>>38 追加
>そして、ZFでは、Vω=∪ k=0〜∞ Vk だ

分かってないらしいので、引用追加する

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.

Indeed, ｱﾚﾌ0 has a constructive axiomatizations involving these axiom and e.g. Set induction and Replacement.

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.

ZF
略（>>38 ご参照）

https://en.wikipedia.org/wiki/Von_Neumann_universe
Von Neumann universe
Finite and low cardinality stages of the hierarchy
The set Vω has the same cardinality as ω.
The set Vω+1 has the same cardinality as the set of real numbers.
(引用終り)

これで
１．Von Neumann universeで、”The set Vω has the same cardinality as ω. ”つまり、Vωの濃度はωで、可算無限
２，だから、Vω=∪ k=0〜∞ Vkで、k=0〜∞は、ちゃんと∞まで渡るよ
３．なお、上記 Hereditarily finite setの”Discussion”に書いてあるように、Axiomatizations Theories of finite sets と ZFでは、扱いが違うってことです