Have you ever heard stories like "Hilbert's Hotel"? Given an infinite hotel (each room labelled with a positive integer), you can empty half your rooms by movi

Have you ever heard stories like "Hilbert's Hotel"? Given an infinite hotel (each room labelled with a positive integer), you can empty half your rooms by moving the person in room x to the room 2*x, which gives you all the odd rooms to fill with a new infinity of people, effectively doubling the size of your infinite hotel? This is very similar -- except it works over the real numbers in 3D space. In particular the cuts are "inifinitely fine". Given any point p, and any distance d, there will be a point less than distance d from p which is in a different "slice". Imagine (while this isn't in banack-tarski) making a "slice" which is all numbers of the form 1/x for all integers. Clearly this "slice" could really exist, but mathematics can of course define it and operate on it.