>>665
なんか
ぐじぐじと、筋の悪いネバり方をするので収束しないね

えーと、下記”∈-induction”を百回音読してね
そして、ZFCの中と外を区別してね

いまの”∈-induction”は、ZFCの中で
ノイマンの正則性公理とともに
選択公理（別名 整列可能定理）がある

(参考)
en.wikipedia.org/wiki/Epsilon-induction
Epsilon-induction
∈-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property.
The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations.

Ordinals
Ordinals may be defined as transitive sets of transitive sets. The induction situation in the first infinite ordinal
ω, the set of natural numbers, is discussed in more detail below. As set induction allows for induction in transitive sets containing
ω, this gives what is called transfinite induction and definition by transfinite recursion using, indeed, the whole proper class of ordinals. With ordinals, induction proves that all sets have ordinal rank and the rank of an ordinal is itself.

The theory of Von Neumann ordinals describes such sets and, there,
y∈x models the order relation
y<x, which classically is provably trichotomous and total. Of interest there is the successor operation
x→x∪{x} that maps ordinals to ordinals. In the classical case, the induction step for successor ordinals can be simplified so that a property must merely be preserved between successive ordinals (this is the formulation that is typically understood as transfinite induction.) The sets are
∈-well-founded.