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

  1. An infinite hotel may not have rooms available if you have infinite people. Take all the hotel's guests and line them up, assigning each a number equal to their place in line. Put person one in room one, person two in room two, and so on. Now, for every room, there is a person. Or, equivalently, it's impossible to name a room for which there is no person. Thus, the hotel is fully occupied. It's also possible to assign people such that any number of rooms, from one through infinitely many, are unfilled, or such that every person gets infinite rooms, or such that every room gets infinite people. The possibilities are literally endless.
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  2. This is not a proper characterization of uncountable infinity. Something like the interval from zero to one, in spite of being bounded in both directions, is uncountable. The set of all integers, despite lacking bounds, is countable. A countably infinite set is simply one that can be mapped to the natural numbers on a one to one basis. An uncountable set is one that cannot be.
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  3. Countably infinite is quite possible. It just doesn't mean what you think it does. A set is countably infinite if you can create some list such that, for any number in the set you name, you can count to it going by that list. So, for the natural numbers for example, the list would just be 1, 2, 3, 4, 5... If you name any natural number you will eventually reach it just by counting upwards like that. Thus, the natural numbers are a countably infinite set. The list need not be a straightforward ascending order one though. The rational numbers are countable, for example, in spite of the fact that a straight up ascending or descending list would be completely impossible. More broadly, this video's presentation of infinity is an accurate one, and infinity is a reasonably comprehended and understood thing in mathematics. If it weren't then we wouldn't even have stuff like calculus. Really, infinity shows up all the time in math, in basically any field you'd care to study. It doesn't constitute any sort of real barrier either.
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  4. Sure, why not?
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  5. The idea of there being infinite numbers is not assumed to any extent. It's pretty trivially provable. After all, for any finite quantity, there is always a greater one. Numbers don't cease to exist when we stop using them. If I only count to 999,999, then I haven't just named the highest number, until I later count to a million when that becomes the highest number. And, no, we are not "creating" a number by moving someone. Assign each hotel room a natural number, and put someone in it. Then, move everyone up a room. What room number is being created? The first room obviously existed from the beginning, so it would necessarily be the "last" room. However, because there are infinite rooms, there is no last room, either before or after the movement.
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  6. He could just give the guests the information about how they're supposed to arrange themselves in one big lump. Just have them keep track of what their seat number and bus number are, and then put up a sign about what those things mean for placement at the front desk.
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  7. For any given infinite set, you can ask, is it possible to create a list of the elements in this set such that I can use the list to reach any arbitrary element in finite steps? For the natural numbers, the method of counting is clear. 1, 2, 3, 4, 5... You'll get everywhere just using that list. For, say, the integers, it's a little less obvious. However, you can do something like 0, 1, -1, 2, -2, 3... The same is true for the primes, the rationals, all kindsa sets. These are countably infinite. However, such a task is impossible for the reals. Thus, it is an uncountably infinite set. And, given that these sets are entirely composed of numbers, I'd say numbers decidedly do not become irrelevant with the concept of infinity.
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  8. By infinitely moving said guests, yes.
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  9. There is no unoccupied room, and thus no highest unoccupied room.
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  10. You can fill the infinite hotel just fine. All you have to do is, for each room, have a person inside. Simple enough.
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  11. I guess that would work. My preference is for a single mapping though. In particular, you can have the guests in the mth seat of the nth bus (counting the initial hotel guests as the first "bus") go to the mth room where the nth prime is that room's smallest prime factor. So, the hotel guests would go to all the evens. The first bus would go to rooms divisible by 3 but not 2. The second would go to rooms divisible by 5 but neither 3 nor 2. Naturally, this mapping still faces issues as regards room 1, which would have to be resolved by, say, keeping guest one where he is and having the count start at room 2, but other than that it's a single mapping that hits every room exactly once.
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  12. You can when X is infinity.
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  13. Here's a basic question. What set is bigger, the set of positive integers, or the set of positive integers without 1? The answer is that the two sets are the same size. Simply pair each element in the positive integers with that element plus one in the other set. So, because adding or removing an element doesn't change the size of an infinite set, we can conclude that infinity+1=infinity.
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  14. There is no open room. The hotel is full.
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  15. I don't think that's how the mapping operates. The buses are the bases, and the seats are the exponents, with, I suppose, the people in the hotel getting assigned to two as a base. So, the first person in the first bus gets assigned to 3, and the first person in the second bus gets assigned to 5. Honestly though, kinda wish they'd gone with a mapping that fills all the rooms. It's definitely possible.
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  16. You definitely have to move infinitely many people, but you don't necessarily have to move any individual person.
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  17. There is indeed room for infinite guests. That's what this proves.
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  18. It's also possible both to fill uncountably many rooms (as long as you have uncountably many people), and to move people around in the fully occupied hotel such that you have at least one room left open. The way this stuff gets constructed is a bit different, and it's kinda weird to consider an uncountable hotel in the first place, but it's kinda doable. The only step that's really problematic is when you have to label each guest with some real number, but the version where a guest is an intrinsic property of a room seems to solve that. We could also just assume that each guest is labelled with a real number of the type we're considering here.
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  19. Which, the uncountable hotel? No, an uncountable hotel would need uncountable people, and uncountable infinities are greater than the countable ones.
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  20. For each room, there is a person inside it.
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  21. ​ @GaryLuKOTH  The hotel also has one room for each natural number. They're the same size of infinity.
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  22.  @GaryLuKOTH  Ah. Sure then.
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  23. I dunno why you think that. This result isn't even taking place in the kind of continuous space that real analysis generally applies to.
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  24. No room requires infinite time, as every room is associated with some finite position. Any given finite quantity of time you name with have infinitely many rooms that take at least that long, but no room takes actual infinity.
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  25. Imagine lining up all the guests and labeling them with their position in line. Now, put each guest into the room that matches their label. What room is unfilled?
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  26. If no room is unfilled, then every room is filled. If every room is filled, then the hotel is full, and thus the hotel is fully booked.
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  27. Y'know, I usually use a mapping for this, but I'ma try something new. Imagine that every room, as part of its design, features a seed at its center. A property of the seeds is that, after 15 minutes, they sprout into guests. So, after that 15 minutes, every room will have exactly one guest in it, and because every room has a guest, the hotel is full.
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  28. The manager doesn't necessarily have to notify people personally. Just assign the first person on each floor the floor yeller. The person in the first room, when a guest comes to claim his room, calls out that it's time to move. He's loud enough, we assume, that everyone on his floor and the person in the room right above his can hear him. Then the person in the room above calls out, and so on. The message is almost necessarily going to travel faster than the moving guests, so this method is only maybe a second or so slower than a version where the manager is capable of magically notifying everyone in the hotel simultaneously.
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  29. There is an infinite amount of people. However, it's more than possible to have infinite people and infinite rooms, and have some rooms left over. What if your pairing starts out with person one assigned to room two, two to three, and so on, leaving room one open? Open room right at the outset.
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  30. You absolutely can empty a room without removing a person from the hotel. Simply put, there are n+1 rooms. N+1 is equal to n, and to n+2, when n is infinity.
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  31. N+1 is meaning a lot of different things right now. I assume you're talking about the guest movement strategy here. Yes, at first, for every room n+1, where n is a given guest's starting position, that room starts out full. However, what is possible is moving every guest up a room simultaneously. Guest one ordinarily wouldn't be able to move to room two, but guest two is going to room three. Guest two wouldn't ordinarily be able to go to room three, but guest three has moved to guest four. And so on. This process never stops working, because the hotel is infinite. The narrative of the word problem is not inconsistent. When the problem says, "The hotel is full," what is meant is that, for every room, there is a person in that room. What is usually meant by that, what is not meant here, is that the hotel cannot accommodate an additional guest.
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  32. Infinity is not a number, but it can tell you the number of elements in a set. So the set of natural numbers, for example, has infinite elements. As for the hotel itself, it can be said to have one room for each natural number.
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  33. The bus actually gives every person a single end-state room that they can move to immediately. There's no need to move repeatedly.
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  34. While the hotel is infinitely large, any given room is only finitely far away. It can still be insanely far away, if this is one of the bus scenarios, and you thus are liable to die before reaching it, but with enough time you can get to your room.
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  35. They aren't adding the people in the hotel together or something. This whole video is about mapping sets, not summing them.
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  36.  @havardmj  I guess? But it makes sense that this "summation" would diverge to infinity. Because the size of the set is infinite. It's not like you can't sum a divergent series. It's just that the result doesn't converge. Summation is a really weird way to approach this though, given it can only tell you that the set is infinite and not the specific cardinality. Mapping is a more logical approach.
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  37. Every room has one of an infinite set of people in it.
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  38. The distance from any given position in the middle of the bus to the front is not infinite, in the same way that no particular natural number is infinite. Only the entire bus, or the set of natural numbers in its entirety, is infinite.
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  39. Your dad's explanation was inaccurate, I think. I'm not precisely sure what you mean by "all numbers", but the set of positive odd numbers, for example, is not smaller than the set of positive integers, in spite of the fact that it seemingly has fewer elements. To show this is the case, you just have to create a one to one mapping from one set to the other. So, simply map each element of the positive integers to 2n-1 in the odd numbers. Every element of each set will be counted exactly once. There are larger infinities, however, For example, the set of all real numbers between zero and one is larger than either the positive integers or the positive odds.
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  40. Yeah, that's what it sounds like. Basically, any set of numbers that you can put in any kind of list is going to be exactly the same size as any other set with that quality. So the whole numbers and odd numbers are obviously like that. The integers are a bit less obviously like that, because you can list them like 0, 1, -1, 2, -2... The fractions are not at all obviously like that. For that ordering, you want to make a chart with the integers hanging out on the X and Y axis, and then the entries in the chart use the X axis as the numerator and the Y as the denominator (the Y axis excludes zero). Then you make this big zig zagging line that goes through every entry in the list, remove any repeats, and you have a full ordering of the rationals. Even less obvious is the set of algebraic numbers. Those are the real numbers that you can write as a finite algebraic expression. So, like the square root of 2 would be an algebraic number. Phi is an algebraic number too. But something like pi cannot be expressed that way, and so is not algebraic. The ordering there is relatively straightforward. You can kinda treat each part of the expression as a digit, and then remove illogical combinations. The reason all these sets have the same size is because you can easily create a mapping of the sort I mentioned. All the elements are already in a list, so there's a first element, a second element, a third element, and so on. All you need to do is pair the nth element of set A to the nth element of set B and the mapping is done. The real numbers cannot be listed this way, and so that is a bigger set.
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  41. There is no end. The hotel is infinite.
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  42. The people aren't finite.
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  43. The part where you tell infinite guests where to go is pretty trivial. The assignment is done by way of algorithm based on the guest's position in one way or another, so you can just tell the guests the algorithm. As for the keys, I guess you could put the keys in front of the rooms. Not like the guests are gonna have much time to get attached to their rooms, after all.
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  44. There is no untaken room. All the rooms start out full.
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  45.  @lionelmizrach2593  Because there are infinite people. Here's one way to think of it. It wouldn't be strange to assume that each room has a bathroom, a bed, and walls, right? Well, it should be equally normal to assume that each room has a person inside it.
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  46. Yeah, the longer version is a straight up bijection between the natural number set of people and the natural number set of rooms. Really, the whole set of results is equivalent to the various mappings between the set of natural numbers and itself.
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  47. There is no specific "room n" that is empty. "Room n" refers to the room "guest n" is in. Thus, "room n+1" refers to the room one greater for every given guest, so room six is "room n+1" for guest five.
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  48. An absolute ton of physics is reliant on calculus, and calculus is all about infinity. Our lives would be quite different if not for people pondering situations involving the concept of infinity.
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  49. No one has a room assignment of infinity. It may take insanely long, but any room is reachable within finite time.
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  50. There is no room infinity, or infinity+1, or infinity+2. There is a room for each natural number. And all of them are full.
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