Describe four different elements of a union of power sets of power sets $P(P(A))\bigcup P(P(P(A)))$
The empty is set one but other than that I'm not really sure what does a power set of a power set means.
Any help would be appreciated.
Edit: Is this how you read this for example: $P(P(\emptyset))= \{\emptyset\{\emptyset\}\}? $
 A: Let $X=P(P(A))\cup P(P(P(A)))$.
Since $\emptyset\subseteq P(A)$, $\emptyset \in P(P(A))$ and thus $\emptyset\in X$.
Since $\emptyset\subseteq A$, $\emptyset \in P(A)\implies\{\emptyset\}\subseteq P(A)\implies \{\emptyset\}\in P(P(A))\implies \{\emptyset\}\in X$.
Again, $\emptyset\subseteq A\implies\emptyset\in P(A)\implies \{\emptyset\}\subseteq P(A)\implies\{\emptyset\}\in P(P(A))\implies \{\{\emptyset\}\}\subseteq P(P(A))\implies \{\{\emptyset\}\}\in P(P(P(A)))\implies \{\{\emptyset\}\}\in X.$
Finally, $\emptyset\subseteq A\implies\emptyset\in P(A)\implies \{\emptyset\}\subseteq P(A)\implies\{\emptyset\}\in P(P(A))$. But also, $\emptyset\subseteq P(A)$ and so $\emptyset\in P(P(A))$. Thus $\{\{\emptyset\}, \emptyset\}\subseteq P(P(A))\implies\{\{\emptyset\}, \emptyset\}\in P(P(P(A)))\implies \{\{\emptyset\}, \emptyset\}\in X$.
A: The power set of the power set of a given set is the set of all subsets of the set of subsets of a given set. Basically you have for a set $X$, $\mathcal{P}(X)= \{A: A \subseteq X\}$. So now the power set of the power set $\mathcal{P}(\mathcal{P}(X))=\{B: B \subseteq \mathcal{P}(X)\}$. One more level above you get $\mathcal{P}(\mathcal{P}(\mathcal{P}(X)))=\{C: C \ \subseteq \mathcal{P}(\mathcal{P}(X))\}$. Now I guess you get the idea how to iterate the power set operator: you just have to keep taking subsets. This is how we build the universe $V=\bigcup V_{\alpha}$
