Hilbert's paradox of the Grand Hotel

Have you ever seen a really big hotel with lots of rooms? Well, imagine a hotel that is so big that it has an infinite number of rooms. This hotel is called the Grand Hotel.

Now, let's say that one day, the hotel manager decides to add even more rooms to the hotel. But, instead of just adding a few extra rooms, the manager decides to add an infinite number of rooms. Yes, you heard me right - an infinite number of rooms!

So, now the Grand Hotel has an infinite number of new rooms. But, here's where things get a little bit tricky. Suddenly, an infinite number of guests arrive at the hotel, each needing a room.

How can the hotel manager accommodate all these new guests? It seems impossible, right? But, thanks to a concept called "transfinite arithmetic," the hotel can actually make room for an infinite number of new guests.

Here's how it works: The hotel manager simply asks all the current guests to move to the room number that is double their current room number. So, the person in room 1 moves to room 2, the person in room 2 moves to room 4, and so on.

Now, all the odd-numbered rooms are empty and available for the new guests. And since there are an infinite number of new guests, there are still an infinite number of empty rooms!

But it doesn't stop there. Let's say an infinite number of new guests arrive again. The hotel manager can simply repeat the same process, but this time the current guests move to a room number that is triple their current room number. And again, an infinite number of rooms become available for the new guests.

This process can continue indefinitely, with the hotel always finding a way to accommodate an infinite number of new guests. This is Hilbert's paradox of the Grand Hotel - a fascinating concept that shows how infinity can create some pretty mind-boggling situations, but also how we can use transfinite arithmetic to navigate them.