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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966: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/20(日) 19:27:34.16 ID:JxJPBISF

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
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

http://rio2016.5ch.net/test/read.cgi/math/1745503590/966

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