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

  1.  @hjl4204  What I'm saying is that, while the video may have mentioned the notion of infinity+1, it was not in reference to any room number. Just saying "infinity+1" does not magically mean it applies in all circumstances.
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  2.  @hjl4204  There isn't a last room, there isn't an empty room, and there is just broadly nothing of the sort you are talking about. There is one room associated with each natural number, and they are all full. If you'd like, they are each full with a guest associated with the same natural number.
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  3.  @hjl4204  You keep talking about this guy at the back. There is no such person. It's an infinite hotel. There is no last person, ever, for any reason. Thus, moving people to the next room never breaks down.
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  4. There are no unused rooms at the beginning. The hotel starts out full of infinite people.
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  5. Not only is it possible to do calculations with infinity. It is frequently easier than doing so with finite quantities. Calculus, for example, uses infinity all over the place, and physics uses calculus all over the place. This in spite of the fact that objects in reality move incredibly small discrete distances rather than the smooth continuous movement implied by continuous spaces. It's way way way easier to just take a derivative than it is to talk about objects moving insane amounts of Planck lengths in each period of time. This video isn't finding some hole in our capacity to perform calculations with infinity. It is doing a calculation with infinity. And mathematicians do those calculations all the time. Just about every field of mathematics you study is going to have infinity. If not in direct form, because you're dealing with infinite things, then indirectly, because it's useful for proving stuff. And, because it's math, you can trust that all of it is fully rational.
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  6. There is no last person. That's what there being infinite people/rooms means.
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  7. While it's possible to put guests into the room such that the hotel never fills, it is also possible to put guests into the hotel such that it does fill. The premise of the scenario is that the hotel is, indeed, full. Maybe the dude gets a bonus if he keeps the hotel full.
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  8. It is impossible to fill the hotel in the sense that you cannot, by some means, accommodate more guests without ditching some. It is very possible to fill the hotel in the sense that every room has a person in it.
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  9. I'm not sure what you mean by that, precisely. Infinity definitionally doesn't have an upper bound, but some infinite sets are the same size as other infinite sets, and in those cases you can map every element from one set to every element from the other on a one to one basis. It is through this sort of mapping that it is possible to fill every room with infinite guests.
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  10. There is no "last available room". There are infinitely many rooms.
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  11. Infinity minus infinity is an indeterminate, so the step where you did that subtraction is a mistake. Infinite does, in fact, equal infinity +1.
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  12. I dunno if I'd really call it a paradox in the first place. To me, it seems more like an intuitive way of getting at the wide variety of mappings from the set of natural numbers to itself.
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  13. Having a set of infinite rooms doesn't mean you get to add new ones. The set of natural numbers, to use an example, is an infinite set, but it's impossible to add new elements to the set without changing what the set is.
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  14.  @liamc3433  Infinite rooms can absolutely all be filled, and statically so. Line up your infinite guests and label them with their position in line. Guest one goes to room one, guest two goes to room two, guest three goes to room three, and so on. Every single room has exactly one unchanging guest, and every single guest is assigned to exactly one unchanging room. There are no gaps anywhere.
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  15.  @liamc3433  There are no more rooms to fill. I filled all of them. If there exists an empty room, even one of them, then name it.
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  16.  @liamc3433  Yes, there can be no more rooms to fill. I just did it. I created a mapping such that there are no empty rooms anywhere.
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  17.  @liamc3433  If you don't have infinite people then the infinite hotel will never be filled. Primarily because finite things are smaller than infinite ones. If you do have infinite people, then the infinite hotel can absolutely be filled. We know this to be the case conclusively because we can just kinda do it. Doing something is absolute proof that it is possible to do that thing.
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  18.  @liamc3433  No, it can be filled. I just explained exactly how it can be filled fully. I'm not really sure why you keep saying otherwise without identifying flaws in my methodology.
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  19.  @liamc3433  Or you could just fill them all simultaneously. Or they could start out full. Or you could have each room filling take half the time of the previous one, causing the overall hotel filling to converge to a finite length of time.
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  20.  @liamc3433  You said it was impossible to have the hotel be full in the first place. This means that the scenario where the hotel just starts out full is a viable counterexample. That said, regarding the bus, the other two methods work just fine. You could teleport all the guests to their rooms simultaneously, or have each guest go faster than the last. This is a mathematical question, not a physics question, so it's perfectly fine to have the guests move as fast as you'd like. As an alternate solution, we could just have all the guests leave the bus at the same time and each take the same amount of time to get to their room. Say it takes any guest five minutes to reach their room. If we have them all go towards their room at once, then the whole process would take five minutes.
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  21.  @liamc3433  You asked why we can't "make a new room for each new guest". The answer is that the hotel started full, and the situation where it just starts full is a valid solution to that. You didn't actually explain why going faster doesn't work. Or teleportation. These are math people, not real people. We can do whatever we want with them. They are not constrained by physical laws, which should be obvious given that there are infinitely many of them. In any case, I have to think the, "They just all move simultaneously" solution would be more satisfying to you. So, do you take issue with that method?
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  22. Infinity does not mean everything. The positive numbers are infinite in quantity, and yet there are infinitely many negative numbers right there that aren't in the set.
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  23. Simply put, for each room have a person be in it. This makes the hotel be full. And, if the hotel if full, then there is no room, at that moment, for an additional guest.
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  24.  @tonyr8371  Imagine lining up all the guests and giving each one a numerical label equal to their place in line. You put guest one in room one, guest two in room two, and so on. You say there will always be empty rooms. Which room is empty?
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  25.  @tonyr8371  Infinity doesn't magically generate rooms. The natural numbers are infinite, and every element of that set is always an element of that set, and always was. The same is the case for the hotel, particularly because the rooms are indexed by the natural numbers. If every natural number is accounted for with a guest, then all the rooms are full. Where would other numbers come from? You don't get to just magically add on negative numbers or something. Basically, you're using a false definition of infinity here. The only requirement is that the rooms have no upper bound, not that new ones sprout into being. The hotel need not have a time element.
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  26.  @tonyr8371  Finite doesn't mean fixed. You're just kinda putting those words together, but they don't have to be. There is also no continuous expansion necessary in infinity. It can all just be there at once. Moreover, infinity is basically always limited. Even the set of all real numbers is limited to, y'know, real numbers. You don't have imaginary numbers in there. You're making up this idea that infinity definitionally cannot be full. There's a ton of study in math into how various infinite sets "fill" other infinite sets (usually called mapping, but still). Again, the only requirement of an infinite set is that it not be finite. The set of natural numbers is not finite. A hotel with no highest floor, even if those floors are all there at once, is not finite.
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  27.  @tonyr8371  This is more straightforward than either of us is making it out to be. What's your definition of infinity?
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  28.  @tonyr8371  The actual definition of infinity is pretty straightforward. A finite set is one that contains a number of elements equivalent to some natural number. The set {1,2,5,6}, for example, is finite because it contains four elements, and four is a natural number. An infinite set is one that is not finite, so therefore one for whom no such natural number can be assigned. Note that nothing in this definition demands that the set be continually growing, or that it grow at all. Nothing of it demands that it never be "filled", or that it contain all possible elements. If a set is not finite, then it is infinite.
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  29.  @tonyr8371  "Tell me, is Infinity of the realm of math?" Yes, and pretty straightforwardly so. "Observe two sets of rooms? An infinite set of empty rooms? And an infinite sets of full rooms? Are those two mutually exclusive?" I'm not sure what you mean here. An infinite set of rooms cannot be simultaneously completely empty and completely full. Two sets of infinite rooms can exist side by side, one full and one empty. "You keep making the mistake of applying one definition and having that as the universal truth of Infinity, when that is a fallacy." No, that is the mistake you are making. If there exists a definition of infinity that does not require continuous expansion, then that could straightforwardly be the definition the video and I are using.
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  30.  @tonyr8371  I am wholly right. If infinity is inclusive of things that do not continually expand and retract, then this hotel can be infinite, not continually expand and retract, and therefore be full. You said that the hotel could not be full. It can be.
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  31.  @tonyr8371  How could a hotel simultaneously have its rooms all be full and all be empty? How do you get to that point? Honestly, even if we assume this bizarre room generation model, it's not hard to get all rooms full. What if a guest is intrinsic to all generated rooms? We assume, on a basic level, that all rooms have walls, and a door, and maybe even a bed. We can assume they also have a person. Done and done.
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  32.  @tonyr8371  Oh, sure, infinite full rooms and infinite empty rooms aren't mutually exclusive. I thought you meant that the hotel as a whole being full and the hotel as a whole being empty are not mutually exclusive. In order for the infinite hotel to be full, every room must have a person in it. Rooms on the left being empty makes the hotel not full.
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  33. The hotel is, in fact, truly full. By which I mean that every room has a person within it. There is thus no later room to put people into. The way this full hotel receives another guest is by the method stated in the video, shifting everyone up a room. Every single guest in the hotel has a room to go to in such a way that each room is filled only once. A useful perspective on this is that the set of all rooms can be mapped one to one with the set of all rooms minus the first room. Or even to only the even rooms, or only the prime numbered rooms.
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  34. First, line up all of the infinitely many guests. Then label each with their place in line. Finally, put each guest in the room that matches their number. So, guest one goes into room one, guest two goes into room two, and so on. There is no room left empty by this method. Alternatively, we accept without question that each room can have a bed, or a bathroom, or even walls. There is no difference between that and accepting that each room just happens to have a guest inside of it.
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  35. The short version is, for every room, there is a person inside. That's the definition of fullness at work here, and it's pretty sensical, I think. To flesh this out a bit, a medium version if you will, I like to consider the nature of the room. Think about one of these rooms. You wouldn't think it wild if I were to say that every single one has a bed inside. A bath. You would be especially unsurprised if I were to say that every room has walls, a floor, a door. This stuff is just to be expected. Why, then, is it odd to say that every room contains a person? Maybe I even do a straight replacement. Where once there was a bed, now there is a guest. It's not like guests are magic. For the marginally long and somewhat more rigorous version, you can just do the mapping straight up. There are infinite guests, yeah? So we line them up, assign each a number based on their place in line. So there's guest one, guest two, guest three, guest four, and so on. Now, we send guest one to room one, guest two to room two, guest three to room three, and so on. This definitely fills the hotel. And provably so. Name any room and I can tell you not only that the room is full, but also exactly which guest is inside. Every room has a guest, the guest with the matching number, and so the hotel is full.
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  36. Infinity-infinity is an indeterminate. That means it can have just about any result. That includes zero, but it also includes infinity, negative infinity, and everything in between.
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  37. How would you go about proving that infinity+1 is equal to infinity? There is no empty room at the beginning. It's possible to produce one by moving the guests. And the video definitely doesn't use incompatible definitions of infinity.
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  38.  @24four18  The definition of infinity is not that infinity+1 is equal to infinity. If you can see different definitions being used, please point it out.
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  39. There is no available room up there. They all start out full.
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  40. The above rooms are being rendered empty by the fact that the guests in those rooms are also moving up.
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  41. There are no unoccupied rooms.
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  42. There is no empty room n+1, because every room is full. Filling the hotel in the first place is pretty straightforward. For every room, put a person in it. Done.
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  43. You're not just moving the new guest to a certain room. You're moving the new guest to a room, moving the guest that was in that room to a different room, moving the guest that was in that room to a different room, and so on. Because the hotel is infinite, this process need never end.
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  44.  @mb2000  Because there's a guest in every room. I assume you wouldn't take issue with each room having a TV or a bed or whatever. Well, instead of a TV, each room has a guest.
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  45. The big question you have to ask of any room assignment is where any given guest will wind up. With the mentioned prime number method, I can, given any guest, tell you exactly which room they're in. Using your method, repeatedly shoving people up rooms, where does even the first person in the first bus land?
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  46. It's a number that has exactly two factors, one and itself. So, like, 6 has four factors, 1, 2, 3, and 6, so it's composite. 2 has two factors, 1 and 2, so it's prime. The primes are like 2, 3, 5, 7, 11, 13, 17 and so on. There're provably infinitely many of them.
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  47. There is no top. The hotel is infinite.
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  48. There isn't room at the beginning because all of the rooms start full. Moving the guests is what empties a room.
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  49. Any prime number^1 equals that prime number. Numbers raised to 0 equal 1.
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  50. You don't have to assign even and odd style though. Future distributions can simply be modifications of existing distributions, using the initial hotel guests as the first "bus", without any issues.
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