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

  1. You can absolutely fully book an infinite hotel if you have infinite guests.
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  2. I'm not sure what you mean by saying it doesn't exist. No, countably infinite things do not exist in the real world. Countable infinity is very much an existing mathematical notion though.
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  3. There is no room infinity, let alone a room infinity+1.
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  4. The hotel starts out full. There are no free rooms.
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  5. Line up all the guests, and number them based on their position in line. Then, put person one in room one, person two in room two, and so on. There is no room without a person, and no person without a room.
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  6. There is no such thing as a singular room n+1. Each room has its own room n+1, which starts out full. The only reason a guest can move up a room is because the person initially in that room is themselves moving up a room. There is no free room at the beginning. Moving the guests generates a free room.
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  7. No. An infinite hotel can be full if there are infinite guests.
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  8. There is no next room that is empty. All the rooms start out full.
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  9. If there's only one bus, then we will definitely reach any given person on that bus in finite time. If there are many buses, or even infinitely many, then we'll have to get a bit creative. My general strategy is using the abacaba pattern. Label each bus with letters of the alphabet, a, b, c...aa, ab, ac... Then, whenever a given label is hit by the following pattern, you remove the next guest from the labelled bus. The pattern in question is abacabadabacabaeabacabadabacaba... While each consecutive label occurs a smaller and smaller percentage of the time, the pattern has infinitely many of every label. So, for any given guest, you will unload them in finite time, regardless of bus quantity.
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  10. Infinity doesn't mean all encompassing. The set of all positive numbers is infinite, but there's still an infinite set of negative numbers to deal with. It is, in fact, possible, to separate any infinite set into infinitely many infinite sets with no elements in common with each other.
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  11. Infinite literally just means not finite. It doesn't mean all. Seriously, we could have infinitely many guests leave their rooms, do that infinitely many times, and still have infinite guests remaining in the hotel. Such is the way of infinity.
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  12. The two infinities are definitely the same size in this case (though not in every case), which is why you're able to provide a mapping in the first place, but that doesn't mean the mapping will always be the same. The basic question you have to answer for a mapping is where any particular guest will end up. So, for your mapping, we can ask what happens in, say, bus three seat five. If you don't have some kinda answer then you probably need a different mapping.
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  13. Infinity pops up all the time in math. Understanding how infinity operates can prove really useful. It's not like this is even all that abstract or weird. It's basically just saying, "There are a lot of mappings from the set of natural numbers to itself."
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  14. There is no highest room. The hotel is infinite.
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  15.  @youse8240  There is also not that. Every room is full, and there are infinite rooms, so there is no highest room that someone is in, or is there a room one greater than that.
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  16. Because those infinities are the same size. You can pair the original hotel guests with the bus guests on a one to one basis, and this is what is actually being proved in this video.
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  17. No, it just means that the guests are equal in number to the hotel rooms. For every hotel room, put a guest into it. Thus, the hotel is full.
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  18. There is no next numbered room available at the beginning. All the rooms start out full.
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  19. You're just misunderstanding what countable means. A countably infinite set is one where you can construct some list such that any element of the set will appear at some finite place on that list. In other words, for any element of the set you can count to it. The canonical countably infinite set is the naturals, which have the obvious list of 1, 2, 3, 4, 5... Lists don't have to be in ascending order though.
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  20. The basic reality is that a mapping from a countably infinite set of guests to a countably infinite set of rooms can have just about any result. You can fill every room with exactly one guest, or you can leave finitely many rooms empty, or you can leave infinitely many rooms empty, or you can put infinite guests in every room, or you can even give infinite rooms to every guest. The adding of infinite guests, using the stated method, just happened to shift things from one of those mappings to another. It is quite possible, however, to fill every single room exactly once using a single mapping from the infinite buses to the infinite hotel.
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  21. People do not end up sharing rooms. The reason moving people "creates" rooms is because, as it turns out, all the rooms except the first room can hold just as many guests as all the rooms can. All the even rooms can also hold just as many guests, as can all the prime numbered rooms. It's all the same size of infinity, in that any one of those sets can be mapped one to one onto any other of those sets. And this is an expression of that fact.
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  22. The circumference doesn't necessarily have a rational length relative to the diameter. Real numbers can always be expressed as fractions. Pi=pi/1. To be rational, it must be possible to write the number as a fraction where the numerator and denominator are both integers.
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  23. There is no last person. The hotel is infinite.
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  24.  @LinxinMusic  The last person in an occupied room means the same thing as the last person. There isn't one.
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  25. There is no next available room. All the rooms start out full.
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  26. All of the rooms can absolutely be full if you have infinite people.
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  27. Eh, ya just need a good way to fill it. Line up all the guests and number them with their position in line. Then, put guest one in room one, guest two in room two, and so on. In this fashion, regardless of the room you select, there will absolutely be a person in it.
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  28. Not necessarily. You could plausibly have every single person start moving at, say, 3 PM, and then, if moving takes an hour, the entire process would be complete at 4 PM. This sort of thing is possible regardless of how many guests you assign at once. After all, the structure of the room assignment is such that everyone knows exactly which room they'll end up in. The amount of time this takes would be exactly as long as the longest single move-in. Which, of course, could itself be unbounded, but it doesn't have to be. Maybe these people can get to literally any room from any location in exactly an hour.
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  29.  @ShadowStorm2001  They could always just have, I dunno, automatically room swapping keys which change the door they're assigned to at the scheduled time.
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  30.  @ShadowStorm2001  Fair, I suppose. In that case, the method would probably have to change to one using increasingly rapid movement. The hotel worker gives the first guest their key in one minute, the second guest in half a minute, the third, in a quarter of a minute, and so on. This infinite series sums to two minutes, so after two minutes every guest will have their key.
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  31. Okay, well, do you take issue with the idea that each room has a bed? A bathroom? Walls? If not, then what's the issue with each room having a person?
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  32. It's pretty straightforward to do both of these things. First, filling the hotel. Line up all the guests in order, and assign them the number equal to their position in line. Then, put guest one in room one, guest two in room two, guest three in room three, and so on. There will be no room left unfilled, and thus the hotel is full. As for adding infinities, the positive evens are infinite. The positive odds are also infinite. Add the sets, and it's incredibly straightforward to identify that you're left with the natural numbers. You're positing here a weird definition of infinity, that it must somehow be unfixed. But that's arbitrary, and not particularly reflective of any truth of infinity. If we agree that the natural numbers are infinite, then obviously it's possible for infinities to be fixed. It is well defined what elements are and are not in that set. There's a fundamental fixedness to the set, and that nature can be utilized for various purposes.
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  33.  @JJokerMoreau  You are mistaken. I'll start with countable infinities. Far from impossible, this categorization deals with the fact that the elements of an infinite set can be ordered such that, starting from the first element, you can reach any given element in finitely many steps. The natural numbers are the canonical example of this. Given the ordering 1, 2, 3, 4..., name an element of the set that cannot be reached. This is, of course, only one of many functionally identical definitions. Perhaps the most classic is that the elements can be matched up one to one with the natural numbers. This is true of, for example, the integers. List the integers as 0, 1, -1, 2, -2, 3, -3... and the pairing is easy to construct. Just pair the first integer, 0, with the first natural, 1, the second integer, 1, with the first natural, 2, and so on. You may note how this definition ultimately works the same as the first, as the method of pairing can always be one of these orderings. From this, we can derive a notion of larger infinities. A larger infinite set is one that can't be ordered in this way. One that can't be paired with the naturals. The canonical example here is the real numbers. Any attempted pairing with the naturals will inevitably fail to account for not just a real number, but infinitely many real numbers. It is in this sense that larger infinities are possible. I'm not really sure what you mean by conceptually impossible. I've listed the concepts, and they operate in a way that doesn't contradict anything external to themselves, and in an internally consistent manner. These aren't games either. The natural numbers are obviously incredibly useful. The real numbers are both very useful, in a pragmatic sense, and a direct extension of those useful naturals.
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  34.  @JJokerMoreau  This isn't part of the list of "unexplainable math". I explained it very thoroughly. That you can't divide by zero isn't really unexplainable math either. It's a well known and understood fact about math that it's an indeterminate.
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  35.  @JJokerMoreau  That's not remotely what it's like, because this isn't unexplainable. You just kinda included the word unexplainable in your analogy. Either way, the infinity stuff is obviously not unexplainable. I literally just explained it.
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  36. Probably no to all of those. Each element of the set has to be such that any exponent of that element takes you outside the set. Perfect squares definitely don't work. There are infinitely many crossovers, because a perfect square squared is itself a perfect square, as is a perfect square to any power of 2. The other two are less trivial, as there is no easy rule, but I see no reason to think there wouldn't be crossover. Fibonacci numbers, for example, face an immediate issue with 8 being 2^3. I can't tell for factorials, because they get big so fast, but it strikes me as unlikely that no two factorials would have this relationship.
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  37. Every room, at the beginning of the scenario, is full. There exists no empty room to move them to, let alone infinity of them. It is, of course, possible to move people before you even have guests in such a way that you wind up with infinite empty spaces, but that's a really similar sorta solution.
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  38. A countably infinite set is one for whom you can develop some ordering such that, if you count along the set by that ordering, you will eventually arrive at any given element in finite time. It is your thinking that is rigid on this matter.
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  39. An infinite hotel can definitely be full if you have infinite guests. The simplest approach is that, for each room, you have a person inside of it. You presumably wouldn't object to the idea that each room contains a bed, a bathroom, and walls. Why not just replace that bed with a person?
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  40. There is no empty room at the beginning. All the rooms are full at the beginning.
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  41. What if you just have all the guests teleport in simultaneously? What if you have each guest reach their room twice as quickly as the last? What if the rooms are just built with the guest inside, the same way you could have each room have a bed inside?
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  42. You have no basis for the idea that infinite things can't be changed all at once. Let's say, for the sake of argument, that you already have everyone in their rooms (as you've already agreed is possible), and there is a sign up in each room that says, "At 12 PM, move up one room." What stops this action from being simultaneous, precisely, and what basis have you for the claim that the process won't be complete by a minute after 12? In any case, again, you've already conceded that a stagnantly full hotel is possible. So, there is no issue. The process of moving guests to accommodate the new guest could theoretically take forever, but nothing in the video falls apart if that process does take forever. The full hotel is not itself a contradiction in terms, and people in the hotel moving, regardless of the time expenditure, is also not a contradiction in terms, so the construction is fully functional.
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  43. I just told you how the process of moving everyone could take finite time. Address this method prior to claiming it impossible, please. Moreover, you haven't addressed the idea that each subsequent guest could take half as long as the last to reach their room. If the first guest takes a minute, the second half a minute, and so on, then would the hotel not be full after two minutes? You haven't even addressed the idea of everyone teleporting to their rooms. If you think I'm wrong, prove me wrong. You can't just keep claiming it without basis.
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  44. The rooms start out taken. Moving people enables those rooms to be opened up.
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  45. There are no free rooms at the top of the hotel. There is no top of the hotel in the first place. Every room is full at the beginning.
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  46. First, we have to ask how to define one set as larger than another. If you have one set with three objects and one with four, how can you generally claim one set has more objects? One method is to take each object from one set and attempt to pair it with one object from another. If you are fundamentally incapable of doing this, then one of the sets, the one with unpaired objects, is larger. So, as an example, we can compare the set of natural numbers to the set of even numbers. Naturals have the immediate appearance of being larger, but if you take each one and multiply it by two, then you will have a perfect mapping from naturals to evens. The same is doable for integers, rationals, and even the algebraic numbers (numbers that can be represented through some finite algebraic expression, like root 2). So, can we map the naturals to the real numbers from zero to one? Let's assume we can, and specifically assume there exists some arbitrary mapping from the naturals to the reals. It'll look something like this. 1: .149874123... 2: .00129384123... 3: .123487102... 4: .981726418... ... In order for this to be a mapping, every real number from zero to one should be on the right side. But, as I will now prove, this is impossible. Draw a diagonal line through all those real numbers, such that it goes through the first digit of the first real, the second digit of the second real, the third digit of the third real, and so on. Now, take each digit that has a line through it, and construct a new number out of it. In this case, the first four digits would be .1037. Finally, increment each digit by one. Now we have a new number .2148... This number cannot be the first real on the list, because the first digit is different, it can't be the second real, because the second digit is different, it can't be the third real, because the third digit is different, and so on. The new number cannot be any number on the list, so the original list did not have all the reals from zero to one. And, in fact, it's possible to generate uncountably many numbers we missed through similar methods. Thus, this mapping, and because it was an arbitrary mapping, any mapping, does not work. This is what is meant by uncountable, that there is no possible mapping from the natural numbers.
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  47. It's just a handy visualization of concepts related to infinity, and infinity, even without representation hotels, is useful. Shows up all the time in math.
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  48. Even with infinite rooms, it is possible to fill every room. And the hotel starts out full.
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  49. I occasionally run through the comments on videos like this using the newest first ordering, cause I don't see that much point in responding to comments from more than a few days ago, maybe longer if the comments on the video are at a low rate.
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  50. All of the rooms start out full. There's no room n+1 for them to go to. N+1 denotes the fact that each guest goes to the very next room, but those rooms are only unoccupied because the person in those rooms also moved up a room. They don't start out empty.
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