A neat article on infinities.
Infinity offers many results that are at first counter-intuitive. A classic example is Hilbert's Hotel, which has infinitely many rooms, each one labeled by a natural number printed on the door: Room 1, Room 2, Room 3, etc., all the way through the natural numbers. One night, a traveler arrives at the front desk only to be told be the clerk that the hotel is full. "But don't worry, sir," says the clerk, "I just took a mathematics course at my local college, and so I know how to find you a room. Just give me a minute to make some phone calls." And a short while later, the traveler has his room for the night. What the clerk did was ask every guest to move to the room with the room number the next integer. Thus, the occupant of Room 1 moved into Room 2, the occupant of Room 2 into Room 3, etc. Everyone moved room, no one was ejected from the hotel, and Room 1 became vacant for the newly arrived guest.