つづき

https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Axiom of dependent choice
Relation with other axioms
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.

＜仏版のgoogle英訳＞
https://fr.wikipedia.org/wiki/Axiome_du_choix_d%C3%A9pendant
Axiom of dependent choice
Relationships with other axioms
Unlike AC in its full formulation, DC is insufficient (in ZF) to demonstrate that there exists an unmeasurable set of reals , or that there exists a set of reals which does not have the Baire property or without the perfect set property .
The axiom of countable choice is easily deduced from the axiom of dependent choice (consider, for a sequence ( A n ) of non-empty sets, the relation R on
⋃n∈N ∏k<n Ak defined by: sRt if s is equal to t minus its last element). It is much more difficult to prove that this implication is strict [ 4 ] .
Notes and references
2.This statement is equivalent to that of (en) Thomas Jech , Set Theory: The Third Millennium Edition, revised and expanded , Springer ,2003, 772 p. ( ISBN 978-3-540-44085-7 , online presentation [ archive ] ) , p. 50, moving from one relationship to the reciprocal relationship .
https://books.google.com/books?id=WTAl997XDb4C&pg=PA50
4.(in) Thomas J. Jech, The Axiom of Choice , Dover ,2013( 1st ed . 1973) ( read online [ archive ] ) , p. 130, Th. 8.12.

https://en.wikipedia.org/wiki/Axiom_of_countable_choice
Axiom of countable choice
(可算選択公理)
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 N (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.
＜仏語＞
https://fr.wikipedia.org/wiki/Axiome_du_choix_d%C3%A9nombrable
Axiome du choix dénombrable
＜google英訳＞
Axiom of Countable Choice
The axiom of countable choice , denoted AC ω , is an axiom of set theory which states that every countable set of non- empty sets must have a choice function , that is, for every sequence ( A ( n )) of non-empty sets, there exists a function f defined on N (the set of natural numbers ) such that f ( n ) ∈ A ( n ) for all n ∈ N.
https://ja.wikipedia.org/wiki/%E5%8F%AF%E7%AE%97%E9%81%B8%E6%8A%9E%E5%85%AC%E7%90%86
可算選択公理
(説明で " f ( n ) ∈ A ( n ) for all n ∈ N"に触れていないからダメ)
(引用終り)
以上