Question

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Answer 1

\[
S = \{1\}\\
\mathcal{P}(S) = \{\emptyset , \{1\}\}\\
\mathcal{P}(\mathcal{P}(S)) = \{\emptyset,\{\emptyset\} , \{\{1\}\},\{\emptyset, \{1\}\}\}
\]It gets pretty crazy pretty fast even for a single element list.

Answer 2

I'd describe \(\mathcal{P}(\mathcal{P}(S))\) as
The set of all subsets of the set of all subets of \(S\)

Answer 3

thanks