Hilbert's Hotel

Hilbert's Hotel, named after mathematician David Hilbert, is a hotel with an infinite number of rooms. Imagine that every single one of them is occupied. What does the manager do when someone else shows up and wants a room? He doesn't need to turn that person away. Instead, he just moves the person in room 1 into room 2, the person in room 2 into room 3, the person in room 3 into room 4, and so on. Room 1 is now vacant for the new guest. Now, a hundred new guests appear. The manager now moves the guest in room 1 into room 101, the guest in room 2 into room 102, etc. and thus creates room for the 100 guests.

Now an infinite number of people show up, all wanting rooms. What does the manager do now? He simply moves the person in room 1 into room 2, the person in room 2 into room 4, the person in room 3 into room 6, and in general the person in room n into room 2n. Now there are an infinite number of rooms free (all of the odd-numbered rooms) for this infinite group of people.

Hilbert's Hotel is paradoxical, but it illustrates an interesting property of infinite sets: An infinite set can be put in one-to-one correspondence with an infinite subset of itself. It also illustrates the seemingly impossible situations that become possible when dealing with infinity.