Let $C$ be a collection of sets, and $\bigcup C \in V_{\omega}$ where $V_{\omega}$ is the collection of hereditarily finite sets. Is it possible to show that $C\in V_{\omega}$?
YES. Because $V_\omega$ models every axiom of set theory except the infinity axiom;
i.e. $\bigcup C \in V_{\omega}\:\Rightarrow\:C \subseteq \mathcal{P}\left(\bigcup C\right)\in V_{\omega}\:\Rightarrow\: C\in \mathcal{P}\left(\mathcal{P}\left(\bigcup C\right)\right)\in V_{\omega}\:\Rightarrow\: C\in V_{\omega}$.
QUESTION What if I instead assume $\bigcup C$ is simply finite. Can I show that $C$ is finite?
Initial Thoughts From the assumption, it follows that every element of $C$ is finite. Is it possible to have an infinite collection of finite sets whose union is finite?
EDIT: $C\subseteq \underbrace{\mathcal{P}(\bigcup C)}_{\text{finite}}$. "Thank you" to a recently deleted answer which brought my senses back.