The classic approach is to move all the existing guests from room n to n+1 and put the new party in room 1.
I'm really struggling to square up my understanding of how the former problem works with my intuition that if every room is in an occupied state, you can't magic up some in an unoccupied state.
That's the problem with infinites. Consider the number of all positive integers. Let's call it a1.
Now consider the number of all even positive integers. Let's call it a2.
Each even integer can be directly mapped to an integer and the other way around by multiplying or dividing by 2, right? All integers can be multiplied by 2, and all even integers can be divided by 2.
So a1 = a2, the number of integers is the same as the number of even integers, even though you'd intuit the number of integers to be twice the number of even integers.
Just as it does when reaching the infinitely small (0.999… is precisely equal to 1), intuition breaks down when reaching into the infinitely big.
> Your intuition is correct.
> The hotel paradox is a trick of symbolic manipulation,
> and the analogy to a real hotel is an amusing deceptive
> misdirection.
I'm no mathematician, but this seems to be an edge phenomenon that everyone is willing to ignore. I suppose you could just as easily tell the new guests to run down the hallway until they find an empty room at the end; out of sight, out of mind...
There is no room 'n+infinity'. There's just not such a room, anywhere, even in principle. Each guest n can move to room n+1 because the guest there is moving to n+2. This works. That's the point.
And you couldn't, in fact, tell the guests to run down the hallway to find an empty room: There is no empty room until you move everyone.
Infinity is counterintuitive, you say? Well, yes, that's the point of this illustration.
You are certainly correct, however, that Cantor's hotel does not need to have any rooms with transfinite room numbers.
What makes it interesting is that we really do mean that every room is occupied to begin with. The surprising result is that we can in fact add another guest by having everyone move over one room, even though we started out with every room occupied. And in fact, we can add infinitely many new guests by telling everyone to move to their room number \* 2.
There is a mapping of every guest to a room: guest n is in room n. Yet somehow there exists a room that has no guest, despite being able to model both rooms and guests with the same infinite set.
Incidentally, this is one way to define infinite sets. A set is infinite if and only if there exists a proper subset that has the same size (cardinality).
That is, unless I am also thoroughly confused.
You are, but that's OK; they call it a "veridical paradox" for a reason.
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?
No, because the former occupants of n+2 are now in n+3.
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 :-)
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.
Deleted comment
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.
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.
So - what do you mean?
Firstly, the whole point of the hotel story is to talk about addition of cardinals. You are moving off topic and to some extent "bike-shedding" - discussing something other than the real point.
In particular, if there are infinitely many rooms then you must be able to create an embedding of the counting numbers into it. Regardless of whether or not that accounts for all the rooms, you then have an infinite chain, one without a concept of a "last room".
It is possible to have infinitely many cycles, and in those cycles the "last room" is followed by the first room, but there are infinitely many of these cycles, so we can order them in a different way, and we get back again the infinite chain with no "last room".
And this matters. We say that two sets are the same size if we can create a matching between them, and we say they are of different sizes of there is no matching. Be careful. Just because your first attempt at a matching doesn't work, that doesn't mean there isn't one. So in this case of the infinite number of rooms, just because one arrangement seems not to work, that doesn't mean that no arrangement will work.
And indeed, the very definition of the number of rooms being infinite means there is a way to arrange the movements of the existing guests to ensure that there are no "last room" problems.
Ponder this if you will: how did the hotel get full in the first place? if you assume that is possible, then you can reverse the original room assignments, and reassign rooms.
The source of your confusion is that you trusted the problem is even possible to set up, without availing yourself of the power that the setup implies.