[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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966(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 07/20(日)19:27 ID:JxJPBISF(4/10) AAS
つづき

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
省8

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967(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 07/20(日)19:36 ID:JxJPBISF(5/10) AAS
>>966 補足

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

だけど、
省3

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