Choice function


A choice function 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 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 = . Then the function that assigns 7 to the set, 9 to, and 2 to is a choice function on X.

History and importance

introduced choice functions as well as the axiom of choice 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 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.
Given two sets X and Y, let F be a multivalued map from X and Y.
A function is said to be a selection of F, if:
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.

Bourbaki tau function

used epsilon calculus for their foundations that had a symbol that could be interpreted as choosing an object that satisfies a given proposition. So if is a predicate, then is one particular object that satisfies . Hence we may obtain quantifiers from the choice function, for example was equivalent to.
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.