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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964: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/20(日) 18:34:52.71 ID:JxJPBISF

>>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 .

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.

http://rio2016.5ch.net/test/read.cgi/math/1745503590/964

965: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/20(日) 19:27:13.97 ID:JxJPBISF

>>964 追加

下記 fr.wikipedia Axiom of infinity（無限公理）
ここでも
　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://fr.wikipedia.org/wiki/Axiome_de_l%27infini
（google翻訳 仏→英）
Axiom of infinity
Statement of the axiom
The axiom is therefore written:
There exists a set to which the empty set belongs and which is closed by application of the successor x ↦ x ∪ { x },
that is, in the formal language of set theory (the calculus of egalitarian first-order predicates with the only non-logical symbol being that for membership, "∈"):
∃A Cl(A)
where Cl( Y ) is the predicate “∅ ∈ Y and ∀ y ( y ∈ Y ⇒ y ∪ { y } ∈ Y )”,
expressing
“  Y is closed under successor and ∅ belongs to it”
(for the abbreviations “∅ ∈ Y  ” and “  y ∪ { y } ∈ Y  ”,
defined from ∈, see Axiom of the empty set , Axiom of the pair and Axiom of the union ).

The set of natural numbers
Definition
To formalize the "and so on", let us define the predicate
Ent(x) as :
∀A (Cl(A)⇒x∈A)

Throughout the following, we will call "natural integers" - or "integers" - the elements x verifying Ent( x ).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1745503590/965

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