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 (437ﾚｽ)
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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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