The mathematician David Hilbert is credited with the concept of an infinite hotel (or Grand Hotel, as he called it), a hotel with an infinite number of rooms. One afternoon, a man went to the desk and asked whether he could book a room for the night. As it turned out, all the rooms in the hotel were taken. The concierge, being a clever fellow, devised a scheme whereby the guest was given a room so that each guest had his or her own room. How did the concierge manage this?
Actually the way it happened was that the Grand Hotel, which is in Somerset, accepted an extra party containing an infinite number of guests. After several jars of scrumpy (proper cider) the concierge (I thought they only had those in France) moved all the guests in the hotel by re-rooming them in a room with a number equal twice their original number. thus the guy in room 1 went to room 2, vacated by the gal who went to room 4 The gal in room 2 went to room 4, vacated by the guy who went to room 8, vacated by the gal who went to room 16.... and so on. At the end of this the concierge put all the new infinite party into rooms with odd numbers. Then he finished the bottle of scrumpy.
Whether or not that scheme will work depends on how you feel about completed infinities. If the shuffling starts with the low (finite) room numbers and if it takes a finite time for a guest to pack up and move to another room, then first guest #1 and guest #2 will be doubled up in room #2. Then guest #2 and guest #4 will be doubled up in room #4. And so on. A never-ending wave of double occupancy will move through the even-numbered hotel rooms. You might as well avoid all the shuffling and just double up the new guest in room #1 with guest #1 and leave it at that. And then the clerk could get back to his bottle of scrumpy in finite time.
The way I see it is that right when guest N moves to room 2N, guest 2N is moving to room 4N. The two guests meet at the door at exactly the same moment, with one coming out as the other is going in. No double occupancy.
The guests would have to be infinitely thin to avoid double occupancy while shifting rooms. Othewise, you might have guest #2 1/3 of the way between rooms #2 and #4, while guest #4 is 1/3 of the way between rooms 4 and 8. Adding up the fractions, you still get double occupancy. If the guests WERE infinitely thin, you could just pack them all into the first room. If you add infinitely many new guests, then you'll have infinite occupancy in the lobby while they're waiting to check in. In the hotel you'll have a finite wave train of double occupancies. But the number of guests packed into the lobby will be always remain infinite, and the number of new guests ensconced in rooms will always remain finite (unless you're comfortable with completing infinities. You could just wave your hand and shift all the guests instantly, but that would not really be a move - it would be a redefinition. It would be like the clerk saying, "We're full up, but our sister hotel across the street has a vacancy in room 1." But that would work equally well for ordinary finite hotels. Technically the wave of double occupancy would not really go on forever. It would die out when it got to the room after which in all subsequent rooms all the guests have died from starvation because of the slow room service. Room service is particulary slow in the infinite-numbered rooms, where just calling room service and giving your room number takes forever.
The Somerset Solution is not the only one. In fact any yarn based on the fact that +n = and/or n times = can be made to work. Come on let's hear some more stories.
I know for a fact that most of the guests at the Hilbert Hotel have starved to death. I have some personal experience with these matters. I once stayed at a Holiday Inn that had infinitely slow room service. Infinity is forever. Here are a couple quotes from the Wikipedia article on "actual infinity:" I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction. ( C.F. Gauss [in a letter to Schumacher, 12 July 1831]) Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking - a remarkable harmony between being and thinking. ( D. Hilbert [6, 190]) However, according to Wikipedia, most modern mathematicians (since Cantor) now believe in completed infinities. I disagree with them. I'm still waiting for my ham-on-rye from Holiday Inn's room service.
All the real numbers from 0 to 1 i.e. in the set 0<x<1 form a completed infinity. Interestingly this set as stated has an upper bound (1) and a lower bound (0), but it has no greatest lower bound or least upper bound. Are you familiar with epsilon - delta mathematical analysis? This is a necessary prerequiste for studying comptedness, limits and infinities.
Here's a link to an interesting story (by Nancy Casey, 1991) about Hilbert's Hotel, Sam Cantori, and Mr Kronecker. http://www.c3.lanl.gov/mega-math/workbk/infinity/inhotel.html Also see Rudy Rucker's math-fiction novel, The White Light.
When Gauss said that Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, Im pretty sure he was talking about epsilon-delta analysis. But Cantor claimed that infinity was more than just a way of speaking. I believe that infinity is a much battered and abused word. I would like to see it replaced in the lexicon by a new word, mornuff (short for more than enough). How many rational numbers are there between 0 and 1? Mornuff. If there are mornuff rational numbers, who needs irrational numbers? Maybe we need another new word, "closenuff." For any irrational number, there is an rational number that is closenuff. "The limit of f(x) as x approaches x-naught = A if f(x) is closenuff to A when x is closenuff to x-naught." How many stars are in the sky? Mornuff. How many people have died in wars? Mornuff.