>>279-780
>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 （＾＾

まとめを書いておこう
可算選択公理さえ仮定しない ZFのみでは・・
1）コーシー列とデデキントカットの同値は言えない（別物？）
2）"This fundamentally contradicts classical intuition about R being "larger" than Q. "
3）”A real number x might be in the closure of a set, but you cannot prove there is a sequence that converges to x.”
4）”Functions that are intuitively continuous (like polynomial or trigonometric functions) might fail to be sequentially continuous (i.e., preserving limits of sequences).”
5）”Countable Additivity of MeasureThe standard way we measure the length of sets (Lebesgue measure) on the real line assumes that the countable sum of measure-zero sets is still measure-zero. Without Countable Choice, you can lose this property, leading to a breakdown in measure theory and probability theory on the real line.”
6）”In summary, working without Countable Choice strips R of its "nicely ordered" limit properties. While you can still define and work with the set, the underlying topology becomes chaotic, and traditional tools of calculus and analysis become difficult to prove or outright false.”

”In summary, working without Countable Choice strips R of its "nicely ordered" limit properties. While you can still define and work with the set, the underlying topology becomes chaotic, and traditional tools of calculus and analysis become difficult to prove or outright false.”