# How to visualize $\mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) )$?

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?

• How exactly does $\mathcal P(\mathbb R^2)$ contain text? – Rahul Nov 12 '14 at 16:27
• @Rahul think of a page of text as a black and white image – hb20007 Nov 12 '14 at 16:30

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.