純粋・応用数学・数学隣接分野（含むガロア理論）21

純粋・応用数学・数学隣接分野（含むガロア理論）21 (275ﾚｽ)
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19: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/21(月) 14:24:20.15 ID:60RWf/A5

>>18
さて 上記を受けて 下記がある
前スレより
https://rio2016.5ch.net/test/read.cgi/math/1745503590/588 2025/06/16 現代数学の系譜 雑談 ◆yH25M02vWFhP
1）すっきりの度合いが違うだろ？
　即ち、和記号Σや積記号Πならば、普通その範囲を明示するべきだろ？
　Σ n=1〜∞とか Σ m,n=1〜∞とかね
　では問う 記号∩について 同じことを要求する
　きちんと、記号∩の定義を書け！
　ここ、ツッコミどころだねｗ
2）”実質同じ”？ 証明は？ 上記1）項のあと 証明やってみてｗ ；ｐ）

https://rio2016.5ch.net/test/read.cgi/math/1745503590/949-957 2025/07/20 ID:2Jr4cGNB
>>588
>2）”実質同じ”？ 証明は？
定義１
　論理式φ(x)を下記で定義する。
　φ(x):={}∈x∧∀y(y∈x→y∪{y}∈x)
　φ(x)を満たすxを帰納的集合と呼ぶ。
以下略

https://rio2016.5ch.net/test/read.cgi/math/1745503590/964-965 2025/07/20 現代数学の系譜 雑談 ◆yH25M02vWFhP
>>949-950
>補題１
>　ωは任意の帰納的集合の共通部分である。

うむ
１）その結論は、正しい。下記の独 de.wikipediaの英訳
　Infinity axiomで、”The natural numbers are therefore defined as the intersection of all inductive sets, as the smallest inductive set.”
　とある通りだ
２）ところで 下記の 独 de.wikipedia Infinity axiom では
　記号∩ 使ってないよ？
　記号∩ は、使わなくてもいいの？
　記号∩ は、使わなくてもいいのならば、その方がすっきりしてないかな？ｗ ；ｐ）

(参考)
https://de.wikipedia.org/wiki/Unendlichkeitsaxiom
（google翻訳 独→英）
Infinity axiom
The axiom of infinity is an axiom of set theory that postulates the existence of an inductive set . It is called the axiom of infinity because inductive sets are also infinite sets .

つづく

http://rio2016.5ch.net/test/read.cgi/math/1753002417/19

20: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/21(月) 14:24:56.15 ID:60RWf/A5

つづき

formulation
There are a lot A, which is the empty set ∅ and with each element
x∈A also the amount x∪{x}contains.
∃A:(∅∈A∧∀x:(x∈A⇒x∪{x}∈A))
The infinity axiom does not merely postulate, as the name might suggest, the existence of any infinite set. It postulates the existence of an inductive set and thus, consequently, the existence of the set of natural numbers according to John von Neumann's model .

Significance for mathematics
Natural numbers
By the existence of at least one inductive set
I together with the exclusion axiom, the existence of natural numbers as a set is also ensured:
N:={x∈I∣∀z(z inductive ⟹ x∈z)}
The natural numbers are therefore defined as the intersection of all inductive sets, as the smallest inductive set.

Infinite quantities
Without the infinity axiom, ZF would only guarantee the existence of finite sets. No statements could be made about the existence of infinite sets. The infinity axiom, together with the power set axiom , ensures that there are also uncountable sets, such as the real numbers.

下記 fr.wikipedia Axiom of infinity（無限公理）も
https://fr.wikipedia.org/wiki/Axiome_de_l%27infini
（google翻訳 仏→英）
Axiom of infinity
Statement of the axiom
略
・let A be a set verifying Cl( A ) whose existence is ensured by the axiom of infinity. Then, the existence of the set ω is ensured by the axiom scheme of comprehension and its uniqueness by the axiom of extensionality , by defining ω as the intersection (therefore the smallest in the sense of inclusion) of all sets containing 0 and closed by successor ( A only intervenes to be able to define ω as a set, but ω does not depend on A ):
ω = { x ∈ A | Ent( x ) } ;
略
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1753002417/20

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