>>222-234
ID:3ZBTbAUT は、御大か
巡回とお天気日誌ありがとうございます

さて、前スレより再録
https://rio2016.5ch.io/test/read.cgi/math/1778894782/831-833
>>830
なるほど・・・
君の論にも一理ある
が >>829にも一理あるぞ

つまり 下記 google検索で
・カントール流のコーシー列による実数構成には、AC_ω つまり Axiom of countable choiceが必要で
　AC_ωを使って、Qを完備したものが実数Rであることが証明できる
・一方、Dedekind Cutsを使うと なんらの選択公理は不要だが
　完備であることの証明はできない
・なお、実数の構成から進んで 実解析などを行うには
　Axiom of Dependent Choice (DC)が必要だと

まあ、おサルさんと痛み分けか・・（＾＾

(google検索)
Construction of real numbers using rational Cauchy sequences and Axiom of countable choice in ZF axiomatic set theory
AI概要
Constructing real numbers as equivalence classes of rational Cauchy sequences formalizes the idea that any real number is the limit of approximations. In ZF, the Axiom of Countable Choice (ACω) is required to prove that these equivalence classes form a mathematically complete field.
1. The Setup: Rational Cauchy Sequences
We begin within ZF (Zermelo-Fraenkel set theory without Choice). We assume the rational numbers, Q, have already been constructed.
・A Cauchy sequence of rationals is defined as a function f: N → Q such that for every rational ε > 0, there exists an integer N such that for all m, n > N:
｜f(m) - f(n)｜ < ε
・Let C be the collection of all such Cauchy sequences. This forms a set via the Axiom of Replacement and the Power Set Axiom.

2. The Equivalence Relation
Intuitively, two sequences represent the same real number if and only if the distance between their terms tends to 0 as n → ∞.
・We define a relation 〜 on C by f 〜 g if and only if lim_n → ∞ ｜f(n) - g(n)｜ = 0).
・This relation 〜 is an equivalence relation.
・The set of real numbers R is defined as the set of equivalence classes:R = C / 〜

3. Arithmetic Operations
The algebraic field operations (addition and multiplication) are defined pointwise on the sequences:
・(𝑓+𝑔)(𝑛)=𝑓(𝑛)+𝑔(𝑛)
・(𝑓⋅𝑔)(𝑛)=𝑓(𝑛)⋅𝑔(𝑛)
We must verify that these operations are well-defined (i.e., if f_1 〜 f_2 and g_1 〜 g_2, then f_1 + g_1 〜 f_2 + g_2). This is entirely proven within standard ZF and requires no choice axioms.

つづく