Your response is a reasonable explanation of Hilbert's hotel paradox, but that paradox is really different from the Banach-Tarski paradox. In particular, the B

Your response is a reasonable explanation of Hilbert's hotel paradox, but that paradox is really different from the Banach-Tarski paradox. In particular, the Banach-Tarski paradox is false in two dimensions. (One technical reason for this is that you can embed a nonabelian free group into SO(3), the rotations on the sphere, but not into SO(2), the rotations on the circle.) This means that you should not think of the Banach-Tarski paradox as being a Hilbert-hotel-like statement about a two dimensional lattice.