Choice function

A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns some element of each set S in that collection to S by f(S); f(S) maps S to some element 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 ${\displaystyle X}$  is a finite set of nonempty sets, then one can construct a choice function for ${\displaystyle X}$  by picking one element from each member of ${\displaystyle X.}$  This requires only finitely many choices, so neither AC or ACω is needed.
• If every member of ${\displaystyle X}$  is a nonempty set, and the union ${\displaystyle \bigcup X}$  is well-ordered, then one may choose the least element of each member of ${\displaystyle X}$ . In this case, it was possible to simultaneously well-order every member of ${\displaystyle X}$  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.)

Choice function of a multivalued map

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

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

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

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.[2] See Selection theorem.

Bourbaki tau function

Nicolas Bourbaki used epsilon calculus for their foundations that had a ${\displaystyle \tau }$  symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if ${\displaystyle P(x)}$  is a predicate, then ${\displaystyle \tau _{x}(P)}$  is one particular object that satisfies ${\displaystyle P}$  (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example ${\displaystyle P(\tau _{x}(P))}$  was equivalent to ${\displaystyle (\exists x)(P(x))}$ .[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: ${\displaystyle A(a)\to A(\varepsilon (A))}$ , where ${\displaystyle \varepsilon }$  is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From nCatLab.