>>214-217
ふっふ、ほっほ

下記”Axiom of countable choice”（可算選択公理）と ”Axiom of dependent choice”（従属選択公理）を 百回音読しよう！ （＾＾
１）”Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a non-zero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ...”
　とありますがなｗ
２）”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. For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S∖{x}, one needs (a weak form of) the axiom of countable choice.”
　とありますがなｗ

”Axiom of countable choice”（可算選択公理）さえ仮定しない ZFのみでは
デデキントカットで なにか実数R らしき集合が出来ても
それと、コーシー列との関係がつかない
そうすると 困るでしょｗ （＾＾

(参考)
https://en.wikipedia.org/wiki/Axiom_of_countable_choice
Axiom of countable choice

https://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Axiom_of_countable_choice.svg/500px-Axiom_of_countable_choice.svg.png
Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a non-zero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ...

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. For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S∖{x}, one needs (a weak form of) the axiom of countable choice.
When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω.
The ability to perform analysis using countable choice has led to the inclusion of ACω as an axiom in some forms of constructive mathematics, despite its assertion that a choice function exists without constructing it.[1]