[leviself]

Yes. The statement as given doesn't work out. If you say "an infinite number of guests and an infinite number of rooms" it works. But when every room is endowed with the property of being occupied there is no room for more guests.

[mistercow]

No, the problem was phrased correctly. What we are saying is that for each natural number n, room n is occupied by a guest. This is the mathematical idea embodied by "an infinite number of occupied rooms". If you say "an infinite number of guests and an infinite number of rooms", then the question becomes obvious and trivial.

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.

[ajanuary]

It brings up an interesting point, though. If I have an infinite number of rooms and an infinite number of guests, then one might intuit that every room is occupied.

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.

[nhaehnle]

There is no empty room initially. However, by having all guests move simultaneously, you can make one room free without having any guest leave the hotel.

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).

[billswift]

That's the sort of thing that happens when semantics meets mathematics; Swizec screwed up his statement of the problem. Swizec and the people agreeing that the answer is yes are confusing the problem's literal statement with the common statement of similar problems.

[nhaehnle]

I am thoroughly confused as to what your point is. What is the problem with Swizec's wording in your view? Should s/he have written "Can you make room for more guests?". I agree that you could argue that right now there is no room free, and that the answer should be No because of that. However, given that you can easily make one room free by having every guest go to the next room, that just seems overly nit-picky to me, and like you are intentionally trying to miss the point.

[damoncali]

The difference is an infinite number of rooms can accommodate an infinite number of guests and still have room for one more. An infinite number of full rooms can accomodate no more guests. They're all full by definition.

That is, unless I am also thoroughly confused.

[mistercow]

>That is, unless I am also thoroughly confused.

You are, but that's OK; they call it a "veridical paradox" for a reason.

[leviself]

The part that makes the assertion false is "...all of which are currently occupied." Moving n to n+1 only works when that restriction is lifted.

[recursive]

No, because the people in n+1 simultaneously move to n+2. (and so on)