Axiom of choice
Axiom asserting existence of a choice function for any collection of nonempty sets.
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The axiom of choice is a principle in set theory that says, roughly, if you have a bunch of non-empty sets, you can form a new set by picking exactly one item from each of them—even if there are infinitely many sets to pick from. The axiom was introduced by Ernst Zermelo to make his proof of the well-ordering theorem rigorous. It asserts that a choice exists, but doesn't tell you how to construct it.
In many situations, you don’t need the axiom. If you only have finitely many sets, you can pick elements using induction. If there’s a clear rule for choosing—like always taking the smallest number from a set of natural numbers—you can define a choice function without the axiom. For example, from the sets {4,5,6}, {10,12}, and {1,400,617,8000}, you can pick {4,10,1} by always choosing the smallest element. This works even for infinitely many sets of natural numbers. But for something like the collection of all non-empty subsets of the real numbers, no such simple rule exists, and the axiom of choice is needed.
Bertrand Russell illustrated this with an analogy. From an infinite collection of pairs of shoes, you can always pick the left shoe from each pair—no axiom needed. But from an infinite collection of identical socks (with no left or right distinction), there’s no natural way to choose one from each pair, so you must rely on the axiom of choice.
Though it was controversial at first, the axiom is now widely accepted by mathematicians and is part of standard Zermelo–Fraenkel set theory with choice (ZFC). Many important results, like Tychonoff’s theorem, depend on it. Some set theorists study alternatives, such as the axiom of determinacy, which conflicts with choice. In constructive mathematics, opinions vary: some reject the axiom, while others accept it.
A choice function is a function f defined on a collection X of non-empty sets, where for each set A in X, f(A) is an element of A. Using this idea, the axiom can be stated formally as: for every set X, either X contains the empty set, or there exists a function f whose domain is X 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
- associated_theory
- 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 states that for every set I and every I-indexed family (S_i)_{i∈I} of nonempty sets, there exists an I-indexed set (x_i)_{i∈I} of elements of ∪_{i∈I} S_i such that x_i ∈ S_i for every i∈I. 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 formally as: ∀X [∅ ∈ X ∨ ∃f [dom f = X ∧ ∀A ∈ X [f(A) ∈ A]]]. Each choice function on a family X of nonempty sets is an element of the Cartesian product of the sets in X, and vice versa.
Reader's Guide
The axiom of choice 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. Although originally controversial, 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. 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, or if a canonical rule on how to choose the elements 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, where one may always select the smallest number. However, for the collection of all non-empty subsets of the real numbers, there is no known canonical rule, and the axiom of choice must be invoked.
Did You Know?
- Bertrand Russell coined an analogy involving pairs of shoes and socks to illustrate when the axiom of choice is needed.
- The axiom of choice was formulated in 1904 by Ernst Zermelo to formalize his proof of the well-ordering theorem.
- For an infinite collection of unordered pairs of socks, there is no natural way of choosing one sock from each pair without the axiom of choice.
- The axiom of choice is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
Frequently Asked Questions
What is the Axiom of Choice?
It is a principle in set theory stating that for any collection of non-empty sets, there exists a function that selects exactly one member from each set in the family. It is most meaningful for infinite collections, where no explicit picking rule may be available.
Who formulated the Axiom of Choice and when?
Ernst Zermelo introduced the axiom in 1904 as part of his rigorous proof of the well-ordering theorem. It subsequently became a standard component of Zermelo–Fraenkel set theory, commonly abbreviated ZFC.
Does the Axiom of Choice tell you how to pick the elements?
No—it is purely an existence claim, guaranteeing that some choice function is available without specifying a constructive procedure. In that sense it is a non-constructive axiom: it says a selection is possible, not how to carry it out.
Why is the Axiom of Choice important in mathematics?
It underlies many central results across analysis, algebra, and topology that cannot be derived from ZF alone. Its inclusion in ZFC makes it one of the most widely relied-upon—and yet philosophically debated—axioms in modern mathematics.
Do you always need the Axiom of Choice to select elements from sets?
Not at all: finite families are handled by simple induction, and infinite families with a natural ordering (such as subsets of the integers) allow you to just take the smallest element each time. The axiom becomes essential when the collection is infinite and no such explicit rule is available.
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