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I'm gunna have to be one of those people that says "really? O_o" when presumably everyone else clearly grasps the concept. I get that if you had an infinite nu
by ajanuary 14y ago
I'm gunna have to be one of those people that says "really? O_o" when presumably everyone else clearly grasps the concept.
I get that if you had an infinite number of rooms, and an infinite number of guests staying, you still have room for an infinite number more guests.
But in your example, surely "occupied" is, more than just an indicator of another guest, a state of the room. You have an infinite number of occupied rooms, but zero unoccupied rooms, so no space for more guests.
- recursive 14y agoThe classic approach is to move all the existing guests from room n to n+1 and put the new party in room 1.
- damoncali 14y agoAren't there already guests in room n+1?
- Cushman 14y agoNo, 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.
- 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 :-)
- plusplus 14y agoYes. They must move to room n+2.
- damoncali 14y agoBut how do you get a situation where there is one more room than occupants, if both are infinite and matched up?
- deleted 14y ago[deleted]
- Evbn 14y agoYou 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.
- damoncali 14y agoThe mixing of the possible with the impossible is certainly jarring.
- leviself 14y agoYes. 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.
- billswift 14y agoThat'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 14y agoI 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 14y agoThe 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 14y ago>That is, unless I am also thoroughly confused. You are, but that's OK; they call it a "veridical paradox" for a reason.
- leviself 14y agoThe 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.
- inportb 14y agoRight, so the occupants of n+1 would have to move to n+2, the occupants of n+2 would have to move to n+3, and so on. Somewhere, the occupants of n+infinity-1 and n+infinity would have to share a room, but I guess everyone accepts this issue because you'd never actually reach rooms n+infinity-1 and n+infinity by counting/visiting. 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...
- Bakkot 14y agoInfinity is not a number. 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.
- Evbn 14y agoNot just counterintuitive, but counterfactual and counterphysical. You can't "do it", you can only "model it in the realm of formal mathematics", which by the way proves that this model is not a model of reality, and infinite hotels cannot exist in physics. If you push the physical analysis further, you can get into the speed of light and bosons.
- nessus42 14y agoYou certainly can't have infinite hotels in physics, but it would be incorrect to assert that we have any good reason to believe that infinite quantities can't exist in nature. For instance, the model of the universe most widely accepted by cosmologists today implies an infinite universe, assuming that our measurements of the curvature of space are correct. And these cosmologists don't believe that these infinities are just a mathematical artifact. Rather, they believe that the universe is likely to be infinitely large, in actuality.
- deleted 14y ago[deleted]
- ajanuary 14y agoI can see how that works if you have an infinite number of rooms and an infinite number of guests, because you still have an unoccupied room to move them up into. But how does that work when you've stated that all rooms are occupied? 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.
- masklinn 14y ago> 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.
- Evbn 14y agoYour intuition is correct. The hotel paradox is a trick of symbolic manipulation, and the analogy to a real hotel is an amusing deceptive misdirection.
- ColinWright 14y ago> Your intuition is correct. No, the intuition is not correct. > The hotel paradox is a trick of symbolic manipulation, > and the analogy to a real hotel is an amusing deceptive > misdirection. No, it's a visualization to help people understand about matchings between infinite sets. It's an accurate analogy.
- jgrahamc 14y agoYou need to go and read up on the transfinite numbers. Swizec is talking about what happens when you deal with a certain sort of infinity (a countable infinity) where there's a one-to-one mapping between the hotel rooms and the numbers. In the transfinite numbers things that at first seem unintuitive happen, e.g.: 1. There are the same number of integers as there are even numbers. And there are the same number of integers as there are odd numbers. 2. There are the same number of rational numbers (fractions) as there are integers. 3. There are more real numbers than integers.
- diminish 14y agoHere are some interesting questions on the Hilbert hotel which boggles my mind. Shortly, If you can keep as many rooms as you wish unoccupied, can you still claim your hotel is full? (1) Example, if you can move everyone from N to N+1 or (N+5); and get an unoccupied room for new reservations, can you still claim anyone that your hotel with infinite rooms is fully occupied? (2) Or is a fully occupied infinite-roomed hotel an oxymoron, or a self-contradictory dasein; which can not exist upfront? (3) Or is it better to call an infinite-roomed hotel both fully occupied and fully available, a superposition of both predicates, or un-predicatable?
- jgrahamc 14y agoI think the mathematical definition of 'the hotel is full' would be 'there is a one-to-one correspondence, also called an isomorphism, between the hotel rooms and occupants'. So, if the hotel has empty rooms, even if there are an infinite number of occupied rooms it is not 'full'. On a side note: no matter how many occupants there are in the hotel one might argue that it was also 'empty' given that it can always accommodate a countably infinite number of new arrivals.
- michaelcampbell 14y agoHonest question here; is the phrase "same number of X and Y" equivalent mathematically to "the cardinality of X and the cardinality of Y are the same"? I was taught "no", since "number of" doesn't apply to any of the infinities. But perhaps I was taught incorrectly, or I have misremembered.
- tokenadult 14y agoThere are many very entertaining discussions of "Hilbert's Hotel" online. I look them up once or twice a year for teaching my math classes. To answer your very pertinent question, I suggest looking at http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/ http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-... for an illustrated, and I think illuminating, discussion of what to do if countably infinite buses, each with countably infinite passengers, drive up to Hilbert's Hotel with infinite rooms, each of which is already full.
- DanBC 14y agoInfinity is one of those things that makes little sense to people who are not mathematicians. Cantor's diagonal argument is something else that I get caught up on. (http://en.wikipedia.org/wiki/Cantors_diagonal_argument http://en.wikipedia.org/wiki/Cantors_diagonal_argument) An explanation of why it's hard: (http://rjlipton.wordpress.com/2010/01/20/are-the-reals-really-uncountable/ http://rjlipton.wordpress.com/2010/01/20/are-the-reals-reall...)