TedEd: The Infinite Hotel Paradox

A TedEd by Jeff Dekofsky

The Infinite Hotel Paradox was created in the 1920's by David Hilbert to show how difficult the concept of infinity is. Imagine a hotel with infinite rooms, all booked for the night. A person walks in and asks for a room. But you don't have to turn him down. You can ask every guest to move over one room, 1 to 2, 2 to 3, and so on until guest number n goes to room number n+1. Then the new guest takes room number 1. Now let's say that a bus with an infinitely countable number of passengers shows up. How do you fit all of them? If the guest in room 1 goes to room 2, 2 to 4, 3 to 6, and so on, each guest will move to room number 2n from n. This will fill all the even rooms, leaving an infinite number of odd rooms open. But now an infinite number of buses with an infinite number of passengers on each arrives. How do you fit all of those people? We can use Euclid's statement that there are an infinite number of prime numbers. So every current guest goes to the first prime number, 2, raised to the power of their current room number, so room 2^n. Then, the passengers on the first bus go to the next prime, 3, raised to the power of their seat number, or 3^n. The next bus, powers of 5, 7, 11, and so on to the last bus. Since each room is a prime to the power of a natural number, each one is unique, and there will be no overlapping rooms. This is only possible, however, with the lowest level of infinity, or the countable infinity of the natural numbers. This is 1, 2, 3 ... infinity. This is also called aleph-zero. With higher orders of infinity, though, our strategies fall apart. For example, the real number infinity hotel would have radical, irrational, and negative rooms. This all shows us just how hard the concept of infinity is.