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.
(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 ).
With this definition, 0 is an "integer" — formally: we have Ent(0) — and the successor x + of any "integer" x is an "integer" — Ent( x ) ⇒ Ent( x + ), and the axiom of infinity is equivalent to ∃ω ∀x(Ent(x)⇔x∈ω), that's to say : The class of natural numbers is a set . Indeed : ・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 ) } ;
・conversely, let ω be a set whose elements are the natural numbers. Then, ω verifies Cl(ω). The very definition of the set ω gives a statement of the principle of recurrence on the integers: any set to which 0 belongs and which is closed by successor is a superset of ω. We can give a slightly more familiar statement but equivalent in set theory by the comprehension scheme, we denote x + the successor of x , we then have for an arbitrary property expressed in the language of set theory by the formula P x a 1 … a k (no other free variable ): ∀ a 1 , … , a k { [ P 0 a 1 … a k and ∀ y ∈ ω ( P y a 1 … a k ⇒ P y + a 1 … a k )] ⇒ ∀ x ∈ ω P x a 1 … a k } (any property that is true at 0 and passes to the successor on integers is true for all integers). For example: every element of ω is a finite ordinal .
The recurrence is valid for any property expressed in the language of set theory. This is not trivial: it makes this recurrence a much stronger property than the recurrence of Peano arithmetic (as a first-order theory), the language of set theory being strictly more expressive than that of Peano arithmetic. (引用終り) 以上 []
fr.wikipedia Axiom of infinity（無限公理）より ”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 ) } ;”
とあるよ ”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 )” とあるよ ”by defining ω as the intersection” とあるよ
