Q&A: Hilbert’s Hotel
Hilbert’s Hotel
Question
(I saw that the Rabbi refers to this.)
I didn’t understand why, when a guest arrives, you need to move the guests from room 1 to room 2 and so on in order to make space. If there are infinitely many rooms, then there is always room for every guest anyway. (In the same place where you put the guests after moving them, you could just put the new guest there from the outset.)
- In the later stages of the story you see the paradox that infinity contains parts that are themselves infinite (prime numbers and the like), but the first part seems to make no claim at all, since if there are infinitely many rooms it can never be full, and there is always room.
Answer
This is a misunderstanding of the definitions of infinity. The assumption is that at the beginning there are no vacant rooms. Everything is occupied, since there is a person in every room (infinitely many people). And nevertheless, it is still possible to make room. You are thinking about infinity like a finite number, but that is not allowed.
Discussion on Answer
I’ll say it again: there is not a single vacant room. There are infinitely many rooms, each assigned a natural number (from 1 to infinity). There are infinitely many people, each of whom is also assigned a natural number. In the initial state, person n is in room n. Is there a vacant room? Please tell me its number. The answer is that there is no vacant room. And nevertheless, you can still carry out the whole process. That is the nature of infinity.
Take all the people in the hotel and add the extra guest. Is there one of them who doesn’t have a room? No, because there are infinitely many rooms.
– What does it help to remove the guests and put them back in? If afterward there is room for everyone, then that should also have been true at the beginning.
Dear friends, I’m not going to teach mathematics here. What I wrote is clear. All that’s needed is to get rid of the general slogans that don’t work when it comes to infinities, and to follow the arguments (a mathematical habit that is worth sticking to in general). All the best.
Eli,
You’re asking why you need to vacate a room if all the rooms are occupied?
Because the answer is pretty simple: all the rooms are occupied.
If you’re trying to say that from the outset it would have been possible to leave a room (or several, or countably infinitely many) vacant, then that’s true—but that is exactly what Hilbert was trying to demonstrate here.
There are infinitely many people, but not all the rooms are occupied, because there are infinitely many rooms and infinity can’t be filled. (After all, when you take them out of the rooms, you can put them all back in, so why not put the guest there from the outset?)