November 20, 2024

Aisha Chloe C. Camaquin

11th Grade

John F. Kennedy High School 

Infinity seems like an easy concept to grasp; it’s just things with no end, right? Well, when you venture deep into the idea, several paradoxes arise, such as the simple question of: are there more counting numbers, or even numbers? The answer is neither. Both sets of numbers are boundless, which means that they can be paired one-to-one. 

A thought experiment, known as “The Infinite Hotel Paradox,” was devised by David Hilbert in 1924. Its purpose was to allow the idea of infinity to be easily understood by the general public, while at the same time exploring the never-ending properties of an endless set. It is as follows: One day, a traveler stumbles upon a hotel while seeking refuge during a storm. The receptionist checks him in and hands him his key card with his room number: “1.” Why not the last one, which appears to be a reasonable choice? Well, because this is Hilbert’s Hotel; a hotel that can house an infinite amount of guests because it has an infinite amount of rooms. Due to its nature, it is impossible to reach the final room. Therefore, to house a new guest, residents are asked to move from their room to the next (n + 1); room one will move to two, room two will move to three, and so on. 

When the storm ends, the traveler’s stay is up. As he walks out the doors of the hotel, a bus arrives. This bus has never-ending seats and unlimited passengers, all planning to book a room. Since it is impossible to move to n plus infinity, they are instead asked to move to their room number times two (2*n); room one will move to two, room two will move to room four, etc. This is because any digit multiplied twice will always be even. By doubling the room number’s value, all odd-numbered rooms are left open for the passengers of the bus. 

The traveler’s journey is up and he must return home. On his return voyage, he stumbles upon the hotel once again. Vehicles identical to the previous arrive, although instead of one, there is an infinite amount of them. Unfortunately, the two previous solutions are not applicable to this scenario, so how will all the new guests fit? Well, similar to even and odd numbers, the quantity of prime numbers is boundless. Additionally, their values do not overlap when increased exponentially. So, by using powers of varying prime numbers, the riders will be given their own unique rooms. 

To house the new guests, the current ones are asked to move to two to the power of their room number (2^n). For example, the resident of room one will move to room 2^1, room two will move to room 2^2, and so on. This is because two is the first prime number. Once all residents have moved, it is time to assign rooms to the new arrivals. Passengers of the first bus will be assigned the room number equivalent to three to the power of their seat number (3^s). For instance, seat one will reside in room 3^1, seat two will be in 3^2, and so on. This process will be applied to the following buses, with the base changing to the succeeding prime number (7, 11, 13, 17, etc.).