Comments by "eggynack" (@eggynack) on "TED-Ed"
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@atheistontheroad4545 I can absolutely move them simultaneously. Try this as a method. At 3:00, every guest leaves their room. At this moment, all the rooms are unoccupied. Then, each guest simultaneously moves to where the next guest, the n+1th guest, was previously standing. This is done by 3:01. Finally, at 3:02, each guest moves into the room they are in front of. The rooms start out occupied, then they are unoccupied, and finally they are reoccupied by different guests.
The difference between your random dream scenario and my scenario is that mine is a mathematical methodology. I am, in a sense, describing a plan. A means of doing a thing. Maybe my means is effective, or maybe it isn't, but either way, things I don't mention are not a part of the methodology. The only valid challenge is stating something impossible about the method. Naming some arbitrary other method is a complete non sequitur.
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@atheistontheroad4545 What you are missing here, and this is the essential thing Hilbert's Hotel was meant to demonstrate, is that there exists more than one mapping from the natural numbers to themselves. More than one injective mapping, in particular. The mapping that is first presented in this video is the most straightforward one. 1 maps to 1, 2 maps to 2, and, broadly, n maps to n.
The second mapping though, that one is more interesting. 1 maps to 2, 2 maps to 3, 3 maps to 4, and, broadly, n maps to n+1. Note here that the first number in each mapping corresponds to the guest and the second to the room. This is a wholly valid mapping. No guest is left out, and no room is double booked. And, of course, you may note that room 1 is left out of the mapping. This is where guest 1 goes.
Even that mapping is relatively straightforward though. It's easy to generate any natural number quantity of vacancies, or infinitely many vacancies. It's possible to assign infinite guests to every room, or infinite rooms to every guest. The video produced a mapping from a infinite quantity of countable infinities to a single countable infinity, but it could have done so without leaving a single room empty, and without leaving a single guest roomless. I could demonstrate any of these mappings, if you like. You're just kinda getting caught up in the very oddity that Hilbert's Hotel is showing to you.
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You move each guest to the room one greater than their own. These rooms were indeed full, but they are rendered not full because the guests in those rooms are also moving to a room one greater. Imagine doing this in a finite hotel, every guest moving up one room. It wouldn't work, because the guest in the top room would have nowhere to go. However, an infinite hotel has no top room, meaning that this process is never stopped.
As for it taking infinite time, it depends on how people move. If you just assume everyone magically learns that they have to move simultaneously, then it'd take as long as it does for one person to change rooms. If you have to send up a signal, then it would take infinite time, but you never really hit any issues there. As long as the signal is sufficiently fast, each person can take the next room near immediately after they get the signal, and subsequent signals could ride in waves after the first. For a guest, it'd be just like for a finite hotel where this occurs.
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The method in the video is that, for bus m and seat n, that guest goes to the nth power of the mth prime. So, bus four seat three would go to 7^3, or 343. You just have to assume that the guests in the hotel constitute the first bus and it works out. I don't actually like that method overmuch though. A big part of the video is eliminating vacancies, and this method produces infinite vacancies.
Instead, I like having each bus still correspond to prime factors, except now the guests go to rooms where that prime factor is the least prime factor. So, guests in bus one (which would be the hotel) go to the rooms where 2 is the smallest prime factor. That's every even number. Then, guests in bus two go to rooms where 3 is the smallest prime factor. So, rooms divisible by 3 but not 2, which means 3, 9, 15, and so on. The next bus goes to rooms where 5 is the least prime factor, and so on. Assign one person to room one, say the first person in the first bus, and every room is full.
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A core question you always have to ask is, where do people wind up? With a single bus, assigning old guests to twice their original room and the new to all the remaining odds, we know where everyone lands. Original guests and busfolk alike. However, with infinite buses, where do they go? Guest one moves to room two, then four, then eight, then sixteen, and so on. Any room you name, it's not where they end up, and there's no room infinity for them to go to. It's thus not an effective mapping of guests to rooms.
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You don't have to finish the first bus before starting to unload the next bus. You could, for the sake of argument, use the following classic pattern, unloading the next guest from a bus when that bus' number is named: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1... In this fashion, every bus pops up infinitely many times.
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You're making a lot of assumptions about how the rooms are being assigned. Neither subtraction nor division need be involved in the process. Instead, let's use the most straightforward bijection imaginable. Label each of the infinite people, 1, 2, 3, 4, 5... Next, label each of the rooms 1, 2, 3, 4, 5... Then, put each person into the room with the matching number. So, end of the day, name me a room that doesn't have a person in it? It is impossible.
I'ma take a bit of extra time here to talk about some of the extra stuff you said. Infinity minus infinity is, in fact, undefined, but infinity divided by infinity is definitely not zero. It is also undefined. Your very website says so, though there are obviously non-website reasons for this. An important thing to note here is that my stated assignment is far from the only one, and far from the only result. It's possible to give people rooms in a way that leaves infinite rooms over at the end. It's possible to assign a countable infinity of rooms to each person. It's possible to assign a countable infinity of rooms to each person and still have a countable infinity at the end. For every natural number, and also countable infinity, you can leave exactly that many rooms open. Of course, you can also leave zero rooms open.
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The short answer is, by the mechanism stated in the video. The longer answer is that, while infinite rooms and infinite guests can lead to infinite full rooms, it can also lead to many other possibilities. Those exact same guests could, just by moving around, leave any finite number of rooms, leave infinite empty rooms, give infinite rooms to every guest, or put infinite guests in every room. This is because all countably infinite sets have the same amount of elements, so the set of all rooms is exactly as big as the set of all even rooms, or all prime numbered rooms.
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You kinda missed the point of a lot of what I said. To your first claim, do you agree that 1+1=2? If so, then proving that infinity+1=infinity would be sufficient to show that infinity is special in this regard. So, we'll start with the set of all positive integers, here assumed not to include zero. Then, we will compare it to the union of the set of positive integers with zero. To prove these sets have the same size, we need merely construct a mapping from one set to the other. The mapping in question will be to take each element from the positive integers plus zero and add one to it to get an element of the set of positive integers. So, 0 maps to 1, 1 maps to 2, 2 maps to 3, and so on. In this fashion, every single element from the first set is paired with exactly one element from the second set. However, the second set is the first set with one additional element, so we have taken the first infinity, added one to it, and reached an infinity of the same exact size.
To your second claim, I think you've just straight up misread me. I didn't say that infinity is a number. I said that, whether infinity is a number or not, we can add to it. If your definition of number doesn't include infinity, then, well, I guess there's just another type of thing we can reasonably add numbers to. Who cares about the word "number" anyway?
To the third, again, I have literally no idea where you're getting the idea that that's what I said. The hotel is not full with no one in the first room. What I said was that you can put infinite guests into the hotel and not necessarily fill it. You referred to putting infinite guests into the infinite hotel as subtracting infinity from infinity, but my point was that this subtraction has a ridiculous variety of results. You can take the same exact quantity of guests and reach a ton of different levels of hotel fullness.
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Well, the core goal is to get each person into a room, yeah? Like, for any person you can name the room they'll land in. So, where does the guest who started in room one end up? They first move to two, then four, then eight, then sixteen, and so on, moving up the powers of two. But, for any single power of two you name, that's not where they'll be, cause they then moved twice as high right after. There is, in point of fact, no room they are in. You moved them up infinitely many times, and there is no room infinity. As a result, this method hasn't actually succeeded in rooming any of the guests. Neither the ones that started there nor the ones that arrived later.
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