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I'm having trouble grasping the concept of a power set. Let's say we have a set P(P(P(A))), what is the minimum amount of elements in that set?

By substituting A = $\emptyset$ I get 4 elements {$\emptyset$, {$\emptyset$}, {{$\emptyset$}}, {$\emptyset$, {$\emptyset$}}} but that is the result when I'm solving/going "from the bottom up" and when I look from the top I kinda get a feeling that it should have only 2 elements meaning {$\emptyset$, P(P(A))} where P(P(A)) is a single element with some subsets inside.

I hope I expressed my problem well enough, this is my first time translating math to other language

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If $A$ has $a$ many elements, then the number of elements of "$P(A)$" is $2^a$.

Therefore the number of elements of $P(P(P(A)))$ is $2^{2^{2^a}}$ which for $a=0$ (i.e. $A=\emptyset$) is $16$

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  • $\begingroup$ Are you sure it's 16 and not 4? $\endgroup$ – Delyew Oct 18 '18 at 15:28

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