Let P(X) denote power set of a set X. If A is the null set, then the number of elements in P(P(P(P(A)))) is

By definition, if set X has n elements, P(X) has $$2^n$$ elements

Now, A is the null set. So, number of elements in set A is zero.

So, P(A) will have $$2^0=1$$ element

or, P(P(A)) will have $$2^1=2$$ elements

or, P(P(P(A))) will have $$2^2=4$$ elements

or, P(P(P(P(A)))) will have $$2^4=16$$ elements