>>38 補足
>遺伝的有限集合、Hereditarily finite set
>「naturally ranked by the number of bracket pairs」で
>そのbracket（カッコ）の深さのシングルトン達、例えば深さ6 with 6 bracket pairs, e.g. {{{{{{}}}}}}とか出てくるよ
>そして、ZFでは、Vω=∪ k=0〜∞ Vk だ
>だから、ω重シングルトン あるんじゃね？

文献を誤読している人がいるので補足する
（>>38 ＆>>46再録）
https://en.wikipedia.org/wiki/Hereditarily_finite_set
Hereditarily finite set

Discussion
A symbol for the class of hereditarily finite sets is H_ｱﾚﾌ0, standing for the cardinality of each of its member being smaller than ｱﾚﾌ0. Whether H_ｱﾚﾌ0 is a set and statements about cardinality depend on the theory in context.

Axiomatizations
Theories of finite sets
The set Φ also represents the first von Neumann ordinal number, denoted 0. And indeed all finite von Neumann ordinals are in ｱﾚﾌ0 and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers. Robinson arithmetic can already be interpreted in ST, the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.

Their models then also fulfill the axioms consisting of the axioms of Zermelo-Fraenkel set theory without the axiom of infinity.
In this context, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.

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