メモ

https://ja.wikipedia.org/wiki/%E9%81%BA%E4%BC%9D%E7%9A%84%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88
遺伝的有限集合（いでんてきゆうげんしゅうごう、英: hereditarily finite set）は有限個の遺伝的有限集合からなる有限集合と定義される。この定義は帰納的である。遺伝的という名称は遺伝的有限という性質がその元に遺伝することによる。

https://en.wikipedia.org/wiki/Hereditarily_finite_set
Hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.

Representation
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:
・{} (i.e. Φ , the Neumann ordinal "0"),
・{{}} (i.e. {Φ } or {0}, the Neumann ordinal "1"),
・{{{}}},
・{{{{}}}} and then also {{},{{}}} (i.e. {0,1}, the Neumann ordinal "2"),
・{{{{{}}}}}, {{{},{{}}}} as well as {{},{{{}}}},
・... sets represented with 6}6 bracket pairs, e.g. {{{{{{}}}}}},
・... sets represented with 7}7 bracket pairs, e.g. {{{{{{{}}}}}}},
・... sets represented with 8}8 bracket pairs, e.g. {{{{{{{{}}}}}}}} or {{},{{}},{{},{{}}}} (i.e. {0,1,2}, the Neumann ordinal "3")
・... etc.
In this way, the number of sets with n bracket pairs is[1] 1,1,1,2,3,6,12,25,52,113,247,548,1226,2770,6299,14426,・・・

Axiomatizations
Theories of finite sets

ZF

See also
Hereditary set
Hereditarily countable set
Hereditary property
Rooted trees
Constructive set theory
Finite set