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

  1. Infinity does not mean all. The natural numbers are infinite, for example, but there are still infinitely many negative numbers not inside that set. And infinitely many irrationals contained in neither of those sets. And so on.
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  2. Yes.
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  3. Not necessarily. As long as you have a single group of infinite people (which is inclusive of that infinite set of infinite buses), it's possible to create a single mapping from guests to rooms. You can tell each person exactly which room they are supposed to end up in, and then they can move from their current room to that room. This is sufficient to accommodate all of those guests. Of course, the distance a guest has to travel may be ridiculously long, but at least the guests don't have to move more than once. Until, y'know, another set of buses comes.
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  4.  @francescobettin4161  You misunderstand. That entire infinite quantity of infinite buses counts as a "group" here. As long as you know what guests you're taking in, even if it's an infinity of infinite buses, you can still get everyone to their rooms in a single movement. The only way to require multiple movements is by having a new group come after people have already moved. Consider, the very structure of the solution is that everyone is assigned a room based on their bus and seat number (and that hotel guests are given assignments as well). All you need to do is put everyone in that assigned room and you're done.
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  5. Why not? Infinity isn't necessarily all inclusive. The integers are infinite but do not include pi, for example.
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  6. Each guest is only able to move to the next highest room because the person in that guest is moving too.
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  7. There is no last guest. There are infinite rooms.
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  8. The hotel starts out full, so you cannot put a guest into any room. However, you are still able to add a person by moving everyone up a room, which is possible specifically because there is no last room, because the hotel is infinite. There are no guests occupying the last room, not because there's some empty room, but because that room is not a thing.
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  9. What room are you shifting the guest into here? The only room that's open is the one made available through the infinite movement.
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  10. All the rooms are full at the beginning. Thus, for empty rooms to be brought into existence, you've gotta move people around a bunch.
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  11. You can always add people, as the video showed, but it's also possible to have each room filled with a person. Simply line up all of the people, and assign each person the number that matches their position in the line. Then, put person one in room one, person two in room two, and so on. Now, for any room you name, there exists a person in that room, meaning every room has a person in it. It's also possible to give each person infinite rooms, or to give each room infinite people, or leave infinite rooms empty, and any arbitrary lesser version of those. Here's an interesting way to disprove your claim without using an example method. Let's assume that, when assigning people to rooms, there will always be rooms left. Now, let's try assigning rooms to people. The two are equivalent, with this situation able to be visualized as dropping some hotel room onto each person in turn (preferably without a floor, so they survive the experience). By your logic, it should be impossible to do this, meaning there are far more than enough people to fill every room.
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  12. Everyone's room assignment is intrinsic to their bus and seat numbers. If each passenger knows those pieces of information, then they can just go to the correct rooms without needing to be individually told.
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  13. It's possible to fill an infinite hotel. Simply line up the guests and put guest one in room one, guest two in room two, guest three in room three, and so on. Every room will have a hotel in it.
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  14. All the rooms start out full.
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  15. I'm not sure what precisely you mean by "dense" in this context. The two sets have the exact same quantity of numbers though.
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  16.  @andrea1854  That's not really a definition of dense I've seen. More common is to say something like, "The rationals are dense in the real numbers." This means that any two reals have a rational between them. Density is generally considered in this sense, with a subset considered dense in the set. Your construction of density is somewhat troublesome. What if the interval is [0,∞]? In this case, there are infinite naturals, infinite evens, and infinite rationals. The same infinity each time as well. However, any smaller interval, say [2,3], will contain finite integers and infinite rationals. Moreover, we can identify some lower key versions of this situation. The odds are going to be more numerous than the primes over most intervals, but if we're working with something like [2,4], then there are more primes than odds, and the interval [2,9] has the same number of each. There's probably some way to formalize this idea of density, but it's pretty tricky to do.
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  17. Feh. If you just have all the people go to their rooms simultaneously, then the process takes finite time regardless of how many people you need to move in.
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  18. Even if the hotel has infinite guests, that doesn't necessarily mean all the rooms are full. For example, if there is a guest in every even numbered room, then you have infinite guests, but all the odd rooms are empty.
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  19. Because it's infinite. There are no topmost floors.
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  20. Not so. We can assume, for the sake of argument, that every guest takes this moving one room over action at exactly the same time. In this fashion it is possible to get everyone inside their room, such that everyone in the hotel, including the new guest, is satisfied.
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  21. Well, it's hard to say that an infinitely large bus will ever be emptied, but we can say that, for any guest on the bus, it will leave the bus after finite time. This is complicated by multiple buses, because a straightforward emptying pattern would never finish the first bus to move on to the second bus, but this issue is resolvable by using some different procedure that hits any given bus any arbitrarily large quantity of times after finite time.
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  22. If we're working with countable infinities, which we almost certainly are? No. A has as many citizens as A, B, and C combined. So do B and C.
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  23. +Mr. Meeseeks You are mistaken (as are a number of people whose replies I was not notified of). You can stick infinite guests in the hotel such that the hotel never fills, sure, but it is also possible to fill the hotel. You have infinite guests, right? Line them up, and give each a label equal to their position in line. Now, put each guest in the room that shares their label. I ask you, if the hotel is not full, name a single room that is empty.
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  24. +Mr. Meeseeks I provided a means by which the hotel can be filled. Thus, I am asking you to either agree that the hotel is filled by this method, or identify a single room of the hotel that is not filled.
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  25. It does not imply a final number in the sequence. It only implies that one infinite set can be mapped to another. Again, if the hotel can not be filled, then my stated method should not be able to fill it. If my method cannot fill the hotel, then there must be at least one empty room. If there is such an empty room, then it should be possible to identify which room is empty. So, do so.
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  26. I understand infinity quite well. From, y'know, studying mathematics a lot. It is entirely possible to fill an infinite hotel. All you need to do is generate a bijective mapping from the set of guests to the set of rooms. Consider a more abstract example. You have two sets, the set of natural numbers, and the exact same set of natural numbers. Can you pair elements of the first set with elements of the second set such that no elements of either set are left over? Of course. Just pair each element in the first set with the same element in the second set. In order to extend this situation to the hotel and guests, you need simply call the first set the guests and call the second set the hotel rooms. By this means, we have our mapping. Every guest has an exact room assignment, and conversely, every room has an exact guest assignment. Room one has guest one, room two has guest two, and so on. This is how countability functions. If you have two countably infinite sets, then, by definition, one can be bijectively mapped onto the other. Both the hotel rooms and the guests, even when the guests are on infinite buses, are countable infinities. Take note, this is not the only possible mapping. It is very much possible to generate a mapping where exactly one room is empty, or where infinite rooms are empty, or where any natural number of rooms is empty, or where every room has infinite guests, or where every guest has infinite rooms. I can provide any of these mappings for you, should you so desire. Their existence is far from theoretical.
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  27. There is no last person. There are infinite people.
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  28.  @94Central  What previous person? The core initial assumption of the hotel is that it starts out full. You don't have to fill the hotel if you don't want to, but you also very much can.
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  29.  @94Central  Jeez, no need to yell. Stick all the infinite guests in a line and number them. You have guest one, guest two, guest three, and so on. Yeah? Now, put each guest in the room that shares their number. What room is empty?
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  30. If there is one room for each natural number, then there are infinitely many rooms.
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  31. There is no "last occupied room". All the rooms start out full. Room 2010039494990101 does have a person inside it at the beginning. The only reason guest two can go to that room is because the person in that room is going to an even higher room.
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  32.  @skyabyss7357  I mean, yeah, if you have a mapping to a set of rooms, and then there are more rooms, then the mapping will no longer work. As long as you're in the same cardinality of infinity though (like, if you add on some finite number of rooms, or add on as many rooms as there are natural numbers), then you can create a new mapping between the original quantity of guests and the new quantity of rooms. Really, you're just asking the original question in reverse. By which I mean you're just asking the original question. You start with n rooms and n guests, you add a guest, and then the n+1 guests are still accommodated. This is the same as starting with n rooms and n guests, adding a room, and then having the n+1 rooms still wind up full, which is the scenario you propose, I think.
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  33. For every room, there is a person in it.
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  34. There is no last person, so there isn't a room that the last person was going to. Even in an infinite hotel, it's possible to fill every room.
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  35. Every person moves up one room, leaving room one empty. The reason it's possible is that every single room is being vacated by the person who was originally in it. Room two is empty because the person in it moved to room three. Room three is empty because the person in it moved to room four. A critical point here is that infinite guests and infinite rooms absolutely does not necessarily lead to a full hotel. If there's a guest in every even numbered room and no odd numbered rooms, then you have infinite guests but infinite empty hotel rooms.
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  36. Every room is full at the beginning. There isn't some singular n+1 room that people are being moved to. N+1 refers to the room one higher for each individual guest. So, the first guest moves into room two, for example.
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  37. It's really not undefined. Doing it division style causes problems, but there's absolutely no problem with putting the people in rooms bijection style.
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  38. Countable infinities are infinities, and the infinity of a giant pile of discrete hotel rooms is a countable one. If the hotel had an uncountable infinity of hotel rooms somehow, then it could only be occupied by uncountably many people, but it could be occupied by uncountably many people. The hotel cannot be full in the sense that people can always be added, but the hotel absolutely can be full in the sense that every room has a person in it. As in, if you name a room then I can tell you both that it is full and who is in it. You say there's no mathematical solution to filling the hotel, but there absolutely is one. As I said, you simply assign each person a natural number, and have them all simultaneously go to the room that matches the number they were assigned. There is no flaw in this construction, and no way that there would remain an unfilled room afterwards. If you think I'm mistaken, please either identify a flaw in the construction, or a room that is unfilled.
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  39. An infinite hotel can be filled by infinite people.
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  40. N+40 is correct. The person that starts in room 1 goes to room 41, the first new guest goes to room 1, the 40th goes to room 40, and broadly we have a unique room for each guest and a unique guest for each room.
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  41. You don't have to finish one bus before starting the others. It isn't, as you note, a super effective strategy, and neither is taking one person from each bus in turn because then you'll never return to the first bus. Instead, you want to use something like the abacaba pattern. So, you label the buses alphabetically (moving onto aa and such when you hit z), and then remove the next passenger when you hit the bus with that label. For the order in which you select the buses, you can use abacabadabacabaeaba... In this fashion, you hit every bus infinitely many times, and across the same sort of time span as you would a single infinite bus. Alternatively, you can use a system like what you use to prove that the natural numbers share a cardinality with the rationals. Label each bus with a number, and each passenger in each bus with a number. Then, set out the natural numbers along an x and y axis, generating a grid where the x axis defines a bus and the y axis a passenger. Finally, you draw a zig zag line that crosses over every number pair, and pick out bus/passenger pairs in an order defined by that line.
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  42. There is no last room either. The hotel is infinite.
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  43. Yes, it can. Simply put, for each room put a person into it. What room is empty? If no room is empty, then every room is full, meaning the hotel is full.
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  44. Person one can move to room two because the person in room two is moving to room three. The person in room two can move to three because the person there is moving to room four. And so on. There is never a point at which a person cannot move to the next highest room.
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  45. There isn't a last floor. There are infinite floors.
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  46. Infinity is a pretty straightforward and practical concept, and it shows up everywhere in math. I mean, jeez, even straightforward division has infinity. How many 3's do you think are in the decimal expansion of 1/3? Infinity: proved.
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  47.  @asagk  Of course I can quantify that assumption. Consider the natural numbers, say. Is there some absurd quantity of them that we just haven't reached? Let's assume there is, for the sake of argument, and assume this quantity is n. Thus, the set is 1, 2, 3, 4... n. Well, n+1 is also a natural number. Thus, there are at least n+1 natural numbers. This means there is a contradiction in our initial assumption that there is some finite quantity of natural numbers, meaning the set must be infinite. As for countability, you're just misunderstanding what that term means. Simply put, if set is countable it means that we can construct some list such that, given any element in the set, we can reach that element by counting along the list. Returning to the natural numbers, I already laid out a basic list of them. 1, 2, 3, 4, 5... Name any natural number and it will show up at some finite position on that list.
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  48. It works fine. In the base scenario, we are specifically and only trying to generate a vacancy in room one. Room one is made vacant by the fact that guest one moved to room two. This would be impossible, but room two is made vacant by the fact that guest two moved to room three. And so on. Every guest can successfully move up a room because the person originally in it has moved.
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  49. It is quite possible to fill all the rooms. Line up all the people and give each the number equal to their position in line. Then put each person in the room whose number matches their own. No room will be empty.
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  50.  @ihold0266  I'm not sure what you mean in saying that no room can be filled. We can obviously fill room one. Just stick a person into it. And we can fill all the rooms, by the method I stated. Infinity doesn't end, but that's also true of the infinity of guests.
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