# What is the minimum amount of elements in nested power sets?

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

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 $$16$$