Английская Википедия:Choice function
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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 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 <math>X</math> is a finite set of nonempty sets, then one can construct a choice function for <math>X</math> by picking one element from each member of <math>X.</math> This requires only finitely many choices, so neither AC or ACω is needed.
- If every member of <math>X</math> is a nonempty set, and the union <math>\bigcup X</math> is well-ordered, then one may choose the least element of each member of <math>X</math>. In this case, it was possible to simultaneously well-order every member of <math>X</math> 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, <math>F:X\rightarrow\mathcal{P}(Y)</math> is a function from X to the power set of Y).
A function <math>f: X \rightarrow Y</math> is said to be a selection of F, if:
<math display="block">\forall x \in X \, ( f(x) \in F(x) ) \,.</math>
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 <math> \tau </math> symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if <math> P(x) </math> is a predicate, then <math>\tau_{x}(P)</math> is one particular object that satisfies <math>P</math> (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example <math> P( \tau_{x}(P))</math> was equivalent to <math> (\exists x)(P(x))</math>.[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]
See also
Notes
References
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ 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: <math>A(a)\to A(\varepsilon(A))</math>, where <math>\varepsilon</math> is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From nCatLab.
