Axiom of choice
Axiom asserting existence of a choice function for any collection of nonempty sets.
The axiom of choice is a principle in set theory, introduced by Ernst Zermelo in 1904 to support his proof of the well-ordering theorem. Put simply, it says that for any collection of non-empty sets—even an infinite one—you can form a new set that picks exactly one element from each of the original sets. More precisely, the axiom deals with existence, not construction: for any set I and any I-indexed family of non-empty sets, there exists an I-indexed set of elements (drawn from the union of all those sets) such that each chosen element belongs to its corresponding set.
In many situations, you can make such a selection without relying on the axiom. For example, if the collection is finite, induction works. Or if there is a clear rule for picking—like "choose the smallest number" from each set of natural numbers—then a choice function exists naturally. Given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, picking the smallest element from each yields {4, 10, 1}. Even with infinitely many sets of natural numbers, this rule works, so no axiom is needed. But for the collection of all non-empty subsets of the real numbers, no such canonical rule is known, and the axiom of choice must be invoked to get a choice function.
Bertrand Russell offered a famous analogy: from an infinite collection of unordered pairs of shoes, you can always pick the left shoe from each pair—no axiom required. But for an infinite collection of unordered pairs of socks (with no left-right distinction), there is no natural way to choose one from each pair, so the axiom of choice is needed.
Though it was once controversial, the axiom is now widely accepted by mathematicians and is part of the standard Zermelo–Fraenkel set theory with choice (ZFC). Many important results, like Tychonoff's theorem, depend on it. Some contemporary set theorists study axioms that conflict with choice, such as the axiom of determinacy. Meanwhile, some forms of constructive mathematics reject the axiom, while others accept it.
A choice function (also called a selector or selection) is a function f defined on a collection X of non-empty sets, such that for every set A in X, f(A) is an element of A. Using this concept, the axiom can be stated formally as: for every X, either the empty set is in X, or there exists a function f whose domain is X and such that for every A in X, f(A) belongs to A.
- field
- Mathematics (set theory)
- known_for
- Axiom of choice, well-ordering theorem
- formulated_by
- Ernst Zermelo
- year_formulated
- 1904
- abbreviations
- AC, AoC
- related_theories
- Zermelo–Fraenkel set theory (ZF), Zermelo–Fraenkel set theory with the axiom of choice (ZFC)
Lore & Background
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. It is now used without reservation by most mathematicians and is included in the standard form of axiomatic set theory, Zermelo–Fraenkel 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.
Reader's Guide
The axiom of choice is a foundational principle in set theory that allows the selection of an element from each set in a collection, even when no explicit rule for selection exists. While in many cases a choice can be made without invoking the axiom—for example, by always picking the smallest element from sets of natural numbers—there are collections, such as all non-empty subsets of the real numbers, where no canonical rule is available. Bertrand Russell illustrated this with an analogy: from an infinite collection of unordered pairs of shoes, one can always pick the left shoe, but for indistinguishable socks, the axiom of choice is needed. Although originally controversial, the axiom is now widely accepted and is part of ZFC, the standard axiomatic framework. Contemporary set theorists also study axioms incompatible with the axiom of choice, such as the axiom of determinacy, and some varieties of constructive mathematics avoid it while others embrace it.
Did You Know?
- The axiom of choice was formulated in 1904 by Ernst Zermelo to formalize his proof of the well-ordering theorem.
- Bertrand Russell coined an analogy involving pairs of shoes and socks to illustrate when the axiom of choice is needed.
- The axiom of choice is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
- Some generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.
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