つづき

2）
(google検索)
what happens in construction of the real number set without countable choice axiom
AI による概要
The core definitions—like Dedekind cuts or Cauchy sequences—still stand, but the relationships between these constructions and certain properties of R change dramatically
The fundamental changes without Countable Choice include:
https://www.youtube.com/watch?v=WaxiL4IRIYw&t=70
MetaMaths
143,161回視聴 2021/03/16

The fundamental changes without Countable Choice include:
1. Countable Unions of Countable Sets Can Be UncountableIn standard math (ZFC), the countable union of countable sets is countable. However, without Countable Choice, it is perfectly consistent with ZF that the real numbers are the countable union of countable sets. This fundamentally contradicts classical intuition about R being "larger" than Q.

2. The Divergence of Different Constructions
・Dedekind Reals: Numbers defined by partitioning the rational numbers into two sets where every element in the left set is smaller than every element in the right set.Without Choice, you may have Dedekind reals that do not correspond to any sequence of Cauchy reals.

3. Disconnected Metric Spaces
Without Countable Choice, standard theorems about limits and continuity fail to hold uniformly.
・A real number x might be in the closure of a set, but you cannot prove there is a sequence that converges to x.
・Functions that are intuitively continuous (like polynomial or trigonometric functions) might fail to be sequentially continuous (i.e., preserving limits of sequences).

4. 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.
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.

If you are exploring the foundations of mathematics, tell me if you'd like to dive into:
・How Dedekind cuts differ from Cauchy sequences without choice.
・Constructive mathematics and how it restricts the use of infinity.
・Specific paradoxes that arise in Lebesgue measure theory without the axiom.
(引用終り)
以上