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

  1. There is no "next floor" that's open. All the rooms start out full.
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  2. There is another room to give specifically because the person that was in that room moved to the next room in the line. Otherwise, there would be no room to give. It is only through infinite such moves that a room is generated.
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  3. Well, there's nothing wrong with every room having a bed, a bathroom, and walls, right? In the same fashion, every room also has a person.
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  4. An infinite hotel can definitely be full. Consider lining all the infinite guests up and giving them a number equal to their place in line. Then, put each guest in the room matching their number. If the hotel isn't full, then what room is empty?
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  5.  @clarkkent3730  Each person in line would indeed be assigned a finite value. However, the set of guests, as is the case for the set of natural numbers, is infinite. Consider that, in spite of the fact that no natural number is itself infinite, there are infinite natural numbers. I have no idea where numbers being divided into infinity comes into this. Infinity divided by infinity, however, is an indeterminate. This means it can take on just about any value, including zero, one, or infinity.
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  6. Straightforwardly, for each room there is a guest inside it. Thus, there is no other room to send a guest to. Building a whole new room is an option, but it's not exactly ideal.
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  7. For every room, you put a guest in it.
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  8. No, Hilbert's Hotel is a hotel with an infinite number of rooms. We know this, definitively, because it is the premise of the scenario. A hotel with infinite rooms that all have a person can absolutely fit another person. That is what the argument presented proves. Every infinite set is not the same size, but all countably infinite sets are, in fact, definitionally the same size. After all, the definition of a countably infinite set is that it has a bijective mapping to the natural numbers.
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  9.  @atheistontheroad4545  You're missing a pretty important point here. Yes, the number of guests and the number of rooms is the same at the start. This is clearly true. However, the number of initial guests with one guest added on is the exact same countable infinity. So is that quantity of guests with two new guests added on. As such, this new quantity of guests is precisely as large as the quantity of hotel rooms. You are correct that the initial mapping accounted for all rooms. However, what's occurring when you move everyone down a room is that you're generating a new mapping. You started out with, say, a mapping from the natural numbers to themselves, and now you're doing the equally valid mapping from the natural numbers plus zero to the natural numbers. You're talking about this last room, one where the guest has nowhere to go. It is true that, if there were a last room, the person in that room would have nowhere to go. This is why you can't add people to full finite hotels. An infinite hotel has no last room though.
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  10.  @atheistontheroad4545  Again, what have I not posted a rebuttal to? I've explicitly mentioned various things you've said, and the specific way my method bypasses those issues. You're acting like I'm not being responsive to you, but that makes no sense in this context. Given that I've addressed double booking, people in hallways, and the non-existent "last guest moved", I must ask, what is it from your previous responses that I am missing?
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  11.  @atheistontheroad4545  Indeed. The problem is that you out and out refuse to put forth an actual issue with the method I've stated. You just keep talking around the issue, talking about unrelated paradoxes under the assumption they'd be applicable and claiming you have some already stated argument hidden among dozens of comments.
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  12. There are not 128 people. Thee are infinite people, one in each and every room.
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  13. You just don't do it perfectly in order like that. The easy solution is to just have everyone leave simultaneously. The destination rooms are designated by math, so you don't even need someone giving directions. A more complex approach is using something like an abacaba pattern. There, you use the following pattern, and, when you say a number, that bus is the one that unloads its next passenger: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5... By this method, you hit every bus arbitrarily many times.
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  14. Infinite literally means endless.
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  15. All the rooms are occupied.
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  16. There's a number of definitions depending on circumstance. However, here's a decent one. If a set has a quantity of elements that is equal to some whole number, then that set is finite. Otherwise, that set is infinite. So, as regards the hotel, we can make each room an element of a set, and, because there exists no whole number corresponding to that quantity of elements, the hotel has infinite rooms.
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  17. There are infinitely many prime numbers.
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  18. There is no last room where someone is staying. There are infinite rooms, and every room has a person in it.
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  19. There does not have to be such a room. There are infinite guests. Line them up and assign each of them a number equal to their position in line. Then, put each guest in the room that matches their number. What is the last room where someone is staying?
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  20. But that situation is not this situation. If there were some last occupied room, then yeah, you could just make use of the next one. But there is no last occupied room. Every room has a person in it.
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  21. That is possible only because the person in that room is also moving to room n+2.
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  22. No, it doesn't. There isn't this one room, n+1, where guests can go. N+1 is a room relative to each already present guest. There is no initially unfilled room. There are only rooms rendered temporarily empty by moving the people in those rooms upwards.
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  23. An infinite hotel can be filled if you have infinite people. Consider putting all the guests in a line, and assigning them the number equal to their spot in line. Then, put each person into the room equal to their number. What room is unfilled?
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  24. For each room, put a person in it.
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  25. Consider taking infinitely many guests and lining them all up. Assign each a number equivalent to their place in line. Then, take each guest and stick them in the room whose number matches their own. What room will be empty? Where can we put a guest without moving anyone? If this infinite hotel is not full, then you should be able to answer these questions. You cannot, because the hotel actually is full. By mapping the infinite guests to the infinite rooms, it is possible to be left with zero rooms empty. It is also possible to create a mapping that leaves infinite rooms empty. Just assign the guests only to even numbered rooms. You can even assign infinite guests to every room, or assign infinite rooms to every guest, or, more mundanely, leave open any given natural number quantity of rooms. I can describe mappings that do any one of these things, and more. All of this is a more concrete expression of a basic fact, that there are many different mappings between the set of natural numbers and itself, as well as the fact that infinity-infinity is not just infinity. It is an indeterminate, meaning it can take on infinity as a value, or negative infinity, or zero, or anywhere in between.
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  26. The question is not whether you can put guests into the hotel such that the hotel will never be full. The question is whether you can put guests in so that the hotel is full, and you can.
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  27. Being able to fill the rooms doesn't suggest an end to the rooms. It simply suggests that the guests are equally endless, and that a one to one mapping between the guests and the rooms is possible. You ask how, but I literally laid out the explicit mapping being used. For me to be wrong, either the mapping fails at some step or it doesn't account for everything. Which of those is the case?
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  28. Infinity minus infinity does not equal infinity. It's an indeterminate, meaning it can take on a variety of values. You're not actually finding any flaws in my logic. You're just saying I'm wrong. Where in my proof did I indicate in any fashion that the hotel is finite? There is one room for each natural number, and that means infinite rooms. And I filled them. You can't call it impossible if I can do it.
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  29. Infinity minus infinity can indeed equal any of those numbers. Infinity itself is not an integer, but subtracting infinity from itself can produce integer values. I'll use a basic example. Consider the set of natural numbers. Now, consider the set of natural numbers with the first 24 elements removed. Subtract all the elements in the second set from the first set. What are you left with? Straightforwardly, what remains is the first 24 natural numbers. However, both the set of natural numbers and the smaller set are clearly infinite. So, infinity subtracted from infinity gives us 24.
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  30. I'm not really sure why infinity-24 has to equal anything but infinity. No step besides the main one, that it's possible to subtract infinity from infinity to get 24, is necessary. I can, in fact, produce any integer result from that subtraction. I can even produce negative infinity. It's easy. You say there's some kind of loop, but I cannot see one. Nothing of what I said has been apparently disproved. As another thing of note, infinity can absolutely be measured. Its measure just can't be given an integer value. The set of all real numbers is provably larger than the set of integers. You could also make use of literal measure theory and note that the Lebesgue measure of the real numbers is also larger than the measure of the integers. The reality is that we can say a ton about infinity. We can map to and from it, and fiddle with it, and work with it, and find properties of it. The idea that it's this unworkable mystery beast that encompasses all things is really not a true one. It just is what it is, which is endless. Doesn't mean a similarly endless thing cannot match up to everything that endless thing is.
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  31. I never really needed to "subtract" 24 from infinity, as it were. The set {25, 26, 27...} is already its own infinite set without ever needing to rely on a second infinite set. Phrasing it in terms of the natural numbers is convenient, but not necessary. Thus, any argument that relies on the presence of this process isn't all that effective. As for measuring infinity, yes you can. It just doesn't look the same as finite measures. For example, we can directly compare the size of the naturals and reals. The classic method is cardinality. Simply put, you attempt to map each element of set A to an element of set B and account for all of set B. If this is possible, then A is at least as big as B. Otherwise, B is strictly larger. Then you do the inverse, mapping B to A. If the first mapping and the second were possible, then the sets are the same size. Otherwise, well, I've already indicated how you identify one set as larger. Correspondingly, it is impossible to create such a mapping from the integers to the reals, but very possible to do so from the reals to the integers. So the set of reals is larger. Not every single thing we associate with measure will necessarily work properly. Percentage, for example, runs into issues. One of those issues is with cardinality itself though. If you take an infinite set, and then consider some finite non-zero percentage of that set, then that new set will always have the same cardinality as the original. Any attempt is necessarily somewhat problematic. You can certainly create something that seems like this 75%-ing though. Line the infinite piece of wood up with a number line, and mark any section that falls between a multiple of 4 and that value plus one. Then, take as your 75% everything left unmarked. What you'll be left with would, again, be as big as your original piece of wood, but it'd also be 75% However, this other type of measure, where we simply compare the size of two infinite sets, works fine. The Lebesgue measure works as well. This is where you create a set of open intervals that covers every element of the infinite set in question that covers the least space. The size of this least quantity of space is the Lebesgue measure. For any subset of the natural numbers, or even any subset of the rationals, the Lebesgue measure will always be zero. However, something like the interval (0,1), which is all the real numbers between those points, will have a Lebesgue measure of 1.
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  32. There is no room to move to initially. A room to move to is produced by the fact that the person who was in that room is also moving up a room. And that person can move up a room because the person who was in the room they're moving to is also moving up a room. And so on.
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  33. In the infinite buses scenario? Looks like 2^128.
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  34. There are infinitely many prime numbers. This is trivially provable.
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  35. You want a clean mapping of guests to rooms. However you map them, you should be able to say, "The person who was in bus 4, seat 3 will go to room n." If each of the infinite buses repeatedly shunts guests up to room 2n, then you don't have a clean mapping for any of the guests. Where does the person who was in room 4 end up? They get pushed to 8, then 16, then 32, and so on. They wind up going infinitely high, but there is no room infinity. Thus they effectively have no destination room.
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  36. There is a need to move guests if all the rooms already have people. Like, take your counting system, and assume that infinite people have already entered such that there is a numbered guest for each numbered room. Once each room has an associated guest, another guest comes in. You gotta move infinite people to make everyone fit, regardless of method.
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  37. There is no last guest. There are infinitely many guests.
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  38. ​ @devapreethi4773  This overall construction, where there is some identifiable guest that arrived immediately prior to the new one, still operates on the assumption that there are finitely many guests. If you say, "There have been x guests so far, and here's a new one," then that's still some natural number guest quantity. But there are infinite guests. Such is the scenario.
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  39. ​ @devapreethi4773  There are infinite rooms, but it is quite possible for an infinite hotel to be full. As you note, we can label each guest with some natural number and assign them to rooms based on that number. Guest one goes to room one, guest two goes to room two, and so on. Well, if we have infinite guests, and therefore one guest for each natural number, then which room winds up empty?
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  40.  @devapreethi4773  Again, there is no last person. This is the essential fact that makes the proof function. The mapping from guests to new rooms never falls apart because every single solitary person has rooms above them.
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  41.  @devapreethi4773  You did say last person. To quote, "The last person wouldn't have a room to move." This statement is the crux of your argument and the reason it doesn't work.
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  42. Isn't that scenario directly addressed in the video? Either way, the hotel is quite capable of accommodating these guests.
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  43. I think the method in the video was assigning each bus to powers of various primes, so the starting guests go to powers of 2, the first bus goes to powers of 3, then next to powers of 5, and so on. However, you get lots of vacancies with that method. So, instead, I'll start by assigning the guests in the hotel to rooms whose smallest prime factor is a 2. So, all even numbers. The first guest gets tossed in room one, cause that room remains unoccupied otherwise, and everyone else lands in the even rooms in order. The first bus is assigned to rooms whose smallest prime factor is 3. So, numbers divisible by 3 that are not even, which is 3, 9, 15, 21... The next bus goes to rooms whose smallest prime factor is 5, then 7, then 11, and so on. This method partitions the hotel into infinitely many sets of infinite size.
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  44. Infinite people.
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  45. While there are infinitely many rooms, every room has a person in it. There doesn't, at the beginning of the scenario, exist a higher numbered room without a person in it.
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  46. There are no free rooms at the beginning. The hotel starts out full.
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  47. People can only move up to other rooms because the people in those rooms have also moved. Without that fact, none of those rooms would be empty.
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  48. That's your prerogative, I guess. Kinda thought that explanation basically covered it, to be honest.
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  49. It's possible to fill an infinite hotel. You just need infinite people.
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  50. Consider taking all the infinite guests, lining them up, and assigning each the number equal to their place in line. Then, put each guest into the room matching their number. After that, if the hotel isn't full, then what room is not full?
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