Hilbert Hotel

Have you ever wondered what would happen if you had an infinitely large hotel with infinitely many rooms, and you wanted to accommodate infinitely many guests? This is the scenario that mathematician David Hilbert imagined in 1924, and it leads to some surprising and paradoxical results.

In this blog post, I will explain the concept of Hilbert's hotel and some of the implications of infinity for mathematics and logic. I will also show you how to use some simple rules to manipulate infinite sets and perform seemingly impossible tasks.

Hilbert's hotel is a thought experiment that illustrates the properties of infinite sets. A set is a collection of distinct objects, such as numbers, letters, or people. A set is finite if it has a fixed number of elements, and infinite if it does not. For example, the set of natural numbers {1, 2, 3, ...} is infinite, because there is no largest natural number.

An infinite set can be either countable or uncountable. A countable set is one that can be put into a one-to-one correspondence with the natural numbers, meaning that each element of the set can be assigned a unique natural number as a label. For example, the set of even numbers {2, 4, 6, ...} is countable, because we can label each even number with its half: 2 -> 1, 4 -> 2, 6 -> 3, and so on. An uncountable set is one that cannot be put into such a correspondence, meaning that there are more elements in the set than there are natural numbers. For example, the set of real numbers (all possible decimal numbers) is uncountable, because there is no way to list them all in order.

Hilbert's hotel is a hypothetical hotel with infinitely many rooms, numbered 1, 2, 3, ... The hotel is always fully occupied by infinitely many guests. One day, a new guest arrives and asks for a room. How can the hotel manager accommodate him without evicting anyone?

The answer is simple: he can ask all the guests to move one room to the right. That is, the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on. This way, room 1 becomes vacant and the new guest can check in. This does not create any problems for the existing guests, because they can always find a room with a higher number than their current one.

But what if infinitely many new guests arrive at once? How can the hotel manager accommodate them all? The answer is again simple: he can ask all the guests to move to the room with twice their current number. That is, the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6, and so on. This way, all the odd-numbered rooms become vacant and the infinitely many new guests can check in. This does not create any problems for the existing guests either, because they can always find a room with an even number higher than their current one.

These examples show that adding one or infinitely many elements to a countably infinite set does not change its size or cardinality. In other words, a countably infinite set plus one or plus another countably infinite set is still countably infinite. This may seem counterintuitive at first, but it follows from the definition of countability: as long as we can find a way to label each element of the set with a unique natural number, we can say that the set is countably infinite.

However, not all infinite sets are countable. For example, suppose that instead of new guests arriving at Hilbert's hotel,