Finding Space at the Hilbert Hotel

                                    To enlarge image, click on it.


The Hilbert Hotel is hotel with a room numbered with EVERY natural number from one up.  It corresponds to Cantor's first transfinite ordinal (lower-case omega.)  Even if every room is occupied, you can fit in an extra guest.  All you have to do is to get all the guests already in a room to move to the room with the next higher room number to their current room.