>>636
>ついでにいうと可算選択公理では可算集合の整列はできない
>なぜなら可算集合の空でない部分集合の全体は、非可算集合だから
>ただし、別のやり方で整列はできる
>可算＝自然数の全体との全単射が存在する
>ということだから、この全単射を使えばいい

そこ 意味不明だよ
ここは、中高一貫校生が来る可能性があるので
赤ペン先生をしておく

１）下記 可算選択公理 Axiom of countable choice ACω は
　”Application of ACω yields a sequence (Bn) n∈N ”
　つまり ω長さの sequence (Bn) n∈N を作る能力がある
２）一方 Axiom of dependent choice DC は 下記
　”The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]
　It is possible to generalize the axiom to produce transfinite sequences.
　If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.”
３）要するに、DC は ACωより強力で ωを超えて ”produce transfinite sequences”だ
　また ”If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.”
　ってこと。つまりは、種々の選択公理の能力は、生成できる列長さで 測ることができる■

なお、下記”every countable collection of non-empty sets must have a choice function. ”
において ”collection of non-empty sets”の素性は不問
可算の集合の collectionであれ、非可算の集合の collectionであれ なんであれ 無問題
問題は ”countable collection”のところ
collectionが 非可算だと 可算選択公理の守備範囲外
下記を百回音読してね　；ｐ）

(参考)
https://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.
Applications
ACω is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers.
Example: infinite implies Dedekind-infinite
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:[2]
Let X be infinite. For each natural number n, let An be the set of all n-tuples of distinct elements of X.
Since X is infinite, each An is non-empty.
Application of ACω yields a sequence (Bn) n∈N where each Bn is an n-tuple.
One can then concatenate these tuples into a single sequence (bn)n∈N of elements of X, possibly with repeating elements.

つづく