Hilbert's Infinite Hotel Paradox
Hilbert's Infinite Hotel Paradox
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By Daniel Lauris Gunawan
December 20, 2023
In the 1920s a German mathemetician David Hilbert devised a thought experiment to show us how hard it is to wrap our minds around the concept of infinity.
Imagine a hotel with an infinite amount of rooms and a hard-working manager. One day the Infinite Hotel is booked up by an infinite number of guests. A man walks in and asks for a room. Rather than turning him down, the night manager decides to make room for him. He asks the guest in room 1 to move to room 2. Guest in room 2 to room 3 and so on. The guests moved from room number “n” to room number “n+1”. Since there is an infinite number of rooms, there is a new room for each existing guest. This leaves room 1 open to the new guest. This process can be repeated for any countable number of new guests. If a tour bus unloads 40 new guests looking for rooms, then every existing guest moves to “n” to room number “n+40”. Opening 40 rooms for the new guest.
But now an infinitely large bus with a countably infinite number of passengers pulls up to rent rooms. It surprised the manager, but he had an idea. He moved the guest from room 1 to room 2. Guest in room 2 to room 4. Guest in room 3 to room 6 and so on. Each current guest moves from room “n” to room “2n”. Filling up only even-numbered rooms. By now he has emptied all of the odd-numbered rooms for the passengers on the infinite bus. Word spreads about this incredible hotel and people pour in from far and wide.
One night the night manager peaked outside and saw an infinite line with an infinitely large bus. If he cannot find ]rooms for them, he will lose out on a fortune. And surely lost his job. Luckily he remembers Euclid proved that there is an infinite quantity of prime numbers. To accomplish this task, the manager assigns every current guest to the first prime number, 2 raised to the power of their current room number. So guests in room 7 will go to room number “2^7” which is room 128. Then the manager takes the people in the first of the infinite buses and assigns them to the room of the next prime number 3 raised to the power of their seat numbers on the bus. The person in seat number 7 goes to room number “3^7”. This continues for all passengers on the first bus. The second bus is assigned to the power of the next prime number 5. The following bus power of 7. Since each of these numbers only has 1 and the natural number power of their prime number base as factors, there are no overlapping numbers. They are all assigned by unique prime numbers. However, many rooms go unfilled, like room 6, since 6 is not a power of any prime number. Luckily his boss isn’t good at math.
While the Infinite Hotel is a logical nightmare. It only deals with the lowest level of infinity, mainly the countable infinity of natural numbers 1, 2, 3, 4, and so on. If we were dealing with high levels of infinity such as real numbers. The strategies would not be possible as we have no way to include every number. Real Number Infinity Hotel has negative rooms in the basement, and fractional rooms, so the guests in room ½ always suspect he has less room than the guests in room 1, square root rooms, and room pi, where guests expect free dessert.
Who would want to work here even for an infinite amount of salary? But over at Hilbert’s Infinite Hotel where there are never any vacancies and always more rooms, the scenario where the diligent and hospitable night manager serves to remind us of just how hard it is for our finite minds to grasp a concept as large as infinity. We may need you to change rooms at 3 am.
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