de.wikipedia.org/wiki/Abz%C3%A4hlbares_Auswahlaxiom 独語（google英訳） Countable Axiom of Choice Of course, for certain (possibly uncountable) sets of nonempty sets, a selection function can also be specified without the (countable) selection axiom, e.g. ・when the cut ∩A is not empty, because then there is a constant selection function, ・if the association ∪A well-ordered , because then the smallest element in terms of well-ordering can be taken from any set, and ・if it is a family of intervals of real numbers, because then the midpoint of each interval can be taken. On the other hand, even for a countable family of two-element sets, the existence of a selection function cannot be proven in ZF.
ja.wikipedia.org/wiki/%E9%81%B8%E6%8A%9E%E5%85%AC%E7%90%86 選択公理 en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice Weaker forms There are several weaker statements that are not equivalent to the axiom of choice but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (ACω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Given an ordinal parameter α ≥ ω+2 — for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ωα, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.
