# Choice function

For the combinatorial choice function C(n, k), see Combination and Binomial coefficient.

A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X.

## An example

Let X = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} is a choice function on X.

## History and importance

Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem,[1] 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 (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, 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 ACω 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 ACω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)

## Refinement of the notion of choice function

A function $f: A \rightarrow B$ is said to be a selection of a multivalued map φ:AB (that is, a function $\varphi:A\rightarrow\mathcal{P}(B)$ from A to the power set $\mathcal{P}(B)$), if

$\forall a \in A \, ( f(a) \in \varphi(a) ) \,.$

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.[2]

### Bourbaki 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 the 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 $(\exists x)(P(x))$.[3]

However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice.[4] Hilbert realized this when introducing epsilon calculus.[5]

5. ^ "Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transfinite axioms are derivable from a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics, namely, the axiom of choice: $A(a)\to A(\varepsilon(A))$, where $\varepsilon$ is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From nCatLab.