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.
Rudy Rucker explored this thoroughly in his novel "White Light" (which is where I first learned of the Banach-Tarski theorem) whilst having with much fun with infinities.
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.