Have you ever thought about a hotel with infinite rooms? No?
David Hilbert, a German mathematician, created a thought experiment to show us how hard it is to wrap our minds around the concept of infinity.
The Infinite Hotel
Imagine a hotel with an infinite number of rooms and a night manager.
One night, the Infinite Hotel is complete, with endless guests and a man walks in asking for a room.
What does the night manager do?
The solution is simple.
The night manager asks each guest to move to the following room number.
Since there are infinite rooms, it creates a new room for each existing guest, leaving room 1 available for the new customer.
The process can be repeated for any finite number of new guests, and everyone can be accommodated.
An Infinitely Full Bus
But what happens when an infinitely full bus of people arrives at the hotel?
At first, the night manager is perplexed.
However, he realizes there’s a way to place each new person.
He asks the guest in room 1 to move to room 2, the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on.
By doing this, he has emptied the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus.
Everyone is happy, and the hotel’s business is booming more than ever.
Infinite Buses
One night, the night manager looks outside and sees an infinite line of infinitely large buses, each with endless passengers.
What can he do? Luckily, he remembers that there is a limitless quantity of prime numbers.
So, to accommodate these seemingly infinite weary travelers, the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number.
The people on the first bus are given the room number of the next prime, 3, and presented with the power of their seat number on the bus.
The passengers on the second bus are assigned management of the next prime, 5. Each bus follows powers 11, 13, 17, etc.
The night manager can accommodate every passenger on every bus by using unique room-assignment schemes based on unique prime numbers.
The Real Number Infinite Hotel
The strategies used by the night manager are only possible because the Infinite Hotel only deals with the lowest level of infinity, aleph-zero. Aleph-zero is the countable infinity of the natural numbers 1, 2, 3, 4, and so on.
If we were dealing with higher orders of infinity, such as that of real numbers, these structured strategies would no longer be possible as we have no way to include every number systematically.
The Real Number Infinite Hotel has a negative number of rooms in the basement and fractional rooms.
So, there you have it, a hotel with infinite rooms!
While it may seem like a logistical nightmare, the night manager can accommodate every guest using prime numbers, even those on infinitely long buses.
Just don’t ask him to deal with the actual numbers.
To access relevant information, check out the following blogs:
- Kangaroo Math Blog for Mathematics
- Kancil Science Blog for Science
- Beaver Computational Thinking Blog for Computer Science
- Kijang Economy Blog for Economics.
