(>>59より）
正則性公理なしでも、自然数が整列集合 or 数学的帰納法成立 （公理として同値） が導けるだろうね
ピエロちゃん、やれよ、その証明を、具体的にさ ｗ（＾＾

ところでさ、下記にご注目ｗ（＾＾
”Axiom of infinity
that these members are all different, because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.”
とあるので、
The axiom of regularityがないと、
できた無限集合が、どんな集合かわけわからんみたい

それで、自然数が出来たということを確認するのが、大変になりそうだよｗ（＾＾
（The minimal set X の確認）

おれ？ おれは、そんな面倒なことはしないよ〜（＾＾
正則性公理採用派だからね

早く、正則性公理無しのZFから、自然数Ｎを構築してさ、整列集合 or 数学的帰納法成立 （公理として同値） の証明頼むよ〜ｗ
みんな、やれるかどうか、あんたの能力を見極めようと、期待して待っているよ〜ｗ（＾＾

どうせ、できないから、ぐだぐだ言い訳しているんだろうがね

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
Zermelo?Fraenkel set theory
(抜粋)
7. Axiom of infinity
Let S(w) abbreviate w∪{w}, where w is some set.
(We can see that {w}is a valid set by applying the Axiom of Pairing with x=y=w so that the set z is {w}.
Then there exists a set X such that the empty set Φ is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.
∃ X [Φ ∈ X Λ ∀ y(y∈ X → S(y)∈ X)].
More colloquially, there exists a set X having infinitely many members.
(It must be established, however, that these members are all different, because if two elements are the same, the sequence will loop around in a finite cycle of sets.
The axiom of regularity prevents this from happening.)
The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers Ｎ .
(引用終わり)