＜補足＞ １）かように、Aのべき集合全体(空集合抜き)の選択関数は不要 ２）Aと同じ順序数（超限帰納）の選択関数で間に合うことを指摘しておく ３）調べると 可算集合Aを整列させるためには、従属選択公理が必要とある （下記の独 de.wikipedia ご参照。en.wikipediaにも類似記載あり。 即ち、”to construct a sequence using countable transfinite recursion” なお、Axiom of countable choice en.wikipedia は、”for every n∈N”つまり、順序数の長さでω（＝N）が限界）
(参考) de.wikipedia.org/wiki/Axiom_der_abh%C3%A4ngigen_Auswahl Axiom der abhängigen Auswahl (google 英訳) axiom of dependent choice use The axiom of dependent choice is a sufficient fragment of the axiom of choice to construct a sequence using countable transfinite recursion .
en.wikipedia.org/wiki/Axiom_of_dependent_choice Axiom of dependent choice Use The axiom DC is the fragment of AC that is required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of those choices cannot be made independently of previous choices.
en.wikipedia.org/wiki/Axiom_of_countable_choice Axiom of countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n∈N, there exists a function f with domain N such that f(n)∈A(n) for every n∈N.
