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

  1.  @siddharthavlash1982  That's also wrong. Mathematicians use infinite sets in calculations all the time. This video is using an infinite set in a calculation, just by describing a bunch of mappings between infinite sets.
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  2.  @randomman6428  I don't think that's even physically possible.
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  3. There is no last room. There are infinite rooms.
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  4. There is no end of the list. The rooms are infinite. There are different ways to do this, but none that don't involve moving infinite people around.
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  5. Every room has a guest in it, so if you pick a random room and try to stick the guest into it then there will already be a guest there.
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  6. I guess you could, though doing a task actually infinitely, instead of doing it some arbitrarily large quantity of times, is tricky.
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  7. The hotel is full in the sense that every room has a person inside. It is not full in the sense that adding more people is possible.
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  8. Yes.
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  9. That's exactly what he did.
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  10. So, you move people around, and create empty space. Such things are possible in infinite hotels.
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  11. At first, yes. Then you render it not full. Fullness isn't some kind of crazy perpetual state. Even a finite hotel can create vacancies by just getting rid of guests. We're not getting rid of guests here, but it's still possible. Really, this whole thing is just a restatement of a basic mathematical fact. That being, you can create a one to one mapping between the natural numbers and the natural numbers with the number one removed.
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  12. The word full means the same thing in math that it does in non-math. Specifically, it means that for every hotel room, there is a person inside. Or, alternately, it means that no room is vacant. In a finite hotel, this would make it impossible to put new people in without taking some out. However, in an infinite hotel, it is possible to move every single guest up a room, thus emptying the first room. The reason this doesn't work in a finite hotel is because the last guest will have nowhere to go, but there's no such thing as a last guest in an infinite hotel.
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  13. Again, you can't just subtract infinity from infinity. This subtraction has an infinity of results, and thus no singular result. The guests can indeed be paired up one to one with the hotel rooms. There is the same quantity of each. However, it is 100% provably the case, and I literally just proved it was the case in that post, that the guests can be paired up with hotel rooms such that there is one hotel room left over. There are different sizes of infinity, yes. However, any two countably infinite sets, for example the set of natural numbers and the set of natural numbers plus or minus some number of elements, will be exactly the same size every time. They are different sets, but one is in no way bigger than the other.
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  14. We do not know that the result will be 0. We know only that the result can be zero. Again, the result can be any number of things, zero included. As for your other statement, set B isn't just set A with all the elements added to. Set B is set A with the element zero added in. In other words, you had infinite objects, and then you added another object. This situation is identical to the hotel scenario. You had infinite guests, and then you add a guest, and you have the same number of guests. In fact, we could just label the new guest 0, and all the old guests as natural numbers, and the proof I presented would show that the number of guests hasn't changed. Infinity doesn't always equal infinity, but every countable infinity, definitionally, does equal every other countable infinity. I can prove this for any two countable sets you care to name, but, given that the definition of countable infinity is the existence of a bijection to the natural numbers, it feels unnecessary.
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  15. No, we don't know the result will be 0, because we can take the exact same number of guests and not fill the hotel. We can fill the hotel, but we don't have to. Not sure what you're missing here. 0 is one possible result that is comparable to not filling the hotel, but putting a guest into only every odd room is an also possible result with the exact same guests that gives the result that infinity minus infinity is equal to infinity. As for the two sets, I did not define them as equal at the start. I started with set A, added an element to get B, and proved from that point that the two sets have the same size. Adding an element provably did not change the size of the set. That was the entire point. You're acting like I'm just assuming these things. I'm not. I'm proving them right here in front of you.
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  16. I'm not sure what precisely I haven't explained here. Infinity plus or minus any finite quantity is the exact same infinity. There are different sizes of infinity, but those sizes of infinity are inevitably, y'know, infinitely different from each other size-wise. Infinity minus infinity is an indeterminate. This means it doesn't have any one particular value. It's akin to 0/0 in this way. That you can fill an infinite hotel with infinite guests does not make infinity minus infinity zero, because you can also fail to fill an infinite hotel with infinite guests. The exact same infinity of guests.
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  17. Okay, start with the hotel full of guests. Guest in every single room. We can take all the guests out of all the rooms, and then put every single one of them back into only odd rooms without leaving any guests over. I'ma change it to even rooms for convenience's sake. Label each guest with the room they're in. Now, take those guests and put them in room 2n, where n is their label. So, guest one goes to room two, guest two goes to room four, room three goes to room six, and so on. Exact same rooms, exact same guests, but now our "subtraction" yields infinity. As for the two sets, it's really pretty straightforward. Set A is the integers from one upwards. Set B is the integers from zero upwards. Set B thus has one more element than A does. Now, we will pair each element of set A with each element of set B. One from set A pairs to zero from set B, two from set A pairs to one from set B, and, in general, n from set A pairs to n-1 from set B. Thus, every single element of A is mapped to exactly one element of B. So, the sets are the same size.
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  18. Infinity isn't switching size. infinity=infinity/2=infinity-1. The reason using algebra doesn't work here is because changing a thing into x doesn't magically change its functioning. Infinity has a set of properties. Finite numbers have a different set of properties. You find those properties somewhat unintuitive, but what I've been presenting is a set of totally accurate proofs. Skip x entirely. If you want to say infinity, say infinity, because infinity is the thing you're working with. In using x, you're trying to generalize something that can't be generalized, and then making it specific to a set it doesn't work for. Straightforwardly, by saying that x/2=x, you're saying something that hasn't been proved, because you only know this to be the case for infinity. X is a thing you solve for, not a thing you put stuff into such that it's true. If x/2=x, then x is 0, or infinity. The reason this doesn't work is the exact reason it doesn't work to say that x/2=4, and then conclude that, because we can replace x with anything, 16/2=4. As for the sets, yes, there is an element in B that is not in A. That's the whole point. In spite of this, it is provably, and I've proved it for you, the case that the sets are the same size. The natural numbers are also the same size as the integers, and the rationals, and the primes. These are different sets, some proper subsets of others, but they are all the exact same size.
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  19. That's the amount of digits that exist, not the amount of numbers that exist. And, even if we are talking about digits, your result is rather arbitrary. In binary there are only two digits, 0 and 1. In hexadecimal there are 16 digits. And, while some bases obviously aren't used all that frequently, or necessarily at all, there's no upper bound to the quantity of digits that can exist. Thus, the answer to your question is still infinitely many.
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  20. It can if you have infinite people.
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  21. Sure, there's a way to just have everyone take up rooms in a clean way that doesn't require anyone moving. But that's boring. Also, I dunno, maybe the hotel is full when you get there, forcing the matter.
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  22. There is no single room n+1. N is the room you're in, so going to room n+1 means moving one room up.
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  23. Infinite and all are different things. The hotel actually contains infinitely many completely disjoint sets of guests that are themselves each infinite. Which means you could have infinitely many guests leave the hotel, leaving infinitely many guests inside the hotel, and do that infinitely many times.
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  24.  @barryisland5942  Eh, it's not a big deal. Lotta ways to do it. There's the magic way, where everyone teleports, the math way, where you have each person move twice as fast as the last person to leave, the boring version, where everyone simultaneously leaves their room out of some back hallways so they travel forever but don't get in anyone's way, or the tragic version, where everyone simultaneously leaps out of a window. Whatever.
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  25. Simply put a person into every room.
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  26. Yeah, I'm not a fan of the mapping for that reason. It's entirely possible to map the infinite buses to the rooms without leaving any rooms empty. For example, have the first guest stay in room one, have the other initial guests go to the even rooms, and then have bus m seat n go to the nth number where the m+1th prime is the least prime factor. So, bus one goes to numbers divisible by three that aren't even, bus two goes to numbers divisible by five that aren't divisible by two or three, and so on. Every room is hit exactly once.
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  27.  @anonygent  My mapping was very explicit about the fact that the initial guests go to the even numbered rooms (plus the first room). They are accounted for.
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  28.  @anonygent  No, no one is taking up the odd numbered rooms. The situation is that the hotel is full, then the buses arrive, and then the initial guests are moved to the even rooms, and then the buses are moved to the other rooms. Based on my stated mapping. This weird middle step where guests take up the odd rooms is arbitrary and irrelevant. If it occurs then the hotel is full and then you can just run my mapping from the beginning. Moving people to p^n is in no regard a more permanent solution than mine. After you're done, and new infinite buses arrive, you still have to move guests around to accommodate the new guests. It's not all that much harder to calculate either. You just have to sort by size the numbers with the stated quality. So you have, say, 11, and 11^2, and 11*13, and 11^2*13 and so on.
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  29.  @anonygent  No, no one is taking up the odd numbered rooms. The situation is that the hotel is full, then the buses arrive, and then the initial guests are moved to the even rooms, and then the buses are moved to the other rooms. Based on my stated mapping. This weird middle step where guests take up the odd rooms is arbitrary and irrelevant. If it occurs then the hotel is full and then you can just run my mapping from the beginning. Moving people to p^n is in no regard a more permanent solution than mine. After you're done, and new infinite buses arrive, you still have to move guests around to accommodate the new guests. It's not all that much harder to calculate either. You just have to sort by size the numbers with the stated quality. So you have, say, 11, and 11^2, and 11*13, and 11^2*13 and so on.
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  30. Full just means that every room has a person in it. This is possible in a finite hotel. The simplest method, of course, being that for every room you put a person in it.
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  31. Tas Family Law Pathways  None of those definitions require the thing that's full be finite. Apart from perhaps the first, every one of those definitions is met. The hotel has no empty space (insofar as a room with a guest is generally considered full), no room is lacking a guest or has a guest omitted, and there is a completeness to the rooms that are full. The only sense in which the hotel isn't full is that you can add more guests. And even then, the hotel starts out with infinite guests, and, after adding guests, you have the exact same sized infinity of guests, so the hotel kinda does hold as many guests as is possible. Which is weird, but that's why it's called a paradox.
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  32. That's not really true, actually. If you were to fill all but the first room, there would be infinite guests and an unfull hotel. In fact, I'm pretty sure there's an uncountably infinite number of hotel fill patterns.
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  33. It holds tons of value in the logical world. Infinity shows up all the time in mathematics, including mathematics used in real world contexts. Calculus is all about infinity, and that's used basically everywhere. Hilbert's hotel is illustrative of some foundational facts about infinity.
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  34. There is no very top, because the hotel is infinite. Filling the hotel is pretty simple. The most basic expression of this is that, for each room, you can have a person in it. It's very possible to render that process more rigorous though.
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  35. Infinite rooms can definitely be filled if you have infinite people. Consider the fact that you wouldn't question each room having, say, a bed. Well, instead of a bed, each room just happens to come with a fancy guest. Every room thus has a person in it, so the hotel is full.
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  36. All the buses have the same amount of guests as one bus, but you still need to determine where any given person goes. You say you're just sending all the bus folk to odd numbered rooms. Where do you put the fifth person from the third bus? You can't associate people in the first bus with odd rooms in sequence, because then the second bus would have no odds left. Similarly, you can't associate the first person in each bus with the odd rooms in sequence, because then no one else on each bus would get a room. Personally, I'm not a big fan of the method in the video. The rest of the video makes a big deal of keeping the hotel fully occupied, and then they ditch that for the infinite buses. My preferred mapping is that people in bus n go to rooms where the n+1st prime is the smallest prime factor, and the hotel folk go to even numbers. Or, the first guest goes to room one, and then they fill up evens, so every room gets filled. So, the fifth bus would have its guests go to rooms that are divisible by 13, but not by 2, 3, 5, 7, or 11. This method should assign every guest to a room and every room to a guest.
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  37. Not particularly. Consider that you wouldn't think it particularly paradoxical if every room had a bed, a bathroom, walls. These are just normal facets of the room. Well, there's no real difference if you replace that bathroom with a person, and if you do that then the hotel is full.
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  38. It can if you have infinite people.
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  39. There isn't some room n+1 that's empty. All the rooms start out full, and a room can only be made available through some infinite set of moves.
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  40. Eh, not really an issue. You might want to precisely define how addition to infinity operates, but that's doable. Say you start with a set containing every integer. This is an object we can reasonably talk about. Now, say we take the union of that set and a set containing 1/2. This can be considered adding a single element to the set. The first set has infinitely many elements, so it makes sense to say that the second set has infinity plus one elements. The two sets share a cardinality, so we can conclude that infinity and infinity plus one are equal.
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  41. Infinite people, one person in each room.
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  42. We also don't have any infinite hotels. They seem like equally strange premises to me.
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  43. There is no next open room. All the rooms start out occupied.
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  44. A lot of math involves the infinite in one way or another, and a lot of important things involve math. Understanding the ways the infinite functions can be helpful for such things.
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  45. If you're not doing something that involves high level math on at least some level, then you're unlikely to require this kinda knowledge. Still, I'd hesitate to call it totally pointless. Math forms the underpinnings for physics, frequently with infinity involved somewhere, and physics dictates the mechanics of everything in existence. Similarly, I'd think that people in these non-mathy careers may take some interest in the theoretically infinitely rough structure of a fractal, which is a concept that arises in most natural constructs. In short, infinity helps us understand reality, and basically anything you do is going to touch upon that reality on some level. So, I dunno, it might come up.
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  46. I dunno why it has to be all philosophical. In math, infinity is a well defined notion that is reasonably well understood, one with a ton of applications. It's really not as mysterious as it's sometimes made out to be.
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  47. It is quite possible for an infinite hotel to be filled if there are infinite guests.
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  48. All the rooms start out full.
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  49. It's not really clear how, exactly, you're filling these rooms that aren't straight up powers. Where does, say, bus three seat five go? That's really the goal here, to get some explicit mapping of people to rooms. That said, it is possible to fill all the rooms. My method is that bus m seat n goes to the nth number where m is the least prime factor. So bus one goes to numbers divisible by 2 (2, 4, 6, 8...), bus two goes to numbers divisible by 3 but not 2 (3, 9, 15, 21...), bus three goes to rooms divisible by 5 but not 2 or 3 (5, 25, 35, 55...) and so on. The only caveats are that we call the hotel bus one, the first bus bus two, and so on, and that we start bus one with room one and then continue the sequence from there. This method creates a one to one mapping between guests and rooms.
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  50.  @ethancharim7321  If you only account for the powers of every number then you're going to miss a lot of numbers. Like 20, for example.
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