Choice function#Choice function of a multivalued map

{{for|the combinatorial choice function C(n, k)|Combination|Binomial coefficient}}

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.

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,{{cite journal| first=Ernst| last=Zermelo| year=1904| title=Beweis, dass jede Menge wohlgeordnet werden kann| journal=Mathematische Annalen| volume=59| issue=4| pages=514–16| doi=10.1007/BF01445300| url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526}} 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.)

Choice 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.{{cite book

| last = Border

| first = Kim C.

| title = Fixed Point Theorems with Applications to Economics and Game Theory

| year = 1989

| publisher = Cambridge University Press

| isbn = 0-521-26564-9

}} See Selection theorem.

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 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 $(\exists x)(P(x))$ .{{cite book|last=Bourbaki|first=Nicolas|title=Elements of Mathematics: Theory of Sets|date=1968 |publisher=Hermann |isbn=0-201-00634-0}}

However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice.John Harrison, "The Bourbaki View" [http://www.rbjones.com/rbjpub/logic/jrh0105.htm eprint]. Hilbert realized this when introducing epsilon calculus."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 [http://ncatlab.org/nlab/show/choice+operator nCatLab].

Notes

References

Category:Basic concepts in set theory

Category:Axiom of choice