The Infinite Hotel Paradox

In the 1920s, David Hilbert created a scenario for people to understand the complexity of infinity. He presented the idea through the infinite hotel paradox.

He began with imagining a hotel with an infinite number of rooms and an infinite number of guests filling up the infinite number of rooms. A man approaches the manager and asks for a room in this infinite hotel, and rather than turning the man down the manager arranges a room for him. The manager asks the guest staying in room 1 to move to room 2, and the guest in room 2 to move to room 3 and so on. Every guest moves from room number n to room number n+1 eventually and as there is an infinite number of rooms, there is a new room for all the existing guests and this leaves room number one open for the man. This process can be repeated for any finite number of new guests. If a bus with 30 people came to the hotel and each passenger was looking for a room, then every existing guest will just have to move from room number n to room number n+30 leaving the first 30 rooms empty for the new guests.

The scenario proceeds to an infinitely large bus with a countably infinite number of passengers that approach the hotel. The manager finds a way to accommodate the new arrivals. He asks the guest in room 1 to move to room 2, the guest in room 2 to move to room 4 and the guest in room 3 to move to room 6. Each current guest moves from room n to room 2n. By doing this only the even infinite numbered rooms will be filled up and all the infinite odd rooms are left empty. These rooms can now be filled by the new infinite number of guests from the infinitely large bus.

Suppose now that there is an infinite line of buses carrying an infinite number of passengers all looking for a room in the hotel. To solve this issue, it is important to recognise that there is an infinite quantity of prime numbers. The manager begins by assigning all the current guests in the hotel the prime number two raised to the power of their current room number. So, the guest in room number 5 now moves to 2^5 which is room 32. The manager then takes the first of the infinitely many numbers of buses and assigns them to the next prime number which is three raised to the number of the seat they are in of this infinitely large bus. The passenger in seat number 5 is asked to go to room 3^5 which are room 243. This continues for all the first bus. The passengers in the second bus are assigned powers of the next prime 5 and the next bus is assigned powers of 7 and so on. Since each of these numbers only has one and the natural number powers of their prime number base as factors, there are no overlapping room numbers and so all the passengers can be assigned rooms. This will, however, mean there will be many empty rooms such as room 6 since 6 is not a power of any prime number.

Once the scenario gets to an infinite number of busses with an infinite number of passengers, there is a range of ways the guests can be organized to accommodate everyone. Along with the prime powered method, there is the interleaving method and triangular number method all which are successful in accommodating the infinite number of guests,

This analogy is able to present the difficulties with our finite minds trying to grasp a concept as large as infinity.

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