Choice function

A choice function (selector, selection) 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(S) 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 that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} 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 (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.)

Refinement of the notion of choice function

A function $f: A \rightarrow B$ is said to be a selection of a multivalued map φ:A → B (that is, a function $\varphi:A\rightarrow\mathcal{P}(B)$ from A to the power set $\mathcal{P}(B)$ ), if

\forall a \in A \, ( f(a) \in \varphi(a) ) \,.

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]

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 the 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 $(\exists x)(P(x))$ .^[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]

Notes

↑ 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.
↑ Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. 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: $A(a)\to A(\varepsilon(A))$ , 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 nCatLab.

References

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[Zermelo,_1904-1] Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300.

[2] Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9.

[3] Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0-201-00634-0.

[4] John Harrison, "The Bourbaki View" eprint.

[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: $A(a)\to A(\varepsilon(A))$ , 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 nCatLab.