Can someone explain to me why $\mathcal P(\mathcal P(\emptyset)) = \{\emptyset,\{\emptyset\}\}$?

Question

Would someone explain the reasoning of the answer of

$$\mathcal P(\mathcal P(\emptyset)) = \{\emptyset,\{\emptyset\}\}$$

I am having trouble understanding this

Answer 1

First of all, there's no such thing as an empty powerset. The smallest powerset is the powerset of the empty set, which is a singleton with one element, namely $\emptyset$. So $P(\emptyset)$ is a singleton, $\{\emptyset\}$. The powerset of a singleton contains the singleton set and the emptyset, thus $P(P(\emptyset))=P(\{\emptyset\})=\{\emptyset,\{\emptyset\}\}$ for the same reason $P(\{1\})=\{\emptyset,\{1\}\}$.