The Hilbert Hotel

The premise: The Hilber hotel is a hotel with an infinite number of rooms that start with room number 1. The Hilbert hotel is currently full of guests, that have completely filled all the rooms that exist.

Problem 1

Let's say a new guest arrives at the hotel, does this guest not have a place in the hotel? He does.

We just ask the person in room 1 to move to room 2 and the person in room 2 to move to room 3 and so on. We every guest to move from their room n to room n+1. Now room 1 is empty, which can be provided to the guest.

This works for any finite number, you just ask the existing guests to move from their room to their room number + number of guests that have to be accommodated.

Problem 2

Let's say a bus of infinite length arrives at the hotel, can this bus be accommodated at the Hilbert hotel? Of course!

Objective: We need an infinite number of rooms that have to be freed up while having an infinite number of rooms for the existing guests. So we need inf + inf number of rooms but lucky for us that just evaluates to inf.

We ask each person to move the room with 2x their room number. For example, the person in room 1 moves to room 2, the person in room 2 moves to room 4, the person in room 3 to 6, and so on.

So we just move the person in room n to room 2n. This way all the even number rooms are taken by the existing guests (an infinite number of even numbers) and the new guests have all the odd number rooms freed up for them (also infinite number).

Problem 3

Let's say an infinite number of buses of infinite length with infinite guests arrive at the hotel, can these guests be accommodated? Hell yeah!

Objective: We need to somehow create an infinite series of infinite numbers from just the natural numbers. Now that sounds like a task...

But that can still be done by using the simple principles: - There are infinitely many primes in the natural numbers. - Every number has a unique prime factorization (to rephrase, no 2 numbers have the same prime factorization.)

To accommodate the guests at the hotel: Each guest is asked to move to the room with the numbers that's the 2 to the power to their room. So guest 1 would move to 2^1, guest 2 would move to 2^2, guest 3 would move to 2^3, and so on.

Now starting with the busses, starting with the first bus, we assign it the next prime that's not already been taken and ask the passenger to go to room prime^seat number

So this would accommodate the entire set of guests. The infinite lists would look like this:

{ 2^1, 2^2, 2^3, 2^4, ...}
{ 3^1, 3^2, 3^3, 3^4, ...}
{ 5^1, 5^2, 5^3, 5^4, ...}
.
.

p^sn would be the room any guest would be assigned where p would be the prime assigned to the bus or the hotel (which would be 2 for the hotel) and sn would be the seat number or the existing guest room number.

Conclusion

All of these problems were solvable since they are related to the lowest form of infinity, the countable infinity. This is called aleph-null which is represented by this symbol:

This now goes into the study of set theory and how to have numbers greater than infinite and now to have larger infinities.