Let $V_\omega$ denote the set of all hereditarily finite sets. A set $S$ is called hereditarily finite if and only if its transitive closure is finite, that is, $TC(S) = \bigcup \{ S, \bigcup S, \bigcup \bigcup S, \dots \}$ is finite. Let $P(S)$ denote the power set of $S$ and let $\omega$ denote the natural numbers.
I am trying to understand what $V_\omega$ looks like and to this end I thought I could work out the relationship between $V_\omega$ and $P(\omega)$:
Of course, since $V_\omega$ is a model of $ZFC$ without the axiom of infinity, neither $\omega$ nor $P(\omega)$ are elements of $V_\omega$. Hence $P(\omega) \nsubseteq V_\omega$.
On the other hand, $\{\{\{\varnothing\}\}\}$ is in $V_\omega$ but not in $P(\omega)$. Hence $V_\omega \nsubseteq P(\omega)$.
So this is not going to give me any information about $V_\omega$. Yet, since hereditary finiteness is a stronger condition than finiteness ($\{\omega\}$ is a finite set that is not hereditarily finite) I am tempted to think that perhaps $V_\omega$ might somehow be in bijection with a subset of $P(\omega)$.
Question: Is there such a bijection? If not: what's a good intuition to think about $V_\omega$? What does $V_\omega$ look like?
Thanks for your help.