Cardinality of fin of superstructure?

Question

The usual superstructure over $\mathbb R$ is defined as $\mathbb V=\cup_{n<\omega}V_n$ where $V_0=\mathbb R$ and $V_{n+1}=V_n\cup \mathcal{P}(V_n)$. Furthermore let $\mathcal{P}_{fin}(\mathbb V)$ be the set of finite subsets of $\mathbb V$. Is it correct that the cardinality of both $\mathbb V$ and $\mathcal{P}_{fin}(\mathbb V)$ is $\beth_\omega$ ?

Answer 1

Yes. It is easy to see that $|\mathbb V_n|=\beth_{n+1}$, so $|\mathbb V|=\beth_\omega$, and for every infinite $X$, $\mathcal P_{\rm fin}(X)$ is equipotent to $X$. At least assuming choice, without choice you can still prove this for this specific case, by noting that $\mathbb V$ is equipotent with $V_{\omega+\omega}$ in the sense of the von Neumann hierarchy, and for the von Neumann hirerachy $\mathcal P_{\rm fin}(V_\delta)\subseteq V_\delta$ for all limit $\delta$.