>>266 補足
>けれどコーシー列や切断から作られるRは
>どちらも実数の公理を満たすので同じもの

可算選択公理さえ仮定しない ZFのみでは どうなるか？
google AIさんの答え下記
シッチャカメッチャカですね （＾＾

1）
(google検索)
what role of Axiom of countable choice in Completeness of real numbers
AI による概要
The Axiom of Countable Choice (CC)
・Axiom of Countable Choice (CC): Given any sequence of non-empty sets, you can simultaneously choose exactly one element from each set in the sequence without needing an explicit rule.

How They Interact
In standard classical mathematics (ZFC), the Axiom of Choice is accepted, so these concepts are trivially equivalent. However, in foundational mathematics, their relationship is deeply nuanced:
・Equivalent Definitions:
To formally prove that every Cauchy sequence of rational numbers corresponds to exactly one Dedekind cut (or real number), you frequently must choose representatives from a countable infinity of sets.
The axiom of countable choice enables this step.
・The Constructive Context:
In constructive mathematics, mathematicians who reject the full Axiom of Choice often accept Countable Choice. In this setting, CC is used to establish the Fundamental Theorem of Algebra and the Bolzano-Weierstrass theorem, which rely fundamentally on the completeness of real numbers.

Where You Need CC in Analysis
If you are working with real numbers and sequential properties, Countable Choice ensures that you can move from a "there exists" statement for each step in a sequence to a single function (sequence) of choices.
・Establishing the equivalence of sequential compactness and topological compactness for the real numbers.

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