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

  1. Not necessarily. What if everyone moves rooms simultaneously?
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  2. There are no empty rooms to move to at first. Empty rooms are generated through the process of people moving.
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  3. "Full" is usually taken to mean two different things that are typically equivalent. First, that no more things can be added. Second, that all available space is occupied. In finite spaces, these are the same. If all available space is occupied, then adding more stuff is impossible. The other direction also holds true so long as there is no other mechanism stopping stuff from occupying the space.. In an infinite space, these things are not equivalent. The hotel is "full" in the sense that every room has a person in it. All available space is occupied. The hotel is not "full" in the sense that you can still add more people. The definition where all space is occupied is a sufficient one, and we could indeed call something that has only this property "full", so a full thing can, in this case, accommodate more guests.
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  4. Room n+1 isn't a thing. It's the room one above any given room you care to name. So, for room number one, n+1 denotes room number two.
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  5. For each room, you put a guest in it.
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  6. What next room? Which room, specifically, are you assigning the new guest?
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  7. Two times infinity is equal to infinity. Consider, say, the set of natural numbers as opposed to even numbers. There would apparently be twice as many natural numbers, but we can map the naturals to the evens by multiplying each natural by two.
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  8. Why would everyone know just because there're infinite people? Instant knowledge transmission isn't a quality of infinitely large person groups as far as I'm aware.
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  9. Those numbers necessarily have infinite digits, and so they are not integers, which have finite digits.
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  10. There are no vacant rooms. All the rooms are full at the beginning.
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  11. One person in each room.
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  12. Why, precisely, can each person not move into the next room? What is stopping that from operating just fine? The reality is that infinityx2 is precisely equal to infinity. There is no need for expansion.
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  13. Infinite people.
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  14. There is no next available room. All of the rooms start out full.
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  15. There are no rooms available. The hotel is full.
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  16. No, all of the infinities involved are exactly equal in size. And I'm being literal when I say all of them. The infinity of the hotel is equal to the infinity of one bus which is also equal to the infinity of the first five buses which are all equal to only the prime numbered rooms, and so on.
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  17.  @themushroomhunter  Because "infinity x infinity" is equal in size to just infinity. The video is actually providing a proof for that being the case, giving a mapping between bus seats and hotel rooms. All countably infinite sets are the exact same size, by definition.
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  18.  @themushroomhunter  It's been a thing since Cantor. None of these infinite sets are bigger though.
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  19. That infinity+1 is still infinity is kinda the point. At the beginning, there are as many guests are there are rooms. If you add on a guest, then there are still as many guests as rooms.
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  20. Every number has a single unique prime factorization. The power of a prime has nothing but that prime number in its prime factorization. Thus, any power of any prime cannot be equal to any other power of any other prime.
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  21. I mean, we're presumably not working with a continuous person blob. As long as there are discrete and listable seats, and discrete and listable people occupying those seats, the infinity we're working with is decidedly countable and thus shares a size with the infinity of the hotel.
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  22.  @sravansuhas2931  It's not a countable number. It's a countable infinity. A countable infinity is basically one where you can put all the elements into an infinite list, such that, for any element, you can point to some position on the list and say, "Here it is." The natural numbers are the canonical countable infinity. The standard list for those is 1,2,3,4,5,6... So you're like, "Where is 123," and I'm like, "It's in the 123rd position on that list." Lots of infinite sets are countable. The integers, the evens, the rationals, and, indeed, the set of people on that bus. And a noteworthy fact about countably infinite sets, the main definitional thing about them even, is that they are all the same size. They all have the same amount of elements.
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  23. Likely infinitely tall, assuming the hotel isn't infinitely wide. And there is no actual infinity or -infinity floor, simply a floor for each natural number. Thus, that is not a valid trip the elevator can take.
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  24. To the first question, imagine lining up the guests and assigning each a number equal to their place in line. Now, put each guest in the room whose number matches theirs. At this point, there will exist no room that does not have a guest, so the hotel is full. To the second question, a countably infinite set is one where you can construct an ordering of that set such that you will arrive at any arbitrary element of that set in finite time. So, for example, the counting numbers are a countable set, because you can order the elements as you would ordinarily count them. If counting n numbers takes n units of time, then reaching some number n takes, obviously, n units of time. Other countably infinite sets include the evens, the integers, the rationals, and even those numbers that can be formed by a finite algebraic expression (such as the square root of 2). However, the set of real numbers is uncountable.
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  25. That infinity equals infinity times 2 or infinity+1 is why you're able to put guests into the full hotel in the first place. However, the hotel can absolutely be filled. Simply put, for each room, put a guest into it.
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  26. The resolution, naturally, is that you do not count the rational numbers by their size. The usual ordering of the rationals is to draw a zig-zagging line through a grid of fractions where the X axis is numerators and the Y axis is denominators (with each axis being an ordered list of naturals or integers, depending on whether you want negatives, and where zero isn't allowed as a denominator). By this ordering, it is quite possible to count from one to three in finite time.
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  27. I guess the use of the word inclusive is weird there. The correct way to put it is that you can count from one to three along the given ordering, meaning you count all the numbers in between as defined by the list itself. Personally, I think that setting two end-points is a little, well, pointless. Why not just say you can count to three along the ordering, starting at the first element listed? As long as we're in a standard numeric structure, the statements should be equivalent.
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  28. Assuming you just need to count all the elements in between according to the ordering, I'm pretty sure that's a necessary and sufficient condition of a set being countable.
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  29. Like, if your ordering is 1, 1/2, 2, 1/3, 2/3, 3, 3/2... to put together something arbitrary, you'd just have to count all those numbers up to 3. The ordering by size wouldn't enter into it, but you can't just say, "1, 3, and I'm done."
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  30. I did, in fact, take that latter meaning from his comment. Specifically that you can count from a rational to another as long as you define some alternative order to count by.
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  31. Just because you don't know how it works doesn't mean that mathematicians don't.
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  32. Infinitely many rooms can be full if you have infinite people.
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  33. There is nothing to the definition of infinity that means you cannot fill it. Infinity is just anything that is not finite, and if you have two infinite sets of the same cardinality then one can "fill" the other just fine.
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  34.  @koko10900  The infinite quantity of rooms exactly equals the quantity of guests. Far from neither being defined, it is actually the case that both are defined. In particular, there is one room for each natural number, and there is equivalently one guest for each natural number. It is, in fact, provably the case that there is a one to one mapping between the two sets. That you can add guests does not prove the hotel wasn't initially full. What the hotel being full means is that there is a guest in each room. The video shows a proof that, in spite of this initial condition, more people can be added. You're just identifying the very thing they're calling weird and agreeing that it's weird.
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  35.  @koko10900  "Once claimed that you have an infinite amount of one substance, all of that substance is now in that infinity." Untrue. Consider that, if I have all the natural numbers, then I have an infinite amount of numbers. However, all of the substance that is numbers is not contained in that infinity. For example, -1 isn't contained in my set. Similarly, if we were to label every person in this imagined infinite person universe with an integer, and then had only the positive labeled people start out in the hotel, then that would leave infinitely many negative labeled guests. "And if it is the fact that the infinite hotel is filled by an infinite amount of people, there is really no point differentiating between the two entities. This is because, as soon as I've room seemingly becomes available, it has to be filled. Unless this is true, at one point in time it has to have been considered empty. If the pairing is instant in which this scenario would seem to suggest, room = guest, which makes them one entity because they are necessary for eachother in this scenario." Also not true. In fact, the whole point of the video is that it's not true (and the video is correct in this). There are many different mappings of guests to rooms. We can assign them in a way that fills the hotel very easily. Person one goes to room one, person two goes to room two, and so on. But we can also assign them in a way that leaves an empty room. Person one goes to room two, person two to room three, and so on. Or we can assign them in a way such that infinite rooms are empty. So one goes to two, two goes to three, three goes to five, and person n goes to the nth prime number. "It is through the language used that you assume you have a defined amount of rooms, "infinite". You will always have the amount of rooms for any amount of guests because any unending space can fit an unending amount of things." Well, this is only true if we're solely considering countably infinite sets. The issue of uncountable sets is bit beside the point though. What's important here is that you are essentially correct. The hotel rooms can indeed accommodate the new guests. That's the point. The hotel can accommodate any countable set of people. It's just that the hotel can also have one person in every room when starting out. "To say that this space is full, is to define it in a measurable way thereby taking away it's quality of being infinite." No. You literally defined infinity up above. Infinite means never ending. It does not mean impossible to measure or define. The basic reality is that there are ways of measuring and dealing with infinity. "Meaning that the infinite hotel will never be full because it simply HAS rooms. Not has 10 rooms, not 11, not 500, just simply HAS rooms." Just magically having rooms is also not a quality of infinity. The rooms are endless, yes, but so are the guests, and it is quite possible to line the endless rooms up with the endless guests.
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  36. ​ @koko10900  "I'm going to be honest this will be my last response because clearly you aren't understanding me." I understand you reasonably well. You are just wrong for the most part. It's your prerogative to respond or not respond. "If you would like to measure infinity, by all means, do what no one has ever done before. Given that any infinite thing has no defined quantity, good luck." Gregor Cantor predates me by about a century and a half, and the broader study of infinity has proceeded just fine both before and after him, so the idea that no one has done this before is wildly inaccurate. "Having a specific set of infinity doesn't by any means, mean you've measured it." No, that alone does not constitute a measurement. What does constitute a measurement is determining the cardinality of the sets involved. Two sets have the same size if you can pair the elements of one to the other on a one to one basis. All countably infinite sets have the same cardinality. Uncountably infinite sets have greater cardinality. "Secondly, once you've made claim that you have an infinite number of people, you've claimed all the whole numbers of people that can exist." Nope. I started with the integers and picked the positive ones, but I don't have to do it like that. Say I start with the natural numbers and pick out only the even ones. Even numbers are infinite, but they do not constitute all the numbers. Nor do they constitute all the people, given I have one for each natural number. "At least we both agree on the fact you can't have negative people or fractions of them counting as people." I think you're misunderstanding the scenario somewhat. I'm saying I'm giving each person numerical labels. This means there can, in fact, be a person -1. My stated scenario used the integers, and so there was a "negative person". It is not strictly necessary that we involve people with negative or fraction labels though. It's actually possible to break any countably infinite set into infinitely many countably infinite sets, so we can narrow things as much as you like and this still works just fine. "You've claimed the basic infinite set of whole numbers. Adding one to it is also within that set." What if I add -1 to it? Or 1/2? I've added a single element, and have something from outside the set. "And again, this is not measured because you haven't given it a defined specific size by which I mean an alloted numbered value by which it can't be anything else." Actually, the cardinalities I talked about above do have specified values. All the infinities we've been talking about have the cardinality aleph 0. The likely next biggest infinite set, aleph 1, includes sets like the set of all real numbers (I say likely because, as I recall, the existence of a set with cardinality between those two is undecidable given standard axioms). "You further haven't answered my main claim in that at some point there will have been a vacant and so non full room, unless of course the existence of a room means it's filled in which case this thought experiment is pointless because again you're creating two things that can't exist separately of each other." This is pretty straightforward. Using this infinite set of people, you can fill all the rooms. By moving the set of people around, you can empty a single room. You never have to have an empty room, but you can have one if you want to. You can produce just about any result you want just by moving people around. You can leave five rooms empty, or infinitely many rooms. You can assign infinite people to every room, or you can assign infinite rooms to every person. The possibilities are literally endless.
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  37. A hotel with infinite rooms can be made full if you have infinite people. Consider taking all the guests and lining them up, assigning them a number equal to their number in line. Then, stick person one in room one, person two in room two, and so on. Every room has a person inside it.
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  38. For each room, put a guest in it.
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  39. There is no highest floor, because the hotel is infinite, and rooms are not constantly spawning.
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  40. There is nothing about infinity that demands it be "circular". If something is not finite, then it is infinite. The natural numbers are not finite. We can prove this to be the case, because if you say there is a given finite quantity of natural numbers then you are always going to be wrong. Lacking a beginning or an end is sufficient, and you could even have both. The real numbers from 0 to 1 have both an upper and lower bound but are decidedly infinite, because the size of the set itself is boundless.
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  41. Infinite does not mean all. The set of all natural numbers is infinite, for example, but -1 is not a part of that set.
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  42. There is no last room. The hotel is full.
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  43. There's an infinite number of rooms, but, at the start of the scenario, no rooms available. Every one of the infinite rooms has a person in it.
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  44. Line up all the infinite people, and give them the number that is their place in line. Then, put person one into room one, person two into room two, and so on. By this method, if you name any room then there will be a person in there, so every room is full.
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  45. There are no empty rooms. They all start out full.
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  46. For infinite people, if you try to use the initial method of moving each person up a room repeatedly, you run into the issue that every single step of this process will leave you with only finitely many available rooms.
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  47. It is possible to fully book an infinite hotel if you have infinite guests. Consider that each room has a bed, a bathroom, walls, and so on. We can equivalently say that each room has a guest.
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  48. There is no last floor. There are infinite floors.
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  49. It was full in the sense that every room contained a person.
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  50. It can with infinite people.
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