Supposing that the set is $A=\{a,b,c\}$. The set of parts of $A$ is the set that has as its elements all possible subsets of the set $A$. By all possible subsets of $A$ we mean precisely all its subsets, including improper ones, that is, the empty set and the set $A$ itself.
We have $$\mathcal{P}(A) = \{ B \mid B \subseteq A \} = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, A\}$$$$ = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, A\}$$
I must to find $\mathcal{P}(\mathcal{P}(A))$. If I indicate the elements of $\mathcal{P}(A)$ with $S_1, S_2, \ldots S_8$, I can build
$$\mathcal{P}(\mathcal{P}(A))= \{S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8, \{S_1, S_2\},\{S_1, S_3\},\\\{S_1, S_4\},\ldots\{S_1, S_8\}, \{S_2, S_3\}\ldots,\{S_2, S_8\},\ldots\}$$
$$\mathcal{P}(\mathcal{P}(A))= \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, A, \{\emptyset, \{a\}\},\\\{\{a\}, \{b\}\},\{\{b\}, \{c\}\},\ldots\{\emptyset, A\}, \{\{a\}, \{b\}\}\ldots,\{\{a\}, A\},\ldots\}$$
and so on. Is it correct? If $|\mathcal{P}(A)|=2^n$ elements how many elements does it have $\mathcal{P}(\mathcal{P}(A))$?
If the definition of $\mathcal{P}(A) = \{ B \mid B \subseteq A \}$ what is the definition of $\mathcal{P}(\mathcal{P}(A))$?
Best Answer
No this is not correct:
$\mathcal{P}(\mathcal{P}(A))=\lbrace\emptyset, \lbrace\emptyset\rbrace , \lbrace \lbrace a\rbrace \rbrace , \lbrace \lbrace b \rbrace \rbrace, \lbrace \lbrace c\rbrace \rbrace,\lbrace \lbrace a \rbrace, \emptyset \rbrace, \ldots\rbrace$
and $|\mathcal{P}(\mathcal{P}(A))|=2^{|\mathcal{P}(A)|}=2^{2^n}$