A choice function (selector, selection) is a
mathematical functionf that is defined on some
collectionX 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 f defined by f({1, 4, 7}) = 7, f({9}) = 9 and f({2, 7}) = 2 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 is a
finite set of nonempty sets, then one can construct a choice function for by picking one element from each member of This requires only finitely many choices, so neither AC or ACω is needed.
If every member of is a nonempty set, and the
union is well-ordered, then one may choose the least element of each member of . In this case, it was possible to simultaneously well-order every member of 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 to Y (equivalently, is a function from X to the
power set of Y).
Nicolas Bourbaki used
epsilon calculus for their foundations that had a symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if is a predicate, then is one particular object that satisfies (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example was equivalent to .[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]
^"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: , where is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From
nCatLab.