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.

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The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident.

For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set—a transversal or choice set —containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences—many indispensable, some startling—and has come to figure prominently in discussions on the foundations of mathematics. It or its equivalents have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it.

In Ernst Zermelo formulated the Axiom of Choice abbreviated as AC throughout this article in terms of what he called coverings Zermelo He continues:. A different choice function is obtained by assigning to each pair its greatest element. AC1 : Any collection of nonempty sets has a choice function. AC1 can be reformulated in terms of indexed or variable sets. AC1 is then equivalent to the assertion. AC2 : Any indexed collection of sets has a choice function.

Informally speaking, AC2 amounts to the assertion that a variable set with an element at each stage has a variable element. Finally AC3 is easily shown to be equivalent in the usual set theories to: [ 2 ]. In a paper Zermelo introduced a modified form of AC. CAC : Any collection of mutually disjoint nonempty sets has a transversal. It is to be noted that AC1 and CAC for finite collections of sets are both provable by induction in the usual set theories. But in the case of an infinite collection, even when each of its members is finite, the question of the existence of a choice function or a transversal is problematic [ 4 ].

For example, as already mentioned, it is easy to come up with a choice function for the collection of pairs of real numbers simply choose the smaller element of each pair.

But it is by no means obvious how to produce a choice function for the collection of pairs of arbitrary sets of real numbers. In the first of these, as remarked above, he reformulated AC in terms of transversals; in the second a he made explicit the further assumptions needed to carry through his proof of the well-ordering theorem. These assumptions constituted the first explicit presentation of an axiom system for set theory.

As the debate concerning the Axiom of Choice rumbled on, it became apparent that the proofs of a number of significant mathematical theorems made essential use of it, thereby leading many mathematicians to treat it as an indispensable tool of their trade.

Hilbert, for example, came to regard AC as an essential principle of mathematics [ 5 ] and employed it in his defence of classical mathematical reasoning against the attacks of the intuitionists. Although the usefulness of AC quickly become clear, doubts about its soundness remained.

These doubts were reinforced by the fact that it had certain strikingly counterintuitive consequences. See Wagon Here is a brief chronology of AC : [ 6 ]. Here by an atom is meant a pure individual, that is, an entity having no members and yet distinct from the empty set so a fortiori an atom cannot be a set.

This had to wait until when Paul Cohen showed that it is consistent with the standard axioms of set theory which preclude the existence of atoms to assume that a countable collection of pairs of sets of real numbers fails to have a choice function. He introduced a new hierarchy of sets—the constructible hierarchy—by analogy with the cumulative type hierarchy. The relative consistency of AC with ZF follows.

The Axiom of Choice is closely allied to a group of mathematical propositions collectively known as maximal principles. Broadly speaking, these propositions assert that certain conditions are sufficient to ensure that a partially ordered set contains at least one maximal element , that is, an element such that, with respect to the given partial ordering, no element strictly exceeds it.

To state it, we need a few definitions. Here is an informal argument. This may in turn be formulated in a dual form.

Call a family of sets strongly reductive if it is closed under intersections of nests. Then any nonempty strongly reductive family of sets has a minimal element, that is, a member properly including no member of the family. The Axiom of Choice has numerous applications in mathematics, a number of which have proved to be formally equivalent to it [ 13 ].

Historically the most important application was the first, namely:. The Well-Ordering Theorem Zermelo , Every set can be well-ordered. After Zermelo published his proof of the well-ordering theorem from AC , it was quickly seen that the two are equivalent. The Multiplicative Axiom Russell The product of any set of non-zero cardinal numbers is non-zero. The Set-Theoretic Distributive Law. Principle of Dependent Choices Bernays , Tarski This principle, although much weaker than AC , cannot be proved without it in the context of the remaining axioms of set theory.

There are a number of mathematical consequences of AC which are known to be weaker [ 14 ] than it, in particular:. The question of the equivalence of this with AC is one of the few remaining interesting open questions in this area; while it clearly implies BPI , it was proved independent of BPI in Bell An initial connection between AC and logic emerges by returning to its formulation AC3 in terms of relations, namely: any binary relation contains a function with the same domain.

The scheme of sentences. Here predicates are playing the role of sets. Up to now we have tacitly assumed our background logic to be the usual classical logic. But the true depth of the connection between AC and logic emerges only when intuitionistic or constructive logic is brought into the picture. The fact that the Axiom of Choice implies Excluded Middle seems at first sight to be at variance with the fact that the former is often taken as a valid principle in systems of constructive mathematics governed by intuitionistic logic, e.

To resolve the difficulty, we note that in deriving Excluded Middle from ACL1 essential use was made of the principles of Predicative Comprehension and Extensionality of Functions [ 18 ]. It follows that, in systems of constructive mathematics affirming AC but not Excluded Middle either the principle of Predicative Comprehension or the principle of Extensionality of Functions must fail.

While the principle of Predicative Comprehension can be given a constructive justification, no such justification can be provided for the principle of Extensionality of Functions. In intuitionistic set theory that is, set theory based on intuitionistic as opposed to classical logic—we shall abbreviate this as IST and in topos theory the principles of Predicative Comprehension and Extensionality of Functions both appropriately construed hold and so there AC implies Excluded Middle.

The derivation of Excluded Middle from AC was first given by Diaconescu in a category-theoretic setting. His proof employed essentially different ideas from the proof presented above; in particular, it makes no use of extensionality principles but instead employs the idea of the quotient of an object or set by an equivalence relation.

Here it is. Now we show that, if AC4 holds, then any subset of a set has an indicator, and hence is detachable. It can be shown Bell that each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent in intuitionistic set theory to a suitably weakened version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles. All of these schemes follow, of course, from the full law of excluded middle, that is SLEM for arbitrary formulas.

This principle, a straightforward consequence of the axiom of choice, asserts that, for any pair of instantiated properties of members of 2, instances may be assigned to the properties in a manner that depends just on their extensions.

Each of the logical principles tabulated above is equivalent in IST to a choice principle. In fact:. In order to provide choice schemes equivalent to Lin and Stone we introduce. These results show just how deeply choice principles interact with logic, when the background logic is assumed to be intuitionistic.

In a classical setting where the Law of Excluded Middle is assumed these connections are obliterated.

Readers interested in the topic of the axiom of choice and type theory may consult the following supplementary document:. The author and editors would like to thank Jesse Alama for carefully reading this piece and making many valuable suggestions for improvement. Origins and Chronology of the Axiom of Choice 2. Independence and Consistency of the Axiom of Choice 3. Mathematical Applications of the Axiom of Choice 5. AC1 is then equivalent to the assertion AC2 : Any indexed collection of sets has a choice function.

AC1 can also be reformulated in terms of relations, viz. Finally AC3 is easily shown to be equivalent in the usual set theories to: [ 2 ] AC4 : Any surjective function has a right inverse.

Hilbert Here is a brief chronology of maximal principles. It seems to have been Artin who first recognized that ZL would yield AC , so that the two are equivalent over the remaining axioms of set theory. Mathematical Applications of the Axiom of Choice The Axiom of Choice has numerous applications in mathematics, a number of which have proved to be formally equivalent to it [ 13 ]. Historically the most important application was the first, namely: The Well-Ordering Theorem Zermelo , Early applications of AC include: Every infinite set has a denumerable subset.

This principle, again weaker than AC , cannot be proved without it in the context of the remaining axioms of set theory. Every infinite cardinal number is equal to its square. This was proved equivalent to AC in Tarski Every vector space has a basis initiated by Hamel This was proved equivalent to AC in Blass Every field has an algebraic closure Steinitz This assertion is weaker than AC , indeed is a consequence of the weaker compactness theorem for first-order logic see below.

There is a Lebesgue nonmeasurable set of real numbers Vitali Solovay established its independence of the remaining axioms of set theory. This was proved equivalent to AC in Kelley This was proved equivalent to AC by Tarski.

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## The Axiom of Choice

The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set—a transversal or choice set —containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences—many indispensable, some startling—and has come to figure prominently in discussions on the foundations of mathematics. It or its equivalents have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. In Ernst Zermelo formulated the Axiom of Choice abbreviated as AC throughout this article in terms of what he called coverings Zermelo He continues:.

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## Axiom of choice

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## Thomas Jech: The Axiom of Choice

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