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

  1. Infinite people.
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  2. That could be done if you started with an empty hotel, but it is assumed that we are starting with a full hotel, and that is a fair assumption to make. What if the dude gets a bonus based on the percentage fullness of the hotel?
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  3. Indeed.
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  4. Yes, it can. Simply put, for each room, put a person into it. This will inevitably render the entire hotel full.
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  5. To start with, there is no room at the very top. The hotel is infinite. Thus, there can be no top to it. I suppose there could be a top if the hotel stretched horizontally, but then there would be no end in that direction. The reason you have to move everyone is because the hotel starts out full. Next, to the buses, first of all, it's really more like there are infinity^2 people in all the buses. There are two integer coordinates, first the bus number, second the seat number. More importantly, you assume that this number of people isn't the exact same infinity. It is. As is proved by the video, there is a mapping from the guests to the rooms that uses up all the guests. It is also possible to do a straightforward one to one mapping, that uses all the guests and the rooms exactly once each. Finally, you are correct that infinity isn't a number. You are incorrect in saying there is no larger or smaller infinity. If it is physically impossible to pair up natural numbers with real numbers such that all the real numbers are used, then the reals are simply more numerous. And it is, in fact, physically impossible. All the infinities talked about in this video are countable infinities, which means they are the same size. The hotel rooms, the initial guests, those guests plus one new guest, the guests plus an infinite bus of guests, or even the guests plus infinite buses of infinite guests, they're all the same. Nothing bigger or smaller. But bigger infinities do exist.
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  6. A hotel with infinite rooms can be full if you have infinite people. And the logic works really straightforwardly because infinity absolutely equals infinity+1.
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  7. The rooms are assumed to start out full. It's not a ridiculous assumption to make, given that we're already working with infinite hotel rooms anyway.
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  8. Yes. The simple version is, for each room, put a person into it.
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  9. It's a weird assumption, but not a ridiculous one for math. I think it's actually pretty misleading to call this a paradox.
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  10. Sure, never gonna come up in reality. Still works reasonably in a theoretical context. And, yeah, I've found that most usages of the term paradox come across as weird in math. Russel's paradox qualifies, but Banach-Tarski and this are just kinda straightforward math claims with zero ambiguity.
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  11. I don't need to prove that infinity is the only thing that has this property, primarily because it's irrelevant. For any sort of finite number, we can prove pretty easily that that sort of finite number changes when you add one to it. For any sort of infinite number, we can pretty easily prove that it doesn't change when you add one to it. For example, for any real number, adding one will change either the ones place, or, if the ones place and some quantity of other places greater than it have 9's, then the first non-9 will change. It's just how addition works. As for infinity being a number, again, not all that relevant. What matters is whether we can add to it, and, y'know, we can. It's not all that difficult. If you want to call it something besides a number, go right ahead, but we can still generally add or subtract with it. Not always, and I'll naturally fall into such a case in a moment, but we know what those cases are. This question is not one of proof but of definition. Define number, and if that definition fits infinity, then we call infinity a number, and otherwise we come up with a different name. That gets to your final post, where you assert that infinity minus infinity equals zero. It doesn't. Infinity minus infinity is an indeterminate, meaning it does not have any one value. In this particular case, that means it can take on any value (mostly, cause countable infinity is inevitably going to be something of a bound). You can put infinite guests into the hotel and get zero vacancies, or you can get exactly one vacancy, if you fill all but the first room, or you can get five vacancies, by skipping the first five rooms, or you can get infinite vacancies, by filling only even rooms, or you could get what we might call negative vacancies, by putting anywhere from two to infinite people in every room, or we could assign everyone infinite rooms, whatever you wanna call that.
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  12. That you don't think we can do it is completely irrelevant when put beside the fact that he just did it. You're correct that we can't reach infinity by multiplying a whole lot, but that's not the method being used. Answer this: If we do label each person with a natural number, taken in order, and then put the person with that number into the matching room, what room is left empty? Alternatively, we can just get to the rooms being full direct style. We haven't yet specified anything about the rooms, so they're not necessarily lacking in stuff. Say, for the sake of argument, that each room has a bathroom, a TV, some lights, and a bed. Seems totally reasonable, right? They're just qualities of this default room. Now, take the TV, and replace it with a person. Not by hand, but by way of room properties. Each room, by default, contains a bathroom, lights, a bed, and a person. Because the property of room fullness refers directly to the presence of a person or the lack thereof, every room is full.
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  13. Why do you assume the people are told one by one?
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  14.  @FistroMan  You are mistaken. Provably. If you'd like to supply your mapping, then I'd be happy to identify its flaw.
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  15.  @clarkkent3730  There is no such thing as "true infinity". There is finite, and there is infinite. The set of natural numbers is infinite. The set of natural numbers less than 100 is finite.
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  16. There is no top. The hotel is infinite.
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  17. Any finite quantity of natural numbers is obviously finite. The set of all natural numbers is decidedly not. After all, for any quantity of natural numbers you can claim there to be, I can say, nope, there's at least one more. The natural numbers are boundless, and so you can continue applying them to guests forever. You'll never label some "last" guest, because there is no such thing, but every guest will have some label, on and on into infinity. The set of natural numbers is the smallest infinity, the countable infinity, aleph_0. The infinity after that (dependent on choice of axioms) is the set of real numbers, the first uncountable infinity, aleph_1. Guests, however, are decidedly not structured like the real numbers, as they are discrete entities, so we're working in a countable space here.
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  18. Infinity is distinct from standard finite numbers. It has a set of properties that are not shared by said finite numbers. One of those properties is that infinity plus one is equal to infinity. You cannot generalize this property to things that aren't infinity. But it's a thing that can be pretty easily proved as regards infinity. You can absolutely add to infinity. That the addition functions differently than addition with finite numbers does not change the fact that you can add to it.
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  19. As you said, there is no final person or room. Talking about all the rooms being full is wholly consistent with the notion of infinity. A last room is not. And you can add people, by the method they spoke of. Yes, the infinity of people already there matches the infinity of rooms, but the infinity of all those people plus an extra person also matches the infinity of rooms.
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  20.  @gaganzanwar  It's not hard to imagine that every room contains a bed, right? A bathroom, a tv, floors, walls. Why, then, is it hard to imagine each room contains a guest alongside those other things?
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  21. There is no "infinite room guy". There is one room for each natural number. You are correct that this wouldn't work in a finite hotel. The reason it does work in an infinite hotel is because you can never do exactly what you just did. You name a room and I will always be able to tell you where they wind up.
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  22. I'm not sure what you mean here. Counting the hotel as the first bus, the nth seat of the mth bus goes to the mth prime raised to the nth power. So, the fifth seat on the second bus goes to 3^5. Cause 3 is the second prime number. The reason they're doing it this way is because any power of any prime number will always have a different result than any different power of any different prime number.
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  23. That's the largest known prime, not the largest prime number. There are provably infinitely many prime numbers.
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  24. I suppose because it highlights a contradiction in our definition of "full". In one regard, the hotel is full. Every room has a guest. In another regard, the hotel is not full. A new guest can arrive and be accommodated. There certainly isn't anything logically contradictory about it though. It's just some standard math stuff.
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  25.  @frskatefun  Each property, countable and uncountable, can be leveraged for different purposes in mathematics (though it's notable that the term would be useful even if only one of the properties provided additional utility). Essentially all of calculus is built on uncountable sets and their odd properties. Similarly, on the other end, countable infinities allow for stuff like mathematical induction. This is a form of proof where you prove something true for a base case, and then prove it true for n+1 if it's true for n, and therefore prove it for all cases. For example, say you want to prove that all the naturals up to n added together equal (n(n+1))/2. You'd prove this where n=1, which is true because (1(1+1))/2 is 1, then you'd prove that, if the sum of numbers up to n is (n(n+1))/2, then the sum of numbers up to n+1 is ((n+1)(n+2))/2. Which, in turn, is easily shown by noting that the above is equal to (n(n+1))/2 +n+1. So, yeah, that's a thing you can do when all the things you're trying to prove stuff for are natural numbers. You need to do some serious legwork if you want to do similar for an uncountably infinite set. Also, sometimes you just want to do some mappings between infinite sets, and you want to know what's possible. The property is intrinsically valuable in this way.
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  26. Dude, dictionary definitions of words are not mathematical proofs. And infinity-infinity is neither zero nor infinity. Or, I guess, both of those things. It's an indeterminate, meaning it can take on infinitely many different values. And you can absolutely add one to infinity. The set of all natural numbers is infinity. -1 is not a natural number. If I add the element -1 to the set of natural numbers, then I will have added one thing to infinity. A basic reality of mathematical proofs is that doing something is the greatest most unassailable proof that that thing can be done. If I create a way to put someone in each of infinitely many rooms, then all the talk about how it's theoretically impossible is clear nonsense. So, again, what's the issue with the methods that have been listed? Any problem needs to be directly with the methods themselves. Once you have proved that the methods do not work, then you can go after theoretical underpinnings.
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  27. It is quite possible for infinity to be "full". The filling set just needs to be infinite as well (and a same or greater size of infinity at that). Consider a set, say the natural numbers. Infinite elements. Now consider a second set that you're going to use to fill the first. Say, I dunno, the natural numbers again. Now, "fill" each element from the first set with the same element from the second set. So five is filled by five, twenty is filled by twenty, and so on. No element of the first set will go unfilled. Notably, it is also the case that no element of the second set will go unused. We can do this, this one to one mapping, with all manner of infinite set.
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  28. Sure, we don't physically see infinite things, but that doesn't mean pondering infinity is useless. This scenario has some pretty direct extensions into mathematics, and math has tons of obvious applications. This is really just a restatement of the general notion of mapping a countably infinite set into itself, which is a pretty straightforward idea when all is said and done. As for whether calculus, and math in general, would be way different if we replaced infinity with massive numbers, the answer, straightforwardly, is yes. Not because we actually need that level of precision in order to have a workable model, but because infinity actually just makes things a lot easier. The derivative of x^2 is 2x. Super easy. Now, instead, imagine figuring out the slope by physically zooming in on that curve and taking measures of slopes that are progressively closer to the one you want to measure, until you get a level of approximation you're comfortable with. The result would be, like, incredibly close to 2x without actually being 2x, and you'd presumably have to determine a result whenever you want to figure out the answer at a different point. And that's one of the more straightforward applications of calculus. There's this weird misconception that calculus makes things harder (and, to be clear, calculus without the infinite really wouldn't be calculus in the first place), but what it really does is take the ostensibly discrete natural world and render it in a much easier to work with continuous way. A lot of things we do with infinity just aren't all that plausible without it.
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  29. You have your infinite rope, right? Well, imagine taking another infinite rope, and putting it right next to the first. They're identical. Every point on the infinite rope is accounted for by some point on the other infinite rope. If we change it so that the original rope is a hollow tube, and the second rope just a regular rope, then we could take the rope and put it in the tube, and the entirety of the tube would be filled. The tube is endless, but every part of it can be accounted for. This is the same as the hotel and the guests.
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  30. Why do you assume that we have to start at the beginning and proceed forward from there? Maybe we just pick up the entire rope, crack open the tube, and drop the rope inside. Similarly, what if every guest is simultaneously teleported into the proper room?
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  31. We're already dealing with infinite objects. This problem is an essentially mathematical one, not a physical one. Consider this as an alternative method though. Instead of having any guest travel from outside the hotel to inside it at first, each room is just built with a guest inside. Each room includes a bed, a TV, a bathroom, and a guest.
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  32. It's not really a riddle. It's mostly an explanation of how mappings function between countably infinite sets. Which, y'know, is useful in math. Math has infinity all over the place,
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  33. The first guest was moved to room two.
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  34.  @sinisterministertv3622  There are more numbers in any interval of real numbers than there are natural numbers. The mode of proving this is Cantor's diagonal argument. First, you need some notion of size that extends to infinite sets. For this, we say that two sets are the same size if you can pair elements of the first set with elements of the second set on a one to one basis. So, {1,2,3} is the same size as {5, -1, pi}. Any pairing with the set {4, 5, 6, 7} would fail to account for at least one element of that set, so we say that this new set is bigger. Now compare, say, the set of naturals to the set of all integers. We may expect the integers to be larger, but this is mistaken. We can order the naturals straightforwardly as 1, 2, 3, 4, 5..., and the integers somewhat less straightforwardly as 0, 1, -1, 2, -2... and creating the mapping is as easy as pairing the first element of the first set with the first element of the second set, second element to second element, third element to third element, and so on. From this we may conclude, and correctly so, that creating an ordering of the elements of an infinite set is equivalent to showing this set is the same size as the naturals, and that the inability to do so means the set is bigger. So, can such a list be created for the real numbers between, say, 0 and 1? Assume, for the sake of argument, that we can. Let's write out such an arbitrary list: 1. .008123491... 2. .423419056... 3. .123412342... 4. .512512341... and so on. This list, by assumption, features every real number between zero and one. I will now generate a new number, and thus demonstrate the premise of the list's existence contradictory. Make a number where the first digit is equal to the first digit of the first number, the second digit is equal to the second digit of the second number, and so on. So, .0235... Now, change each digit. I'll add one to each, and 9's become 0's. So, .1346... This number cannot be on the original list. It can't be the first number, because the first digit is different. It can't be the second number, because the second digit is different. And so on. Any list will have this problem. Thus, this interval of reals is bigger than the naturals, and the same argument applies to any such interval. Infinity is weird. This video was about how sets with the same size can be mapped to each other, but, as you can see, it's not trivial to identify that the sets in question are the same size. The infinite buses are as big as the hotel, but this other set is bigger.
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  35. Seven is just the exponent used when someone's seat is seven. All exponents get used.
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  36. All the rooms start out full. The only reason you can move existing guests into these full rooms is because you're moving the people in those rooms to previously full rooms.
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  37.  @PlayzBlanston  Infinite rooms can be full. Simply put, for each room put a person into it. Or even have a person be in it. You're presumably comfortable with the stipulation that each room have a bed, a bathroom, and a painting of a horse in it. What's wrong with the stipulation that each room have a person?
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  38. Simply put, for each room put a person in it.
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  39. I'm not entirely sure what you mean. Each initial hotel guest is placed in room 2^n, where n is the room number. This is the map for every single hotel guest. The first bus is mapped to 3^n, where n is the seat number, and then 5^n, and so on. Where's the crossover here? My big problem with this mapping is that it fails to hit all the rooms, when a stated goal has been maximum occupancy. I prefer mapping each initial hotel guest to successive rooms with 2 as the smallest prime factor. Then the first bus gets mapped to rooms with 3 as the smallest prime factor, then 5, then so on. Pick some guest to assign to room one, and every room is accounted for exactly once.
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  40.  @clarkkent3730  I have no idea what you're saying with this "finite reasoning" stuff. Hilbert's Hotel is ultimately just an analogy for the fact that there are multiple possible mappings from the set of natural numbers to themselves. Mappings between infinite sets are a thing, as much as you might think of it as a finite notion. You say that you cannot have infinitely many finite things, but this is blatantly untrue. Most infinite sets are constituted of some quantity of finite things. I see this kinda thing a lot. People make more out of infinity than it really is, turning it into some sort of bizarre philosophical object. It's frankly pretty straightforward. If there is not a natural number of things in a set, then the set is infinite. There is no natural number that can describe the number of guests. That is all that is required for infinite guests.
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  41.  @atheistontheroad4545  There is no one displaced by this process. Tell me this: if there is a guest that does not have a room, which guest is it? In other words, name a single guest for whom I cannot tell you what room they are in? Infinite sets are analogues to infinite hotels in the sense that they are both infinite. The fact that the hotel is infinite absolutely does make the scenario work, precisely because it means there is no last room. This is why the hotel needed to be infinite in the first place, and not just mysterious or something.
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  42.  @atheistontheroad4545  The first guest I move winds up in room two. Thus, this is not the guest without a room. The guest in room two actually moves to room three, not room four, but that's kinda irrelevant. The process does continue forever, but you don't really need anyone hanging out in the halls. You could have every single guest simultaneously teleport to the next room and it would work just fine. There is no guest that moved to a room that was not initially unoccupied. Every room begins occupied. This method of person movement works just fine regardless. Every single solitary person, after every movement is done (which can be accomplished via teleportation if you like), winds up with exactly one room. That includes the first guest, the new guest, and every other guest.
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  43.  @atheistontheroad4545  Room two isn't occupied, because the person who was there is now in room three. The room starts out occupied, certainly, but we can easily make it not occupied. Person one can move to room two, meaning that person one has a room. This is possible because person two can move to room three, which means that person two is also not roomless.
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  44.  @atheistontheroad4545  Room two starts out occupied. It's just that that guest moves. There's no need to kill guest two. He's just moving to room three. You're also just wrong about countable infinity. If only every even room were filled, that would still be a countably infinite set. There are infinitely many mappings between a pair of countably infinite sets that miss elements from one set, the other, or both. What's important isn't that every mapping be bijective, but that there exists a bijective mapping. If guest two were missing, we could still make a mapping from that set of guests to the whole hotel.
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  45.  @atheistontheroad4545  Of course guest 2 can move to room 3. Guest 3 is moving to room 4. There is no empty room in the hotel at the beginning. No occupancy. It is impossible, at that moment, to add a person. The process of moving every person up a room is generating an empty room.
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  46.  @atheistontheroad4545  I'm telling you the exact way to move literally everyone. You're just arbitrarily calling it magic. I can tell you exactly which guest is in exactly which room at the beginning. Name a room and I'll tell you which guest is inside it. This means the hotel starts out occupied. After adding a guest, I can, again, tell you exactly which guest is in exactly which room. Name a room and I'll name the guest, or name the guest and I'll name the room. No guest is left out, and no guest is forced to roam the halls. This means that a guest can be successfully added. There's really not much else to it.
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  47.  @atheistontheroad4545  Of course. At the beginning of the situation, the hotel does not have any vacancies. That's what I meant when I said that every room has a guest and every guest has a room (where I am capable of naming any room's guest and any guest's room).
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  48.  @atheistontheroad4545  Sure.
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  49.  @atheistontheroad4545  There is no contradiction here. You are misstating what I agreed to. Hilbert's Hotel does not have any vacant rooms at the beginning. Later, after taking a series of actions, Hilbert's Hotel does have a vacant room. Look, I told you I could do something earlier. In particular, after everyone has moved, I can tell you exactly where every guest is. This can be with either a vacant room one, or with a room one occupied by the new guest. Do you think I cannot do this? If you think there is a roomless guest, name that guest. If you cannot name a roomless guest, then it is absolutely the case that a vacant room can be produced. I am floating no one, and we know this to be the case because the location of no guest you name will be "in the halls".
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  50.  @atheistontheroad4545  You have it exactly opposite. I can absolutely move guest one to room to room two because I can move guest two to room three because I can move guest three to room four and so on. No person is floated. As you yourself note, the task I have set before myself, telling you which guest is in which room, is trivial. Of course it's trivial. This isn't a particularly hard problem. You say that, if I tell you which guest I have moved, then you can tell me what room is double booked. I have moved every guest, and I have moved them each one room upward. Now, what room is double booked? Which guest is being "floated"?
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