[MATHS] The Infinite Hotel Paradox by David Hilbert

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Recently I watched a YouTube video about the infinite hotel paradox which was introduced in 1920s by a German mathematician, David Hilbert. In case you’re curious about he video, just search on YouTube using “The Infinite Hotel Paradox” keyword.

I found this paradox interesting as it shows how hard it is for our mind to grasp the concept of infinity. To understand the video better, I did a quick review of Euclid’s proof for infinite number of prime numbers and the difference between countable and uncountable infinity. I’ve actually written about this Euclid’s proof on another post here.

Basically, the video shows some cases where the hotel’s manager wants to add new guests presuming that the infinite hotel is full.

Case I. How to add one new guest?

Easy. Every existing guest in room number N goes to room number N + 1.

Case II. How to add M new guests (M > 1)?

Still easy. Every existing guest in room number N goes to room number N + M

Case III. How to add an infinite number of guests?

Starts to be a bit complicated, but still solvable. Every existing guest in room number N goes to room number 2N. This fills up all the even-numbered rooms. All the odd-numbered rooms are now empty.

Case IV. Now, an infinite number of buses, each with an infinite number of passengers arrive. How to allocate a room for every new guest?

More maths involved here. Every existing guest in room number N goes to room number (P0^N). Then, every new guest from bus 1 with seat number M goes to room number (P1^M). Every new guest from bus 2 with seat number M goes to room number (P2^M). The same rule applies to bus 3, 4, and so forth. Here, P0, P1, and P2 refers to the first, second, and third prime numbers respectively. Therefore, P0=2, P1=3, P2=5, P3=7, P4=11, and so forth.

In this 4th case, the Euclid’s proof that there is an infinite number of prime numbers is applied.

By the way, I laughed when watching the video’s introduction as the narrator of the video said something like “one night, the infinite hotel is completely full”. Well, how come an infinite hotel can be completely out of rooms?