TITLE: Is power set of a power set of a set equal to the power set of the same set?
QUESTION [8 upvotes]: I have to decide whether this statement is true, I think it is not.
Since the power set of a set with cardinality $n$, will have $2^n$ subsets, however the power set of this set will include the subsets themselves and subsets of the subsets.

REPLY [2 votes]: All the proofs given above are for finite sets, but what if it is an $\textit{arbitrary}$ set?
Well, $\mathcal{P}(\mathcal{P}(X))=\mathcal{P}(X)$ is still false if you were to assume a set theoretic axiom called Regularity Axiom, which forbids the existence of sets that belong to themselves. Since  $\mathcal{P}(X)\in\mathcal{P}(\mathcal{P}(X))$, then we would have $\mathcal{P}(X)\in\mathcal{P}(X)$, a contradiction  by this axiom.