plusplus's comments

On: Does infinity exist?

[plusplus]

You 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 :-)

[priv_acy → plusplus]

You cannot move someone to a full room. All of the rooms are full. Explain that part.

[plusplus → priv_acy]

I think nhaehnle explained it very nicely.

On: Does infinity exist?

[plusplus]

Yes. They must move to room n+2.

[priv_acy → plusplus]

But how do you get a situation where there is one more room than occupants, if both are infinite and matched up?

[Evbn → priv_acy]

You rematch. Mathematically, not physically! There isn't' "one more", all finite differences between infinite sets are equivalent to each other, including zero.

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.

[priv_acy → Evbn]

The mixing of the possible with the impossible is certainly jarring.