純粋・応用数学・数学隣接分野（含むガロア理論）20

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）20 (1002ﾚｽ)
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920: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/19(土) 15:34:03.80 ID:jT6bEcWg

>>874 戻る
>Informally と intersection が同一文内にある。だから∩を使った構成は間違い。

えーと >>867 より再録
　>>852-853より
https://en.wikipedia.org/wiki/Axiom_of_infinity
Axiom of infinity
Extracting the natural numbers from the infinite set
Φ(x) be the formula that says "x is inductive"; i.e.
Φ(x)=(∅∈x∧∀y(y∈x→(y∪{y}∈x))).
Informally, what we will do is take the intersection of all inductive sets. More formally, we wish to prove the existence of a unique set W such that
∀x(x∈W↔∀I(Φ(I)→x∈I)). (*)
For existence, we will use the Axiom of Infinity combined with the Axiom schema of specification.
Let I be an inductive set guaranteed by the Axiom of Infinity. Then we use the axiom schema of specification to define our set
W={x∈I:∀J(Φ(J)→x∈J)}
– i.e. W is the set of all elements of I, which also happen to be elements of every other inductive set. This clearly satisfies the hypothesis of (*), since if x∈W, then
x is in every inductive set, and if
x is in every inductive set, it is in particular in I, so it must also be in W.
For uniqueness, first note that any set that satisfies (*) is itself inductive, since 0 is in all inductive sets, and if an element
x is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set
W′ that satisfied (*) we would have that
W′⊆W since
W is inductive, and
W⊆W′since
W′is inductive. Thus W=W′.
Let ω denote this unique element.
This definition is convenient because the principle of induction immediately follows: If
I⊆ω is inductive, then also
ω⊆I, so that I=ω.■
(引用終り)
１）”Informally, what we will do is take the intersection of all inductive sets.”
　intersection：共通部分 英: intersection（下記）ね
２）で、これ ”Informally”とあるよね。つまり、
　”∩{x⊂A|{}∈x∧∀y[y∈x→y∪{y}∈x]}”>>727 は、”Informally”なんだよ
　ここを勘違いした人が ja.wikipediaに >>847の”ペアノの公理”を 書いたんじゃないの？
３）さて、Formallyには ”Let I be an inductive set guaranteed by the Axiom of Infinity. Then we use the axiom schema of specification to define our set
　W={x∈I:∀J(Φ(J)→x∈J)}
　– i.e. W is the set of all elements of I, which also happen to be elements of every other inductive set.”
　だよね。ここに、”∩”は 使われない
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1745503590/920

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