4 ms·
No, those guests now stay in room n+2.
by Cushman 14y ago
No, those guests now stay in room n+2.
- damoncali 14y agoI don't get it. Can't I just as easily say that n+2 is also occupied? After all, they're all occupied by definition, aren't they? For example. All blocks are either red or blue. You have an infinite number of blocks that are all red. Do you have any blue blocks? No. Where is the mistake in my logic?
- icebraining 14y agon+2 was occupied, but it's not anymore because you moved them to n+3.
- damoncali 14y agono, n+3 is also full. See? Why do you get to win that back-and-forth and not me? After all, the problem stated that all rooms are full, even N+3, 4, 5, etc. Why do you get to construct a new room without also constructing a new pre-existing occupant?
- nhaehnle 14y agoYou don't have to do that. Just use following simple steps (assuming that the hotel rooms are all along one infinitely long corridor): 1. Have all guests of the hotel simultaneously pack their things and move out of their room onto the hallway. 2. Have all guests simultaneously move along the hallway to be standing in front of the room next door (in the same direction, obviously). 3. Have all guests simultaneously move into the room they are now standing in front of. The first room is now empty, with nobody having had to leave the hotel, and every guest still has his/her own room.
- damoncali 14y agoThanks. I can understand that.
- deleted 14y ago[deleted]
- mistercow 14y agoYou don't need a hallway as a holding area. Everyone in the hotel simultaneously moves one room up. Talking about it sequentially as "1 moves to 2 and 2 moves to 3 etc." is convenient for describing it, but the process does not actually happen sequentially; otherwise it would take an infinite amount of time. But since each person is capable of moving at the same time, and since their movements need not interfere with each other, the whole process can happen instantly. If you are visualizing a traditional hotel, then this means that everyone walks out of their room into the hallway and then moves one door down. But the hallway is irrelevant. The rooms could instead be directly connected by doors, and everyone would still just simultaneously move to the next room. No holding area needed.
- damoncali 14y agoOne more question. What if the rooms are not in an infinitely long hallway (with or without interconnecting doors), but a ring of infinite circumference? Or is there such a thing?
- mistercow 14y agoIt depends on what you mean by "ring of infinite circumference", because you're going to have to discard some conventional ideas associated with a "ring" in order to make the definition work. If you just mean "the rooms extend infinitely in both directions", then yes, the same reasoning applies; we'll just have an arbitrary "room 0", and then "room -1" on one side and "room 1" on the other. Then you tell everyone in a negative room to move to room n-1 and everyone in a positive room to move to room n+1. However, this definition does away with the idea of a ring meeting itself on the other side. We cannot say that "room -∞" is the same as "room ∞" for example, because there neither of those rooms actually exist.
- ColinWright 14y agoWhen dealing with infinities you need to be really precise, and very careful. When you ask about "a ring of infinite circumference" I have to ask - what do you mean by that? The only thing I can think of that's related is a "circle if infinite radius", but the usual interpretation of that is a straight line. Then we're back to an infinitely long hallway. So - what do you mean?
- mistercow 14y ago>Can't I just as easily say that n+2 is also occupied? No, because the former occupants of n+2 are now in n+3.
- plusplus 14y agoYou do it for all natural numbers n=1, 2, 3 , ... (those are countably infinately many numbers). Because every natural number n has a successor n+1, you can do it for all of them. Each n except 1 has a predecessor n-1 from which guests where moved to n. Nobody moved into 1, therefore you can put the new arrivals there. After that, there are still (countably) infinite many guests. They are just matched up differently with the rooms/numbers. Because you have infinitely many rooms, you were able to accomodate one more by matching them up differently. That's sort of the point :-)