# Is there a difference between using $\alpha \cup \{\alpha\}$ and $\mathscr{P}(\alpha)$ in constructing ordinals in the cumulative hierarchy of sets?

In constructing the sets in the cumulative hierarchy from the empty set, why is the power set operation even needed?

Let $V_{0} = \emptyset$, and inductively define $V_{\alpha+1} = V_{\alpha} \cup \{V_{\alpha}\}$. By using the union operator we can keep on constructing larger and larger sets. Why do we even need the power set operator here?