So,$ \ \mathcal P \ ( \ \mathbb R ^2 \ ) $ , the power set of the set of all ordered pairs of real numbers, contains every imaginable (2D) function, black and white image and text as per its elements.
But how do we go a step further in understanding what the power set of that, $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $ is about?
On
Here is an intuition I made to think of this. Let's first understand just $P(X)$, it looks like this:
It's a union of set with no elements of $X$ kept together (empty set), set of $1$ elements of $X$ paired kept together, 2 ..., till size increases that we get $X$ as a whole.
Now, what happens when we do $P(P(X))$? We can understand it by thinking of what the generic element should look like. The generic element of $P(P(X))$ is a subset of the elements of $X$, so that's a subset maybe containing the empty set, maybe 1 element of $X$ kept together, $2$ element of kept together and so on.
For any set of points of $\mathbb R^2$ we can draw the set in 2-D by visualizing $\mathbb R^2$ as a sheet of paper and printing a pixel black if it belongs to the set or leaving it white if it doesn't belong (keep extra toner handy, as every page is infinite in size and there are uncountably many sheets). Print every set in this fashion and you'll have quite a large stack of paper; this is $\mathcal P(\mathbb R^2)$.
Now you're going to make portfolios by compiling every conceivable combination of drawings from your stack (don't ask where the binding goes). That is, your sets are 'substacks' of paper, or sets of sets of points. The resulting collection of portfolios is $\mathcal P\bigl(\mathcal P(\mathbb R^2)\bigr)$.
Now the publisher wants an anthology of all possible collections of portfolios: $\mathcal P\bigl[\mathcal P\bigl(\mathcal P(\mathbb R^2)\bigr)\bigr]$. And so on.