>>176 補足
>超限帰納法

ZFCの”Axiom of regularity”
基礎の公理とか正則性公理とか言われる

その意味は、ZFCで出来る集合を、
整列集合にして、帰納法を使えるようにすることが、大きな目的の一つです

最初、大小記号 ＜ などはまだ定義されていないとき、
代わりの順序として ∈（epsilon）を使う

これぞ、Epsilon-induction なり〜！（＾＾
日本で、意味が分かっている人少ないかもね（＾＾；

https://en.wikipedia.org/wiki/Axiom_of_regularity
Axiom of regularity

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:
∀ x,(x≠ Φ → ∃ y∈ x,(y∪ x=Φ )).
However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on {(n,α )｜ n∈ ω ∧ α is an ordinal }.
Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.

See also
・Epsilon-induction

https://en.wikipedia.org/wiki/Epsilon-induction
Epsilon-induction
In mathematics, ∈-induction (epsilon-induction or set-induction) is a variant of transfinite induction.
Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction.
It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction.
(引用終り)
以上