English Dictionary

axiom of countable choice

名詞

(set theory) A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.

日本語の意味

可算選択公理とは、集合論における選択公理の弱い形であり、非空集合からなる可算個の集合族に対して、各集合から一つずつ要素を選ぶ選択関数が存在すること、すなわち可算個の非空集合の直積が非空であるという主張に同値する。

When teaching introductory set theory, I always emphasize the axiom of countable choice and how it differs from the full axiom of choice.

入門的な集合論を教えるとき、私はいつも可算選択公理とそれが完全な選択公理とどのように異なるかを強調します。

意味（1）

(set theory) A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.

The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable. One example is the axiom of countable choice, which states that if A_0,A_1,…A_n… form a denumerable set of nonempty sets, their product is nonempty. […] The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow confusion in the minds of the students (who are inclined to accept anything desired) or of the professors (who do not like to shake the foundations of the discipline).