Comments by "eggynack" (@eggynack) on "TED-Ed" channel.

  1. No. After all, my described method just filled it.
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  2. You say that it is not full, but you have no proof for that. I say it is full, and I do have proof for that. That being, if you name any arbitrary room, it will have a person in it. You can definitely "equal" two infinities. There is nothing to the definition of infinity that would give rise to the properties you're citing. If you were to just put people into the hotel one after the other, then, no, the hotel would not fill, but by generating a mapping from the set of people to the set of rooms, and ignoring transit time (say for the sake of argument that each person reaches their room after a second), we find all hotel rooms full. Infinity isn't just a concept. It's a well defined mathematical construct, and nothing of that construct is incompatible with this result.
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  3. What is your precise definition of infinity that stops my proof from working? And, more specifically, how does my proof not work?
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  4. I don't see how that's pertinent. The hotel construct is something of an arbitrary one that could never exist in the real world. That being said, it's not a box, because a box has a top face. The hotel just extends out into the sky forever. In any case, you said that you disagreed with my proof, and had a different definition of infinity. You've yet to define either the basis of the disagreement or this alternate infinity.
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  5. Oh this is definitely math. Hilbert's hotel is more or less a restatement of a basic math result, that all these sets, the naturals, the naturals plus one, the integers, even the rationals, all share a size. And they do so specifically by means of this mapping notion. It's kinda weird to think that the set {1, 2, 3, 4, 5, 6...} has exactly as many numbers as {0, 1, 2, 3, 4, 5, 6...}, but it does. This gets even weirder when you take the extra step and note that the set of all real numbers is strictly larger than any of these. Neat stuff. As for your issue with the scenario in itself, you don't have to keep anyone in transit between the rooms. Just have everyone move simultaneously. At, say, three o' clock, everyone simultaneously leaves their room and then goes into the next room. Every single guest will be inside a room by four and room one will be empty. You note the contradiction between full and not full, but this is actually quite explicable. The hotel is full in one sense, as there is a person inhabiting every room, but not full in another sense, because the hotel can accommodate more guests. As it turns out, these are distinct concepts in an infinite hotel. And if we define "full" the first way, which is a fair way to define it, then the full hotel can absolutely accommodate guests.
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  6. The hotel starts out completely full. Then, by moving infinitely many guests, you render the hotel not completely full.
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  7. I mean, it's not possible in reality because there's no such thing as an infinite hotel or infinite guests. If there were an infinite hotel and infinite guests somehow, this would indeed be possible in reality.
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  8. The person who was initially in room 128 goes to room 2^128.
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  9. Seven.
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  10. There is no end. Cause the hotel is infinite.
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  11. It is possible to fill an infinite hotel if you have infinite people. Simply put, for each room put a person in it. What room is empty?
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  12. There isn't some singular room n+1. Room n+1 is relative to each individual guest. So, for guest 1, room n+1 is 2, and for guest 100, room n+1 is 101. No room starts out unoccupied. However, each destination room is rendered unoccupied for the guest going to it because the person who was once in the room is going to their room n+1. It's worth thinking about what would happen with finite hotels. Say one with 100 rooms. Person one goes to room two, person two goes to room three, and so on. Every single step here works just fine, and it does so until you have to deal with guest 100. They have nowhere to go, and it is here where the movement breaks down. In an infinite hotel, however, there is no top room. The process never breaks down.
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  13.  @jayjeckel  The movement at no point premises on the existence of some initially unoccupied room. Person one goes to room two. This room was, at the beginning, occupied by guest two, but it is rendered unoccupied by the fact that guest two went to room three. Room three was initially occupied by guest three, but they went to room four. Here's yet another way of thinking about it. After the new guest comes in, everyone has exactly one room. Every single solitary guest. If you name a guest, I can tell you exactly what room they're in, and if you name a room, I can name the guest. Thus, the hotel can accommodate the extra guest.
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  14.  @jayjeckel  You reached room six without ever needing there to be an unoccupied room. Where, exactly, do you think this process stops working? I am by no means disputing the definition of "all". Every last room starts out full. It's just that that doesn't stop me from successfully adding guests. Shuffling does create unoccupied rooms. We know this because I can do a shuffling that creates unoccupied rooms.
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  15.  @jayjeckel  No, I can do it because I already did it. It is what is called in math a constructive proof. You prove that a thing can be done by doing it. In this case, the question is whether there is a bijective mapping from all the guests in the full hotel plus one additional guest to the rooms in the full hotel. And there is one. In particular, this mapping is that guest n, where the new guest is guest 0, goes to room n+1. Clean bijective mapping. No new rooms or initially unoccupied rooms needed. It might be easier for you to get a handle on it if we ditch the hotel for something more abstract. There are infinitely many primes, right? It makes sense, then, to talk about the first prime, the second prime, the fifth prime, the 100th prime. In fact, no matter which prime you want, it'll be there. Because there are infinitely many of them. What this means, however, is that there is a one to one mapping from natural numbers to primes. The first prime goes to one, the fifth prime goes to five, the 100th prime goes to 100, and so on. Every natural and every prime is accounted for exactly once each. We can, if we want, extend this reasoning back to the hotel. There's a guest for each natural number, because they're assigned to the natural numbered rooms. We can, therefore, cleanly map the guests in this full hotel to only the prime numbered rooms. This leaves us with infinitely many empty rooms, all the composite numbered ones (which are also infinite), and all just by moving people.
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  16.  @jayjeckel  It obviously does not fail immediately. The new guest successfully enters the first room because the first guest leaves it. The first guest successfully enters the second room because the second guest leaves it. The process has not failed yet and we are already past the beginning.
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  17.  @jayjeckel  You can actually just do all the movement simultaneously, thus causing there to be no one in the halls. Assume we attempt that, having everyone leave their rooms, move to the next room, and then enter their new room. Who is not in a room, exactly? The hotel is full in the sense that every room has a person in it. Nonetheless, in an infinite hotel, this property does not mean that new guests are impossible. I would ask that you respond to the straightforward numerical version. I've found that explanation sometimes more effective.
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  18.  @jayjeckel  The basic issue here is that there actually is not one more person than there are rooms. The infinity of the initial hotel guests is exactly equal to the infinity of the initial guests plus an additional one, which is also exactly equal to the infinity of those infinite buses. That's what this hotel scenario, as well as my prime number scenario, is proving. These infinities all share a cardinality, which means they can be freely mapped back and forth like this. The numerical version I proposed is actually 100% identical to the hotel version. It's just put slightly differently, and I use prime numbers so that I can leverage the way we refer to prime numbers for greater understanding. It's all the same thing. Different approaches are just more or less intuitive for different people. The hotel in particular is sometimes presented in a way that highlights how weird and contradictory the situation appears. It's good for getting people curious about math, but maybe not so good all the time for getting a deeper understanding. But there is nothing logically inconsistent about the hotel version, I promise you that.
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  19.  @jayjeckel  Of course I can map them freely back and forth. Again, I can literally take the guests of the full hotel, move them to only prime numbered rooms, and have infinite vacant rooms. This doesn't require creating rooms, and neither does it require a person shell game. Tell me, if I simultaneously have every guest, call them n, move to the nth prime, then what guest is left roaming the halls? Which guest am I incapable of telling you the location of? The hotel is full in one sense and not full in another sense. It's full in the sense that every room has a person in it. It is not full in the sense that you can still add more people. As it turns out, our general definition of "fullness" is inconsistent when dealing with infinite sets.
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  20. There is no last guest. The hotel is infinite.
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  21. Infinity-infinity is an indeterminate. It can take on infinitely many possible values, including zero, infinity, negative infinity, and everything in between.
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  22. A hotel with infinite rooms can be full when there are infinite guests. This necessitates this sort of infinite guest movement if you aren't just removing a guest.
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  23.  @oliversobhizakhary6169  It doesn't matter that it's infinite. An infinite hotel can be full.
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  24.  @oliversobhizakhary6169  That's a true statement, yes. I'm not really sure what problem is occurring that you think shouldn't be occurring.
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  25. Put one person in each room.
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  26. Infinity minus infinity is not zero. It's an indeterminate. I agree that it's not really a paradox though. More like a straightforward quality of countable infinities.
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  27. There's no such thing as making a new room. There are the rooms there are. Those rooms are infinite, but you don't get extra ones on top of that.
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  28. A fancy light in their room that blinks when they're supposed to move? You wouldn't get simultaneous movement, but it'd be fast enough that there wouldn't be much in the way of issues.
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  29. Simply put, infinity is useful. Basically any type of mathematics you care to name will make heavy use of infinity at some level. Understanding infinity, then, has a lot of functionality to it. This hotel is essentially just a restatement of the various ways we can map the natural numbers to themselves, and that kind of thing can be pretty useful, especially in, say, abstract algebra. Or, hey, maybe this leads you to a question, like whether there is a quantity of guests it would be impossible for the hotel to accommodate, and you wind up at Cantor's diagonal argument, which is a foundational result in mathematics.
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  30. Because you're not specifying which even rooms they're going to to any extent. If you're just assigning people the order, finishing a bus before moving on to the next (which is a natural method of assignment), then every even room will have a person in it from the first bus alone.
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  31. Sure, but you need to know where everyone's going. We know for a fact that it is possible to assign all the buses of guests to rooms. But, under your plan, where do you put, say, the third person on the third bus? You should know the answer to this question.
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  32.  @roodepoo  No, the guest in room 3 goes to room 6, the guest in room 4 goes to room 8, and so on.
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  33. There is no n+1st room that's empty at the beginning. Every room starts out full. In order to make room, you necessarily have to make infinitely many moves of some variety.
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  34. Of course you can have a one to one mapping of guests to hotel rooms. Line up all of the infinite guests and assign them a number equal to their place in line. Guest one goes to room one, guest two goes to room two, guest three goes to room three, and so on. Done. Every guest is assigned to exactly one room and every room is assigned to exactly one guest.
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  35. You can. That would require infinite person moving as well though. For example, have every person move from their room, n, to a new room, 2n. So the person in room 100 moves to room 200. No two people land in the same room, and an infinite amount of rooms are now unfilled. You're just kinda offloading the weirdness to an earlier step.
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  36. An infinite hotel can definitely be full. You just need infinite people. You wouldn't question each room having, say, a bed in it. Why can't each room have a person instead?
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  37.  @luckystar3641  I am asserting that the person also comes with the room. Why not? Mathematical people don't have some special quality that means they can't just be built in.
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  38.  @luckystar3641  He states that people arrive to go into the rooms after the hotel is already full. Nothing wrong with this as a method of starting the hotel out full. Your initial issue was that having all rooms be full is impossible. This resolves that issue. This does indeed lead to an endless cycle of people moving over to the next room. In particular, every single person is, under this system, moving one room over. Guest one can move to room two because guest two is moving to room three, and they can do that because guest three is moving to room four, and so on. It doesn't have to be endless in a time sense though. If you'd like, you can have everyone move one room over simultaneously. Infinitely many people would be moving, but the entire process would take only as long as one person moving would be.
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  39.  @luckystar3641  There is no "very end". The hotel is infinite.
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  40.  @luckystar3641  There would not be. For every single person, we know exactly where their new room is.
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  41.  @luckystar3641  There's always someone in the next room, but that person always has a room to go to, which was occupied until they moved. Look at it from the inverse perspective. If there is a guest being left without a room, where are they? There is a guest for each natural number, and a natural number for each guest, so, if someone isn't getting a room, it should be possible to identify a natural number such that that guest won't get a room.
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  42. There is no empty room at the beginning. They all start out full.
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  43. It is quite possible to have all the rooms be filled. You presumably don't view it as contradictory to say that each room has a bed, or a mini-fridge, or a floor. It is identical to say that each room just so happens to contain a guest.
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  44. There aren't enough rooms at the beginning. You effectively produce open rooms by moving guests.
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  45. No. You just need infinite guests. Instead of filling the rooms one at a time, in order, what you need to do is just say that, for each room, you put a person in it.
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  46. Infinite people.
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  47. There is no unfilled room. They're all full.
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  48.  @JohnPaulBuce  There is no last guest. There are infinite rooms and therefore infinite guests.
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  49. The hotel is infinitely full in the sense that there is a person in every room. You can't put anyone into the hotel without either removing a guest, or moving infinitely many guests.
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  50. Infinity does not mean no beginning or end at all. The set of natural numbers is absolutely infinite. If they are finite, then how many are they?
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