Choice function

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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#edit

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#edit

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 (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 ${\displaystyle X}$ 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 AC_ω 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 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#edit

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

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

{{mathimg|7183511b19eefb9b372a5c3d35b5bd1ab8235623|width=22.811ex|height=2.843ex|va=-0.838ex|alt={\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.

A survey of selectors was published by Claude Ambrose Rogers and John E. Jayne in 2002.^[3]

Bourbaki tau function#edit

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 ${\displaystyle (\exists x)(P(x))}$ .^[4]

However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice.^[5] Hilbert realized this when introducing epsilon calculus.^[6]

Notes#edit

^ Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300.
^ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.
^ Jayne, John E.; Rogers, C. Ambrose (2002), Selectors, Princeton University Press, ISBN 978-0-691-09628-5, MR 1915965
^ Bourbaki, Nicolas (1968). Elements of Mathematics: Theory of Sets. Hermann. ISBN 0-201-00634-0.
^ John Harrison, "The Bourbaki View" eprint.
^ "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.

References#edit

This article incorporates material from Choice function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Categories Basic concepts in set theory Axiom of choice

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