This is one of the hardest concepts that mankind has ever tried to understand. And, why is it so complicated? Because it is just too unusual! No one has ever seen an infinite road, and it is very hard to imagine! How do you do it? Do you imagine yourself walking on that road? Do you imagine it on the space? Because, the earth do not have space for a thing like that! But, does it means that space has space for it? If we consider that the universe is finite, how can an infinite thing be inside a finite one? If we consider that the universe is infinite, things go so crazy that you might need the help of the infinite “yous” that exist out there just to help you understanding it.The Hilbert Hotel is a paradox that was created to make us see the complexity of the concept of infinity.
For a moment, imagine an infinite hotel, and that hotel is full. Suddenly a guest arrives and asks for a room. What does the hotel’s manager do? He can’t just say that they don’t have a room for him. And he can’t also put him in the last room for two reasons: the first one is that he can’t find it (since it is an infinite hotel), and second is because it is already occupied. So, what is the solution? He thinks and concludes that he can ask to each guest to go to the next room. For example, the guest in room number one goes to room number two, the guest in room number two goes to room number three, and so on. That is going to free room number one, and the new guest can now sleep in peace. This solution work of any finite number of guests, so, everyone can have a room in that hotel, even if it is full.
But, what happens if an infinite number of guests ask of an infinite number of rooms? Now, the manager can’t ask to the person in room number one to go to the room number infinite* (*infinite is not a number, just a concept) because that person will never find that room.
It seems that there is nothing he can do, or is it? Well, he can ask to the guests that he already have on the hotel to go to a room that is not a room with an even number on the door. And, one that it is done, he can now put the infinite guests on the infinite rooms that are free (because the even numbers are infinite). Problem solved!
But, what does he do if an infinite number of cars (each one with its infinity number of people inside) arrive and everyone asks for a room? The first thing to be done is to label the cars and the seats with infinite numbers. Once that is done, he can now ask to the person on the first seat of the first car to go to room number two, the person on the second seat go to room number four, the person on the third seat go to room number eight. What we is doing is asking to the people on the first car to go to de first number prime (two) powered by their seat number. So, a person who is seat on the fifth seat of the seventh car goes to room number 1419857 (17^5). With this system, everyone can have a room, leaving free an infinite number of rooms, because not every number is affected by this system.
We are starting to see the cool things that would happen if we live on an infinite earth with infinite people and infinite hotels, here, even full, there is always room for you.
Here we just looked to the natural numbers, but we can’t forget that there are a lot of numbers we don’t talk about, like the negative ones, the fractions, those that are irrational, etc. So, we can say that we only talked about one infinite part of an infinite group.
