Comments by "eggynack" (@eggynack) on "TED-Ed"
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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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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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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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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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@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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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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@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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@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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@tonyh8371 The actual definition of infinity is pretty straightforward. A finite set is one that contains a number of elements equivalent to some natural number. The set {1,2,5,6}, for example, is finite because it contains four elements, and four is a natural number. An infinite set is one that is not finite, so therefore one for whom no such natural number can be assigned.
Note that nothing in this definition demands that the set be continually growing, or that it grow at all. Nothing of it demands that it never be "filled", or that it contain all possible elements. If a set is not finite, then it is infinite.
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The short version is, for every room, there is a person inside. That's the definition of fullness at work here, and it's pretty sensical, I think. To flesh this out a bit, a medium version if you will, I like to consider the nature of the room. Think about one of these rooms. You wouldn't think it wild if I were to say that every single one has a bed inside. A bath. You would be especially unsurprised if I were to say that every room has walls, a floor, a door. This stuff is just to be expected. Why, then, is it odd to say that every room contains a person? Maybe I even do a straight replacement. Where once there was a bed, now there is a guest. It's not like guests are magic.
For the marginally long and somewhat more rigorous version, you can just do the mapping straight up. There are infinite guests, yeah? So we line them up, assign each a number based on their place in line. So there's guest one, guest two, guest three, guest four, and so on. Now, we send guest one to room one, guest two to room two, guest three to room three, and so on. This definitely fills the hotel. And provably so. Name any room and I can tell you not only that the room is full, but also exactly which guest is inside. Every room has a guest, the guest with the matching number, and so the hotel is full.
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It's a number that has exactly two factors, one and itself. So, like, 6 has four factors, 1, 2, 3, and 6, so it's composite. 2 has two factors, 1 and 2, so it's prime. The primes are like 2, 3, 5, 7, 11, 13, 17 and so on. There're provably infinitely many of them.
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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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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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@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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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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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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@sravansuhas2931 It's not a countable number. It's a countable infinity. A countable infinity is basically one where you can put all the elements into an infinite list, such that, for any element, you can point to some position on the list and say, "Here it is."
The natural numbers are the canonical countable infinity. The standard list for those is 1,2,3,4,5,6... So you're like, "Where is 123," and I'm like, "It's in the 123rd position on that list." Lots of infinite sets are countable. The integers, the evens, the rationals, and, indeed, the set of people on that bus.
And a noteworthy fact about countably infinite sets, the main definitional thing about them even, is that they are all the same size. They all have the same amount of elements.
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To the first question, imagine lining up the guests and assigning each a number equal to their place in line. Now, put each guest in the room whose number matches theirs. At this point, there will exist no room that does not have a guest, so the hotel is full.
To the second question, a countably infinite set is one where you can construct an ordering of that set such that you will arrive at any arbitrary element of that set in finite time. So, for example, the counting numbers are a countable set, because you can order the elements as you would ordinarily count them. If counting n numbers takes n units of time, then reaching some number n takes, obviously, n units of time. Other countably infinite sets include the evens, the integers, the rationals, and even those numbers that can be formed by a finite algebraic expression (such as the square root of 2). However, the set of real numbers is uncountable.
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Like, if your ordering is 1, 1/2, 2, 1/3, 2/3, 3, 3/2... to put together something arbitrary, you'd just have to count all those numbers up to 3. The ordering by size wouldn't enter into it, but you can't just say, "1, 3, and I'm done."
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@koko10900 "Once claimed that you have an infinite amount of one substance, all of that substance is now in that infinity."
Untrue. Consider that, if I have all the natural numbers, then I have an infinite amount of numbers. However, all of the substance that is numbers is not contained in that infinity. For example, -1 isn't contained in my set. Similarly, if we were to label every person in this imagined infinite person universe with an integer, and then had only the positive labeled people start out in the hotel, then that would leave infinitely many negative labeled guests.
"And if it is the fact that the infinite hotel is filled by an infinite amount of people, there is really no point differentiating between the two entities. This is because, as soon as I've room seemingly becomes available, it has to be filled. Unless this is true, at one point in time it has to have been considered empty. If the pairing is instant in which this scenario would seem to suggest, room = guest, which makes them one entity because they are necessary for eachother in this scenario."
Also not true. In fact, the whole point of the video is that it's not true (and the video is correct in this). There are many different mappings of guests to rooms. We can assign them in a way that fills the hotel very easily. Person one goes to room one, person two goes to room two, and so on. But we can also assign them in a way that leaves an empty room. Person one goes to room two, person two to room three, and so on. Or we can assign them in a way such that infinite rooms are empty. So one goes to two, two goes to three, three goes to five, and person n goes to the nth prime number.
"It is through the language used that you assume you have a defined amount of rooms, "infinite". You will always have the amount of rooms for any amount of guests because any unending space can fit an unending amount of things."
Well, this is only true if we're solely considering countably infinite sets. The issue of uncountable sets is bit beside the point though. What's important here is that you are essentially correct. The hotel rooms can indeed accommodate the new guests. That's the point. The hotel can accommodate any countable set of people. It's just that the hotel can also have one person in every room when starting out.
"To say that this space is full, is to define it in a measurable way thereby taking away it's quality of being infinite."
No. You literally defined infinity up above. Infinite means never ending. It does not mean impossible to measure or define. The basic reality is that there are ways of measuring and dealing with infinity.
"Meaning that the infinite hotel will never be full because it simply HAS rooms. Not has 10 rooms, not 11, not 500, just simply HAS rooms."
Just magically having rooms is also not a quality of infinity. The rooms are endless, yes, but so are the guests, and it is quite possible to line the endless rooms up with the endless guests.
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@koko10900 "I'm going to be honest this will be my last response because clearly you aren't understanding me."
I understand you reasonably well. You are just wrong for the most part. It's your prerogative to respond or not respond.
"If you would like to measure infinity, by all means, do what no one has ever done before. Given that any infinite thing has no defined quantity, good luck."
Gregor Cantor predates me by about a century and a half, and the broader study of infinity has proceeded just fine both before and after him, so the idea that no one has done this before is wildly inaccurate.
"Having a specific set of infinity doesn't by any means, mean you've measured it."
No, that alone does not constitute a measurement. What does constitute a measurement is determining the cardinality of the sets involved. Two sets have the same size if you can pair the elements of one to the other on a one to one basis. All countably infinite sets have the same cardinality. Uncountably infinite sets have greater cardinality.
"Secondly, once you've made claim that you have an infinite number of people, you've claimed all the whole numbers of people that can exist."
Nope. I started with the integers and picked the positive ones, but I don't have to do it like that. Say I start with the natural numbers and pick out only the even ones. Even numbers are infinite, but they do not constitute all the numbers. Nor do they constitute all the people, given I have one for each natural number.
"At least we both agree on the fact you can't have negative people or fractions of them counting as people."
I think you're misunderstanding the scenario somewhat. I'm saying I'm giving each person numerical labels. This means there can, in fact, be a person -1. My stated scenario used the integers, and so there was a "negative person". It is not strictly necessary that we involve people with negative or fraction labels though. It's actually possible to break any countably infinite set into infinitely many countably infinite sets, so we can narrow things as much as you like and this still works just fine.
"You've claimed the basic infinite set of whole numbers. Adding one to it is also within that set."
What if I add -1 to it? Or 1/2? I've added a single element, and have something from outside the set.
"And again, this is not measured because you haven't given it a defined specific size by which I mean an alloted numbered value by which it can't be anything else."
Actually, the cardinalities I talked about above do have specified values. All the infinities we've been talking about have the cardinality aleph 0. The likely next biggest infinite set, aleph 1, includes sets like the set of all real numbers (I say likely because, as I recall, the existence of a set with cardinality between those two is undecidable given standard axioms).
"You further haven't answered my main claim in that at some point there will have been a vacant and so non full room, unless of course the existence of a room means it's filled in which case this thought experiment is pointless because again you're creating two things that can't exist separately of each other."
This is pretty straightforward. Using this infinite set of people, you can fill all the rooms. By moving the set of people around, you can empty a single room. You never have to have an empty room, but you can have one if you want to.
You can produce just about any result you want just by moving people around. You can leave five rooms empty, or infinitely many rooms. You can assign infinite people to every room, or you can assign infinite rooms to every person. The possibilities are literally endless.
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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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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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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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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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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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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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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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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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So, if I'm getting you right here, then you're asking what's the minimum area you could enclose with an infinite quantity of fence, assuming you can't just use a single straight line. You need some caveats to even work with this in the first place. The biggest, naturally, that such a fence can't exactly be said to enclose anything, because one of the sides will be at infinity and thus not exist. That being said, we can define the area of this object as the space where you can draw a line from one piece of fence to another and go through that space, and assume for the sake of argument that the shape is convex.
The straightforward answer here is infinite area. You can make the area as narrow as you want, but an instantiation of this situation will inevitably have all of that area. A more complex question is what the lower bound on the set of possible areas is. After all, you can keep reducing the area of the fence, regardless of its size. I'm inclined to say that the lower bound is infinity, because all elements of the set are greater than any finite quantity, rather than zero, because you can keep reducing the size. Notably, if you have a given finite quantity of fence, all possible areas would be positive, and the lower bound would be zero.
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A hotel with infinite rooms and infinite guests doesn't have to be full. For example, we could put a guest in every even room, leaving every odd room free. In any case, the hotel does start out full. It is by moving every guest up a room that a free room is produced.
As for countable infinity, you are misunderstanding what that means. A countably infinite set is one where we can put the elements of that set in some order such that, counting by that order, we can reach any given element of the set in finite time. The natural numbers are the canonical countable infinity, and both the hotel and guests begin labelled by the natural numbers.
Meanwhile, just because there are infinite guests in the hotel doesn't mean there can't be infinite guests outside the hotel. Say, for the sake of argument, that the guests in the hotel are associated with the natural numbers, and the guests outside the hotel are associated with negative numbers. Or, returning to my first point, maybe the guests in the hotel are even numbers and the guests outside are odd. It is, in fact, possible to have infinitely many countably infinite sets that are wholly disjoint from themselves and each other.
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Still a resolvable problem. Let's assume that the guests can reach any given room in any desired finite length of time, and that they do indeed need to enter one at a time (which, honestly, not sure why that'd be the case. The lining up thing was just to assign rooms, not a method of entrance). The first person reaches their room in a second, the second person reaches theirs in half a second, the third person in a quarter second, the fourth person in an eighth second, and, generally speaking, the people reach their room in 1/2(n-1) seconds, where n is their place in line. After two seconds, every person will be inside their room.
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There's no division involved. Put all the people in a line, and label each person with a number based on their position in the line. Send person one to room one, send person two to room two, send person three to room three, and so on. In this fashion, for each room there will exist one person occupying that room.
The impossibility of division here doesn't imply that it's impossible to assign people to rooms such that this specific thing can happen. What it implies, if it implies anything, is that you can assign people to rooms such that anything you want can happen. There are assignments that put infinite people in each room, assignments that give infinite rooms to each person, assignments that leave infinite rooms empty, and assignments that leave any finite quantity of rooms you desire empty.
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If each of the infinite buses has a countable infinity of people, then both my method and the video's method will successfully house any arbitrary person in any arbitrary bus after finite time without any room ending up with more than one guest. My method will also not have any room with less than one guest.
I don't know where all this stats stuff could possibly fit in. There's no randomness involved in this process. Every person, even when infinite people are present, is assigned to a very specific room. Name any room and I can tell you exactly which guest my method will assign that guest to, and, name any passenger, and I can tell you where my method assigns that passenger. The video only accounts for some of the rooms, so for that mapping I can tell you whether a room has a guest, and, if it does, where that guest came from.
Room 128, in both mappings, contains exactly one guest. In the video's mapping, it has the guest that was originally in room 7. In mine, assuming you assign the guest in room one to room one, it has the guest that was originally in room 65.
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@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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@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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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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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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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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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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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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You say that it is not full, but you have no proof for that. I say it is full, and I do have proof for that. That being, if you name any arbitrary room, it will have a person in it. You can definitely "equal" two infinities. There is nothing to the definition of infinity that would give rise to the properties you're citing. If you were to just put people into the hotel one after the other, then, no, the hotel would not fill, but by generating a mapping from the set of people to the set of rooms, and ignoring transit time (say for the sake of argument that each person reaches their room after a second), we find all hotel rooms full.
Infinity isn't just a concept. It's a well defined mathematical construct, and nothing of that construct is incompatible with this result.
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Oh this is definitely math. Hilbert's hotel is more or less a restatement of a basic math result, that all these sets, the naturals, the naturals plus one, the integers, even the rationals, all share a size. And they do so specifically by means of this mapping notion. It's kinda weird to think that the set {1, 2, 3, 4, 5, 6...} has exactly as many numbers as {0, 1, 2, 3, 4, 5, 6...}, but it does. This gets even weirder when you take the extra step and note that the set of all real numbers is strictly larger than any of these. Neat stuff.
As for your issue with the scenario in itself, you don't have to keep anyone in transit between the rooms. Just have everyone move simultaneously. At, say, three o' clock, everyone simultaneously leaves their room and then goes into the next room. Every single guest will be inside a room by four and room one will be empty. You note the contradiction between full and not full, but this is actually quite explicable. The hotel is full in one sense, as there is a person inhabiting every room, but not full in another sense, because the hotel can accommodate more guests. As it turns out, these are distinct concepts in an infinite hotel. And if we define "full" the first way, which is a fair way to define it, then the full hotel can absolutely accommodate guests.
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There isn't some singular room n+1. Room n+1 is relative to each individual guest. So, for guest 1, room n+1 is 2, and for guest 100, room n+1 is 101. No room starts out unoccupied. However, each destination room is rendered unoccupied for the guest going to it because the person who was once in the room is going to their room n+1.
It's worth thinking about what would happen with finite hotels. Say one with 100 rooms. Person one goes to room two, person two goes to room three, and so on. Every single step here works just fine, and it does so until you have to deal with guest 100. They have nowhere to go, and it is here where the movement breaks down. In an infinite hotel, however, there is no top room. The process never breaks down.
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@jayjeckel No, I can do it because I already did it. It is what is called in math a constructive proof. You prove that a thing can be done by doing it. In this case, the question is whether there is a bijective mapping from all the guests in the full hotel plus one additional guest to the rooms in the full hotel. And there is one. In particular, this mapping is that guest n, where the new guest is guest 0, goes to room n+1. Clean bijective mapping. No new rooms or initially unoccupied rooms needed.
It might be easier for you to get a handle on it if we ditch the hotel for something more abstract. There are infinitely many primes, right? It makes sense, then, to talk about the first prime, the second prime, the fifth prime, the 100th prime. In fact, no matter which prime you want, it'll be there. Because there are infinitely many of them.
What this means, however, is that there is a one to one mapping from natural numbers to primes. The first prime goes to one, the fifth prime goes to five, the 100th prime goes to 100, and so on. Every natural and every prime is accounted for exactly once each.
We can, if we want, extend this reasoning back to the hotel. There's a guest for each natural number, because they're assigned to the natural numbered rooms. We can, therefore, cleanly map the guests in this full hotel to only the prime numbered rooms. This leaves us with infinitely many empty rooms, all the composite numbered ones (which are also infinite), and all just by moving people.
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Simply put, infinity is useful. Basically any type of mathematics you care to name will make heavy use of infinity at some level. Understanding infinity, then, has a lot of functionality to it. This hotel is essentially just a restatement of the various ways we can map the natural numbers to themselves, and that kind of thing can be pretty useful, especially in, say, abstract algebra. Or, hey, maybe this leads you to a question, like whether there is a quantity of guests it would be impossible for the hotel to accommodate, and you wind up at Cantor's diagonal argument, which is a foundational result in mathematics.
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You can always add people, as the video showed, but it's also possible to have each room filled with a person. Simply line up all of the people, and assign each person the number that matches their position in the line. Then, put person one in room one, person two in room two, and so on. Now, for any room you name, there exists a person in that room, meaning every room has a person in it. It's also possible to give each person infinite rooms, or to give each room infinite people, or leave infinite rooms empty, and any arbitrary lesser version of those.
Here's an interesting way to disprove your claim without using an example method. Let's assume that, when assigning people to rooms, there will always be rooms left. Now, let's try assigning rooms to people. The two are equivalent, with this situation able to be visualized as dropping some hotel room onto each person in turn (preferably without a floor, so they survive the experience). By your logic, it should be impossible to do this, meaning there are far more than enough people to fill every room.
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@andrea1854 That's not really a definition of dense I've seen. More common is to say something like, "The rationals are dense in the real numbers." This means that any two reals have a rational between them. Density is generally considered in this sense, with a subset considered dense in the set.
Your construction of density is somewhat troublesome. What if the interval is [0,∞]? In this case, there are infinite naturals, infinite evens, and infinite rationals. The same infinity each time as well. However, any smaller interval, say [2,3], will contain finite integers and infinite rationals. Moreover, we can identify some lower key versions of this situation. The odds are going to be more numerous than the primes over most intervals, but if we're working with something like [2,4], then there are more primes than odds, and the interval [2,9] has the same number of each. There's probably some way to formalize this idea of density, but it's pretty tricky to do.
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I understand infinity quite well. From, y'know, studying mathematics a lot. It is entirely possible to fill an infinite hotel. All you need to do is generate a bijective mapping from the set of guests to the set of rooms.
Consider a more abstract example. You have two sets, the set of natural numbers, and the exact same set of natural numbers. Can you pair elements of the first set with elements of the second set such that no elements of either set are left over? Of course. Just pair each element in the first set with the same element in the second set.
In order to extend this situation to the hotel and guests, you need simply call the first set the guests and call the second set the hotel rooms. By this means, we have our mapping. Every guest has an exact room assignment, and conversely, every room has an exact guest assignment. Room one has guest one, room two has guest two, and so on.
This is how countability functions. If you have two countably infinite sets, then, by definition, one can be bijectively mapped onto the other. Both the hotel rooms and the guests, even when the guests are on infinite buses, are countable infinities.
Take note, this is not the only possible mapping. It is very much possible to generate a mapping where exactly one room is empty, or where infinite rooms are empty, or where any natural number of rooms is empty, or where every room has infinite guests, or where every guest has infinite rooms. I can provide any of these mappings for you, should you so desire. Their existence is far from theoretical.
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@skyabyss7357 I mean, yeah, if you have a mapping to a set of rooms, and then there are more rooms, then the mapping will no longer work. As long as you're in the same cardinality of infinity though (like, if you add on some finite number of rooms, or add on as many rooms as there are natural numbers), then you can create a new mapping between the original quantity of guests and the new quantity of rooms.
Really, you're just asking the original question in reverse. By which I mean you're just asking the original question. You start with n rooms and n guests, you add a guest, and then the n+1 guests are still accommodated. This is the same as starting with n rooms and n guests, adding a room, and then having the n+1 rooms still wind up full, which is the scenario you propose, I think.
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Countable infinities are infinities, and the infinity of a giant pile of discrete hotel rooms is a countable one. If the hotel had an uncountable infinity of hotel rooms somehow, then it could only be occupied by uncountably many people, but it could be occupied by uncountably many people. The hotel cannot be full in the sense that people can always be added, but the hotel absolutely can be full in the sense that every room has a person in it. As in, if you name a room then I can tell you both that it is full and who is in it.
You say there's no mathematical solution to filling the hotel, but there absolutely is one. As I said, you simply assign each person a natural number, and have them all simultaneously go to the room that matches the number they were assigned. There is no flaw in this construction, and no way that there would remain an unfilled room afterwards. If you think I'm mistaken, please either identify a flaw in the construction, or a room that is unfilled.
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You don't have to finish one bus before starting the others. It isn't, as you note, a super effective strategy, and neither is taking one person from each bus in turn because then you'll never return to the first bus. Instead, you want to use something like the abacaba pattern. So, you label the buses alphabetically (moving onto aa and such when you hit z), and then remove the next passenger when you hit the bus with that label. For the order in which you select the buses, you can use abacabadabacabaeaba... In this fashion, you hit every bus infinitely many times, and across the same sort of time span as you would a single infinite bus.
Alternatively, you can use a system like what you use to prove that the natural numbers share a cardinality with the rationals. Label each bus with a number, and each passenger in each bus with a number. Then, set out the natural numbers along an x and y axis, generating a grid where the x axis defines a bus and the y axis a passenger. Finally, you draw a zig zag line that crosses over every number pair, and pick out bus/passenger pairs in an order defined by that line.
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@asagk Of course I can quantify that assumption. Consider the natural numbers, say. Is there some absurd quantity of them that we just haven't reached? Let's assume there is, for the sake of argument, and assume this quantity is n. Thus, the set is 1, 2, 3, 4... n. Well, n+1 is also a natural number. Thus, there are at least n+1 natural numbers. This means there is a contradiction in our initial assumption that there is some finite quantity of natural numbers, meaning the set must be infinite.
As for countability, you're just misunderstanding what that term means. Simply put, if set is countable it means that we can construct some list such that, given any element in the set, we can reach that element by counting along the list. Returning to the natural numbers, I already laid out a basic list of them. 1, 2, 3, 4, 5... Name any natural number and it will show up at some finite position on that list.
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Again, you can't just subtract infinity from infinity. This subtraction has an infinity of results, and thus no singular result. The guests can indeed be paired up one to one with the hotel rooms. There is the same quantity of each. However, it is 100% provably the case, and I literally just proved it was the case in that post, that the guests can be paired up with hotel rooms such that there is one hotel room left over. There are different sizes of infinity, yes. However, any two countably infinite sets, for example the set of natural numbers and the set of natural numbers plus or minus some number of elements, will be exactly the same size every time. They are different sets, but one is in no way bigger than the other.
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We do not know that the result will be 0. We know only that the result can be zero. Again, the result can be any number of things, zero included. As for your other statement, set B isn't just set A with all the elements added to. Set B is set A with the element zero added in. In other words, you had infinite objects, and then you added another object.
This situation is identical to the hotel scenario. You had infinite guests, and then you add a guest, and you have the same number of guests. In fact, we could just label the new guest 0, and all the old guests as natural numbers, and the proof I presented would show that the number of guests hasn't changed. Infinity doesn't always equal infinity, but every countable infinity, definitionally, does equal every other countable infinity. I can prove this for any two countable sets you care to name, but, given that the definition of countable infinity is the existence of a bijection to the natural numbers, it feels unnecessary.
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Okay, start with the hotel full of guests. Guest in every single room. We can take all the guests out of all the rooms, and then put every single one of them back into only odd rooms without leaving any guests over. I'ma change it to even rooms for convenience's sake. Label each guest with the room they're in. Now, take those guests and put them in room 2n, where n is their label. So, guest one goes to room two, guest two goes to room four, room three goes to room six, and so on. Exact same rooms, exact same guests, but now our "subtraction" yields infinity.
As for the two sets, it's really pretty straightforward. Set A is the integers from one upwards. Set B is the integers from zero upwards. Set B thus has one more element than A does. Now, we will pair each element of set A with each element of set B. One from set A pairs to zero from set B, two from set A pairs to one from set B, and, in general, n from set A pairs to n-1 from set B. Thus, every single element of A is mapped to exactly one element of B. So, the sets are the same size.
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Infinity isn't switching size. infinity=infinity/2=infinity-1. The reason using algebra doesn't work here is because changing a thing into x doesn't magically change its functioning. Infinity has a set of properties. Finite numbers have a different set of properties. You find those properties somewhat unintuitive, but what I've been presenting is a set of totally accurate proofs. Skip x entirely. If you want to say infinity, say infinity, because infinity is the thing you're working with. In using x, you're trying to generalize something that can't be generalized, and then making it specific to a set it doesn't work for.
Straightforwardly, by saying that x/2=x, you're saying something that hasn't been proved, because you only know this to be the case for infinity. X is a thing you solve for, not a thing you put stuff into such that it's true. If x/2=x, then x is 0, or infinity. The reason this doesn't work is the exact reason it doesn't work to say that x/2=4, and then conclude that, because we can replace x with anything, 16/2=4.
As for the sets, yes, there is an element in B that is not in A. That's the whole point. In spite of this, it is provably, and I've proved it for you, the case that the sets are the same size. The natural numbers are also the same size as the integers, and the rationals, and the primes. These are different sets, some proper subsets of others, but they are all the exact same size.
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@anonygent No, no one is taking up the odd numbered rooms. The situation is that the hotel is full, then the buses arrive, and then the initial guests are moved to the even rooms, and then the buses are moved to the other rooms. Based on my stated mapping. This weird middle step where guests take up the odd rooms is arbitrary and irrelevant. If it occurs then the hotel is full and then you can just run my mapping from the beginning.
Moving people to p^n is in no regard a more permanent solution than mine. After you're done, and new infinite buses arrive, you still have to move guests around to accommodate the new guests. It's not all that much harder to calculate either. You just have to sort by size the numbers with the stated quality. So you have, say, 11, and 11^2, and 11*13, and 11^2*13 and so on.
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@anonygent No, no one is taking up the odd numbered rooms. The situation is that the hotel is full, then the buses arrive, and then the initial guests are moved to the even rooms, and then the buses are moved to the other rooms. Based on my stated mapping. This weird middle step where guests take up the odd rooms is arbitrary and irrelevant. If it occurs then the hotel is full and then you can just run my mapping from the beginning.
Moving people to p^n is in no regard a more permanent solution than mine. After you're done, and new infinite buses arrive, you still have to move guests around to accommodate the new guests. It's not all that much harder to calculate either. You just have to sort by size the numbers with the stated quality. So you have, say, 11, and 11^2, and 11*13, and 11^2*13 and so on.
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All the buses have the same amount of guests as one bus, but you still need to determine where any given person goes. You say you're just sending all the bus folk to odd numbered rooms. Where do you put the fifth person from the third bus? You can't associate people in the first bus with odd rooms in sequence, because then the second bus would have no odds left. Similarly, you can't associate the first person in each bus with the odd rooms in sequence, because then no one else on each bus would get a room.
Personally, I'm not a big fan of the method in the video. The rest of the video makes a big deal of keeping the hotel fully occupied, and then they ditch that for the infinite buses. My preferred mapping is that people in bus n go to rooms where the n+1st prime is the smallest prime factor, and the hotel folk go to even numbers. Or, the first guest goes to room one, and then they fill up evens, so every room gets filled. So, the fifth bus would have its guests go to rooms that are divisible by 13, but not by 2, 3, 5, 7, or 11. This method should assign every guest to a room and every room to a guest.
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If you're not doing something that involves high level math on at least some level, then you're unlikely to require this kinda knowledge. Still, I'd hesitate to call it totally pointless. Math forms the underpinnings for physics, frequently with infinity involved somewhere, and physics dictates the mechanics of everything in existence. Similarly, I'd think that people in these non-mathy careers may take some interest in the theoretically infinitely rough structure of a fractal, which is a concept that arises in most natural constructs.
In short, infinity helps us understand reality, and basically anything you do is going to touch upon that reality on some level. So, I dunno, it might come up.
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It's not really clear how, exactly, you're filling these rooms that aren't straight up powers. Where does, say, bus three seat five go? That's really the goal here, to get some explicit mapping of people to rooms.
That said, it is possible to fill all the rooms. My method is that bus m seat n goes to the nth number where m is the least prime factor. So bus one goes to numbers divisible by 2 (2, 4, 6, 8...), bus two goes to numbers divisible by 3 but not 2 (3, 9, 15, 21...), bus three goes to rooms divisible by 5 but not 2 or 3 (5, 25, 35, 55...) and so on.
The only caveats are that we call the hotel bus one, the first bus bus two, and so on, and that we start bus one with room one and then continue the sequence from there. This method creates a one to one mapping between guests and rooms.
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Infinity-infinity is not necessarily zero. It's an indeterminate, meaning it can have basically any result. You've provided a good example of this. If you have a person for each natural number, and you remove all the odds, then you've subtracted infinity from infinity and been left with infinity. You could also remove all but the first guest and thus have the subtraction yield the result of one.
The notion of "half" is weird with infinity, and you really have to rigorously define how you're using it. After all, there are just as many odd numbered rooms, countably infinite, as there are rooms in general. The hotel is neither empty nor full. It is not empty because there are full rooms, and it is not full because there are empty rooms. That part, at least, is relatively straightforward.
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@drawingkong5091 There is no spare room. The infinitude of the hotel allows the guests to go up in rooms even despite that absence. Look, consider a finite hotel. 100 rooms, all full. A new guest arrives. Guest one goes to room two, guest 50 goes to room 51, guest 99 goes to room 100, and then guest 100 has nowhere to go. The chain breaks down specifically and only because guest 100 has no room 101 to go to. Or, to be more specific, because the hotel is finite. After all, the 101 room hotel just kicks the can down the road by a room. With a finite hotel, the chain terminates at the last room.
In an infinite hotel though, there is no last room. Every guest has a place to go. And we can observe this fact directly. Name a guest and I can tell you exactly where they wind up. Guest one winds up in room two. Guest 1000 lands in room 1001. Guest 712341 goes to room 712342. It never stops working. And this is in spite of the fact that there is no empty room, no specific place that accommodates a new guest. We can add one new guest or a thousand or infinitely many and it works out just fine.
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@tonyh8371 Empirical evidence for infinity, especially this weird philosophical version of infinity, doesn't exist. That quote wasn't the only thing I said regarding ants. If an ant told me that TED Talks were an objective thing in the real world, I would ask what basis it has for that idea. After all, it has no observation it's using to make that claim. The ant arbitrarily happened to be right in this case, but it was a random guess on the ant's part, and not some ultimate truth, that lead to that reality. If we are as an ant to YouTube as regards our relationship to infinity, then what you're saying is that, when you say infinity is a real objective thing, you are making an arbitrary guess that has an incredibly small chance of being correct.
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@davidb.854 I have presented you a process whereby every person, including the new guest, winds up in a room. Could you please identify a flaw in the process? To be clear, the process is that person 0 goes to room 1, person 1 goes to room 2, person 2 goes to room 3, and so on, and this all happens simultaneously. Which person does not have a new room assigned to them? Where does this mapping fail? My claim is that, for any room, I can tell you exactly which guest is in it, and that, for any guest, I can tell you which room they are in.
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