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Choice function

Mathematical function

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Source updated June 23, 2026

Let X be a set of sets none of which are empty. Then a choice function (selector, selection) on X is a mathematical function f that is defined on X such that f is a mapping that assigns each element of X to one of its elements.

01An example

Let X = { {1,4,7}, {9}, {2,7} }. Then the function f defined by f({1, 4, 7}) = 7, f({9}) = 9 and f({2, 7}) = 2 is a choice function on X.

02History and importance

Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (AC_ω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or AC_ω, some sets can still be shown to have a choice function.

If $X$ is a finite set of nonempty sets, then one can construct a choice function for $X$ by picking one element from each member of $X.$ This requires only finitely many choices, so neither AC or AC_ω is needed.
If every member of $X$ is a nonempty set, and the union $\bigcup X$ is well-ordered, then one may choose the least element of each member of $X$ . In this case, it was possible to simultaneously well-order every member of $X$ by making just one choice of a well-order of the union, so neither AC nor AC_ω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)

03Choice function of a multivalued map

Given two sets $X$ and $Y$ , let $F$ be a multivalued map from $X$ to $Y$ (equivalently, $F:X\rightarrow {\mathcal {P}}(Y)$ is a function from $X$ to the power set of $Y$ ).

A function $f:X\rightarrow Y$ is said to be a selection of $F$ , if:

$\forall x\in X\,(f(x)\in F(x))\,.$

The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics. See Selection theorem.

04Bourbaki tau function

Nicolas Bourbaki used epsilon calculus for their foundations that had a $\tau$ symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if $P(x)$ is a predicate, then $\tau _{x}(P)$ is one particular object that satisfies $P$ (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example $P(\tau _{x}(P))$ was equivalent to $(\exists x)(P(x))$ .

However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice. Hilbert realized this when introducing epsilon calculus.

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Sources and credits

This article is adapted from the Wikipedia article “Choice function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Choice function

01An example

02History and importance

03Choice function of a multivalued map

04Bourbaki tau function

Sources and credits

Related topics

Axiom of countable choice

Axiom of dependent choice

Hausdorff paradox