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic , are provable in ZF if and only if they are provable in ZFC. When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof.
The axiom of choice is not the only significant statement which is independent of ZF.
Axiom Of Choice
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Axiom of Choice
In mathematics , the axiom of choice , or AC , is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. The axiom of choice was formulated in by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of sets is finite, or if a selection rule is available — some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers.