Hilbert's Infinite Hotel - 60-Second Adventures in Thought (4/6)
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- Опубликовано: 14 окт 2024
- A never-ending hotel, always full of guests, helps to explain the nature of infinity.
(Part 4 of 6)
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The paradox is that one infinity can be contained in another infinity, or in other words some infinities are larger than the others.
There is an infinity of prime numbers within an infinity of natural numbers. So when the guests move to their new room numbers which is double their current one, there is created a smaller infinity of odd prime numbers (vacated rooms).
Those infinities are the same. They are all aleph-0, the number of natural numbers. The next infinity, aleph-1, is in some sense much bigger. Understand that I'm not using "bigger" in the usual way. It does no harm to think of aleph-1 as the number of real numbers.
How can an infinity be larger than another infinity?
@@michaeldrew3292speed/comprehension. This is all a very confusing way of explaining how added dimensions get bigger even when they are at the maximum of their current restrictions. An infinite space for existing without time isn't as big as an infinite space for one week, from our perspective at least.
@@joshanderson1685 No, infinity is always infinite, and dot not depend on time. It's infinite a every time. We can't add infinity. Maths try to move infinity in a mathematical operation by create notion like "'natural numbers", etc... but infinity is not a part of any operation.
Infinity could contain infinity, but then it would be full and could contain nothing else. Or not. Semantics, really. It couldn't possibly ever actually matter, and I think that could be proven formally.
New Guest : "Hi, I need a room."
Clerk: "We have infinite rooms: here is your key for room [Infinity + 1]"
New Guest: "Where is that room? I can't find it."
Clerk: "Never mind. I will shift the other guests to the next highest room number and give you room #1."
New Guest: "Do you have HBO?"
The clerk cannot have provided that key - it doesn't exist.
Haha that solution works also! Nice and simple!
Clerk: "No, but we have HB∞"
Chris Seib this hotel don’t have key...?
its a funny joke but the guests are moving 2x their number in one effort, which can fit infinity amount of people after. The video doesn't suggest moving one room down, for each new visitor.
It seems logical that an infinitely large object cannot be contained.
It seems logical that an infinitely large container can contain anything and everything.
So we have a paradox.
Hilbert's argument begins after he has taken the liberty of containing an infinitely large object, the aggregate body of the guests, within an infinitely large container, the hotel.
Herr Hilbert is a very naughty boy!
Actually, not really; the notion of an "infinitely large container" contradicts itself because it applies the concept of limitation (a "container") to infinity, which by its very definition has no limit. That's precisely the fallacy that sends this contrived house of cards so tenuously called a "thought experiment" crashing to the ground.
@@MisterVercetti Hi, your explanation really interests me but I can't wrap my head around it. Do you have time to expand a bit more?
you have made the same mistake by supposing infinity is logical. it is by no means logical.
Another paradox if you are smoking a lot of pot, I suppose. Infinity is not a number. Attempting to treat it as one is the source of the problem.
Yeah, who needs limits and calculus.
CookinginRussia
The video uses a different definition than the one used in calculus
I always fail to understand this one. Where is the paradox? Infinite people can be accommodated in a hotel with infinite rooms...
+1m2a3t4t5 he sort of phrased it wrongly. its supposed to be that there are an infinit amount of rooms. and in every room there is a person. thats what he means with infinit guests. so that means that all rooms are taken and no other guests can join the hotel. however if you do move guest 1 to room 2 it will free up the first space. however all rooms are full so guest number one cant go to room number 2. BUT SINCE ITS INFINIT they could always move 1 number up because you can always take the last one and move him 1 up because there will always be a number after the last room.
1m2a3t4t5 You didn't notice the "no vacancies" sign. The hotel is full, yet it can accommodate new guests, even infinitely many new guests. That's the (so-called) paradox.
+1m2a3t4t5 All of the guest simultaneously leave there rooms and go into the corridor. Now all the rooms are empty. The guests now walk to the new (empty rooms) and enter. Now all the rooms are full again, except for room 1.
See that is where you got lost. The hotel has that many rooms, and the bus has that EXACT same people to occupy, so 100% occupancy. Each room number is taken!!! No matter what the number is, the EQUAL amount came from the bus. So if you choose a number, it is taken. So if you get ANOTHER bus with infinities, you have to FIND make those numbers to put them in!
Unless there's an uncountable infinity of people. Then Hilbert is straight out of luck.
Hilbert's Infinite Hotel for you. Get down early for breakfast...(1 min)
The solution is that infinity continues moving and never ends. The guests would be constantly switching rooms forever.
but what is the difference then between when no one is switching and the infinite switching process
Actually, that does seem like a valid solution. Just keep moving them one up! Haha. Nice.
Love the video! It makes me smile :)
Hilbert's Infinite Hotel paradox with adding a guest (alternatively adding a finite number of guests) is clearly a play on the mathematical concept of "almost all".
Adding an infinite number of guests is in turn a play on the intuition that e.g. there is "more" natural numbers than there is even numbers, while according to the definition of two sets being equal in element count, there is just as many even numbers as there is natural numbers (this breaks only from rational numbers onward).
Such a lovely place (Such a lovely place)
Such a lovely face
Welcome to the hotel infi-nity.. Such a lovely place..
...You can check in anytime you like, but you can never leave...
[2] If you can say "We have an infinite number of people in the hotel and it is full" it means you've been able to process an infinite number of things in a finite amount of time. The paradox only exists because we assume something much more extraordinary that simply an infinite amount of things.
A lot of people in the comments are makung the mistake saying each person moves down one but this is not the case as it would only free up one room in the end.. This process would have to be done an infinite amount of times and wouldn't be efficient. Rather every occupant moves 2x their number. ex) 1 --> 2 , 2 --> 4, 3 -->6, 4 --> 8, 5 --->10. It's amazing because this process fills in every single even room without any vacant, while every odd room remains open. This process only takes 2 times infinity (time) while moving them up one at a time takes infinity times infinity (time).
The number of points between 4 and 5 is infinite, the number of points between 5 and six are equally infinite and the number of points between 4 and 6 are likewise infinite. But since the difference between 6 and 4 is two, and the difference between 5 and 4 and 6 and 5 are both 1, the infinite set of points between 6 and 4 is double the number of infinite points between 5 and 4 and 6 and 5. However they are both still infinite.
Mind......blown
Keith Fife because infinite is not a number.
Derederi Yes I know that... That is what I was demonstrating...
I did so be clever use of set theory, it's not that hard to see...
Keith Fife Although mainstream set theory disagrees with your argument... I agree.
adamtract9 How so? The logic is fairly axiomatic and it is know that other infinite sets are larger than others.
i think is impossible operate with the number infinity because the infinity is all. If you worked with infinity, you are operate with amount concrete. And if you are operate that amount concrete, it is a mystery concrete. Infinity is a incognited as X.
00 = X
You've described how the guests already in the hotel change rooms but you haven't described how the people on the buses will be awarded room numbers in a way that each individual will find a room in a finite amount of steps.
The hotel can't be full in the begining as it is always in the process of putting people inside room since you can't reach a state when you say "there's an infinite number of people in the hotel". You'll barely say that there's an infinite line of people entering the hotel.
You're on the right track. But you need to consider what happens to the second person on the first bus. If the first passengers on each bus get to the hotel first, we've already filled infinitely many rooms which means the second, third, fourth, etc. passengers on each bus are left out in the cold. They need to all find a hotel room in finitely many steps. They need a finitely numbered hotel room.
I really like the style of this animation. So cute
my opinion...guests will go to the rooms 2^n(2,4,8,16...) people from first bus will go to the rooms of 3^n(3,9,27,81...)from the second bus to rooms 5^n...third bus>7^n..//prime numbers etc etc... i dont know just thinking about that..but the first room will be empty ..
sorry for my english:)
The reason it's a paradox: Rooms and guests will stay at the same number for infinity, there will never be room for new guests. Whenever a room is available, one of the infinite guests will take that room, and this will continue at very high speed.
Or you can look at it this way, infinite plus 1 (room) = infinite+1, then infinite+1 minus the free room the guest will take. When the room gets "available", the new guest will have to take it at the same time it gets available.
The equation "x-x=0" works since x represents a fixed arbitrary number. But infinity is not a fixed arbitrary amount.......When we have the eqn "infinity - infinity = 0 " , the first infinity symbol is bounded to be no larger or smaller than the second, contradicting the unboundedness of infinity. Hence the equation does not work.
What if there is no corridor? Surely we are using the old junior school remainder format. Without a corridor, room 1 cannot move out until room 2 is empty etc ad infinitum so nobody moves. At least one occupant has to move into the remainder corridor to vacate a room.
10 divided 3 equals 3 remainder 1 carry the zero to make 10. 10 divide by 3 equals 3 remainder 1. Or you leave the sum undone as 3 1/3.
Infinity both exists and doesn't exist.
@StrikaAmaru People tend to perceive infinity as a number which is not true, Infinity+1=? Infinity-1=?
It doesn't if you to ask person move to n2 of his room number or 10n, if each person to move 10n then 1-10 2-20 3-30.... all integrals which are divided by 10 with fraction of as a result will be available, what im trying to say is that you cannot apply any arithmetics to infinity.
I think it shows 2 infinities can exist separately at the same time or merge together to form one infinity. So suppose you have 2 universes both stretching infinitely in all directions, then a wormhole opens up opening a pathway from one universe to the other, now since both are connected by space both infinities are existing in the same space, and thus become the same, though cut off this worm hole and they become separate.
We cannot say that the hotel has infinitely many rooms unless there were infinitely many builders building it. But we cannot say that infinitely many builders built it because that would imply that the hotel was built at all. We cannot say it was built at all because that would imply "completion". Completeness implies wholeness or in the mathematical sense, whole numbers. Since we can count whole numbers, this contradicts the idea of infinity. Therefore such a hotel cannot be built.
"`A` grand hotel with an `infinite` number of rooms" -> contradiction
+chigoze123 Nope. You have stretched "complete" out of all recognition. It is not complete in the sense of a "whole" number. It is complete in the sense that it is actually infinite. The hotel is not physically real, so you don't need to worry about how it got built. It cannot really be built.There is no contradiction, but there is a paradox. The hotel is full in the sense that every room is occupied, but more guests can be accommodated (without sharing rooms)..
What a foolish assumption, trying to twist words to change the point of the argument. It doesn't even have to be a "hotel" you moron. It is just there as metaphor or visual example. Small brained children.
nonsense. You can't add to infinity. If you add to it, you are proving that it wasn't infinity in the first place.
You would empty the odd numbers rooms as before. Then, you would put the passengers from the first bus into the rooms representing the powers of 3 so rooms 3, 9, 27, etc would become occupied. Then you would put the passengers from the second bus into the rooms representing the powers of 5 so rooms 5, 25, 125, etc would become occupied. Repeat this process for each prime number, and since you have an infinite amount of prime numbers and since they share no common factors you would be able to fit
This one gets an instant subscription. TedED had an animation about this but they made a really shitty boring video. This channel did it even better and they did it way faster.
But what is the "next room?" If you were to pick any room number it must be occupied by one of the infinite guests. Think of it mathematically. If you have a sequence 1,2,3,4... it contains all natural numbers, so no natural number exists that is not in this sequence (ie no free rooms). If you double each number, you get the sequence 2,4,6,8... Now there are no odd natural numbers in the sequence. This infinite number of "rooms" can be filled by an infinite number of "guests"
You need to provide a process by which each person on each bus can find a unique room number. Essentially, we need a way of ordering everyone off the buses and line them all up in such a way that that every finitely-labeled person on a bus will be finitely-labeled in the single-file line to the hotel.
Look at how mathematicians represent the infinite set of rational numbers as a countable sequence. You can make a table of the columns representing your hotel room or your seat on the bus, and the rows represents your bus number (0 can represent the people already in the hotel). Each person can be assigned a specific spot on this table, and then to get their final room number, start drawing diagonals on this table like they do here: watch?v=elvOZm0d4H0 at about 3:10
You can use the dovetailing method.
Let (m,n) represent the m-th passenger on the n-th bus. If we order all the passengers in the following way, we can accommodate everyone into the hotel.
(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4) (2,3) (3,2) (4,1) (1,5) (2,4) (3,3) (4,2) .... etc.
Umm....no, I think you misunderstand. As I said before, not all infinite sets are equal. But, if they are equal, they will always be equal. The initial statement is that the "room" set is equal to the "guests" set. This is the given initial condition, so no more guests can ever be accommodated, since the two sets are equivalent. For example, two equal sized circles. They are infinite sets since both have infinite points, and yet one will completely fill the other, with no extra room left.
In order to try to free up rooms, the owner asks every guest to call the room twice their number and see if it is free to move to. No rooms become free, because all the rooms are full.
Other solution:
1st: everybody in the hotel moves to double is number.
2nd: start with first bus 1 st seat, then second bus 1st seat then 2nd bus 2nd seat 1st bus 2nd seat; 3rd bus 1st s; 2nd b 2nd s; 1st b 3rd ..... therefore you now can count the new guests...
...continued
If it had all fractions as a room, then doubling each person's room number wouldn't matter, because every room has a room half of itself. When you are limited to integers (no fractions) odd numbers cannot be divided by 2 into another integer, thus providing the "illusion" of empty space. When in fact, if the hotel REALLY had infinity rooms, even odd numbered rooms would be filled when fractions are doubled.
the amount of space available in the universe is finite and can't ever reach infinity. space itself is stretched toward infinity thus always getting larger. also the argument that each object is half the size of the previous is bogus because one plank length is as small of a unit that is possible.
You forgot the third part : If an infinite amount of people came on a infinite amount of buses each one containing an infinite amount of people. Possible if there is an infinite amount of prime numbers which there is
Correct. Wiki "Aleph numbers" and "Georg Cantor" for more information and about the concept of the sizes of infinity, one-to-one correspondence, and set theory.
@oriocoookie No, that is not the definition. The definition of a Dedekind-infinite set is that it has a proper subset of the same cardinality. The notion of a set which its own proper subset is absurd. If A is a proper subset of B, then in particular A is not equal B. If A were a proper subset of A, then A would have to be different from A, which would violate the axiom of extensionality.
Have anyone ask, do infinity even exist? considering it not a number, but an idea.
The only reason infinity exist is because no one or machine is capable of comprehending & obtain the finite limit of the universal scale. Infinity is an assumption if infinity exist at all, just like "monkey theorem" or "fractal".
The paradox arises from the impossible assumption that some people showed up at our door even though everyone in the universe was already here, and has nothing to do with infinity. In a universe where people magically appear, there are two sizes of infinity, but then they would have ALL KINDS of counting problems, wouldn't they? Bad experiments lead to bad conclusions.
Imagine seven people suddenly appearing all around you. I guess they would be babies? Hi! This is the universe where there are two sizes of infinities!! It's kuhraaaazy! If there were infinite universes, Rick and Morty would be a documentary series in an infinite number of them. Oh, pinheads and their little math ideas!!
So tell me if this argument holds water.
Lets say we follow the same procedure, but instead of moving the guest from room 1 to start, we instead have the guest from room 1 wait until the guest from room 2 moves first. That guest has to wait for the next guest and so on so that we only free room N when room N+1 has be vacated. Since there are infinite rooms this process will never be completed, as each guest will always be waiting on the next guest to move. Therefore the new guest never actually gets a room.
It's always seemed to me that this only works because we start at room one which is suddenly empty, but infinite room shuffling happens afterwards. If we require the infinite shuffling to happen FIRST, then it becomes impossible to complete, and the hotel indeed has no vacancies.
I'm sure I will be proven wrong somehow, but I'm curious to see the reasoning. :)
You nailed it. If you reverse the domino effect you have at the start what in this problem happens towards the end (the last guest never gets to the last room cause there isn't a last guest nor a last room)
It's not a domino effect if they all move at once or teleport, remember, thought experiment...
So it really depends on what assumptions you take on "how" the people move. Which is actually quite interesting!
Kyle. As the rooms become vacated, the new guests can go in. For instance as soon as the guest who was in room 1 leaves to start going to room 2, the new guest can enter room 1. He doesn't have too wait for the "ripple" to finish.
This paradox is only sound if you believe it is possible to have an infinite amount of objects. This is almost certainly untrue, while infinity can easily apply to concepts (like mathematics) and possibly even space and time, it is hard to see how you could have an infinite amount of physical objects.
Holy ! ! ! Is that David Mitchell's voice? This must be good.
Each room in your only hotel in town, can be made into a new hotel, by dividing it up into new rooms. So you can have a hotel, full of infinite hotels. Ergo, there is no limit to the number of hotels you can have in town.
[2] .. "in an ordered set, x - y means the predecessor of x by y unit". Now, if I accept the definition of equality and if we agree that infinity is equal to all its successor and predecessor then infinity = infinity, infinity - 1 = infinity and infinity - infinity = the null element, which is abreviated in all set as "0". Matrix A and B can be of infinite order and yes, A - B = O where O is the null element. Moreover, google that : Extended real number line.
Mathematically that doesn't work, because when speaking of what rooms to be filled you have to deal with countable numbers. After all what are we supposed to make of room number ∞+1 ? There are an infinite number of rooms but also an infinite number of guests. Given this there is no room number you can actually choose that will not be occupied.
The infinity paradoxes seem particularly odd to me. Infinity has no sum, no last number so you can’t add anything to it. No guests are moving nowhere. Paradox solved
This example misses the point by failing to continue the thought experiment with infinite buses containing infinite guests (who can be accommodated in Infinite Hotel) and then showing that the hotel cannot accommodate infinite guests denoted by decimal numbers. Thus showing that some infinities are larger than others. Hilbert would turn in his grave if he watched this (possibly continuously)
This problem addresses the lowest level of infinity. Other infinities cannot be grasped with this example, but I can give you a new one if you like. Take a piece of paper and cut it in two infinite times (disregarding actual physical boundaries such as particles) and you will end up with an infinity that adds up to a finite thing (the piece of paper at the start). Fascinating ain't it?
The infinite buses of infinite people is solved by listing them diagonally on a square grid (like with the rational numbers), getting the guests to move to double their room number and assigning the newcomers the empty odd number rooms.
Infinite ships of infinite buses of infinite people would need a 3D grid and list the series of triangular planes from the start corner.
This should work for any finite n powers of infinity with an n-dimensional grid.
More simply assigning to any person a prime number each bus elevated to the persons seat number.
[cont.] Since every hotel contains an infinitude of rooms, we needn't be concerned how each hotel owner will accommodate it's busload of people; however, how do we ensure that each bus finds a hotel?
The problem with his hyposthesis is that you cant take into acount formulas involving the concept "infinite" because its not a variable that can be added. Its merely a concept. But if we were to take "infinite" as a variable in a formula it would be like this:
THERE IS INFINITE ROOMS; AND INFINITE GUESTS; ANOTHER BUNCH OF INFINITE GUESTS COME ALONG
infinite guests in the hotel are represented with (A) + infinite guests that come along (B).
A+B=infinite. so there is enough space for everyone.
What most people fail to realize is that infinite sets are NOT ALL EQUAL. Take, for example, all the numbers in-between 0 and 1. this is an infinite series that is smaller than the numbers between 0 and 2. Originally the hotel had two equal infinite sets: guests and rooms. The initial statement of "No Vacancies" indicates that the room and guest sets (although infinite) are fixed. The reason the idea of the hotel works is because the hotel is cheating by expanding its room set.
RUclips recommended me this video after I solved Hilbert hotel question at codeforces.
I think the answer will be the same. Since there are going an infinite no. of odd room number to hold all infinite passengers.
You're right, that's why we use another set : google Extended real number line., read the Arithmetic operations in wikipedia.
@Swolf943 thats why it equals infinity because infinity is not a finite number
I tottaly agree with this idea.I allso want to add , that if the qifsha is one with the motren then and only then the infinity will be quaninty.Only the qifsha motren matters...be aware of that
Great video except for the end which referred to this as a "paradox". It isn't. It is counter-intuitive, but it is the foundation for a non-paradoxical ways to talk about the infinite.
It's not quite that simple :P
It's true that in Cantor Theory, infinity^2 = infinity; however, it can also be shown that 2^infinity > infinity.
If the buses park simultaneously given they have infinite parking space too. Then we can group all the passengers into one infinity. And that is solvable.
That doesn't really prove it. Not all infinities are equal after all. In order to show that infinity^2 = infinity, you need to establish a bijection between NxN and N. In the Hotel Infinity model, that means you have to show a way of moving all the people in the buses into the hotel in a systematic way.
But wouldn't it take an infinite amount of time to carry out the instructions to the guests?
You will have an infinite number of porters ;)
in any normal hotel too it does...
It's mind candy. The hotel cannot physically exist.
What "indetermination" ? In function analysis it is of course, as you can't say that x^2 - x = 0 when x becomes arbitrarily big which is another way of saying that x goes to infinity. Because you never actually calculate "infinity^2 - infinity" but always a real arbitrarily big x. But, what we mean by "infinity - infinity = 0" is "x - x = 0" which is true for any element of R. In closed R, infinity is an element and of course infinity - infinity = 0 because of the definition of equality " = ".
i may need some help with this one... if there is a infinite number of rooms therefore there is a infinite number of next rooms that can be ocuppied by the infinite people on the infinite bus, am i wrong? so why is there a need to change the guests that are already in the hotel??
Paradox: A seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.
I think this is very much a paradox.
You're saying that infinity (how the hell do you put that sign anyway XD ?), infinity - 1 = infinity which is "true" except that "infinity - 1" has no meaning. All I was saying in the first place is that infinity - infinity = 0 otherwise infinity is different from itself. This is just a traduction of "a mad mathematician with a gun kills ALL the guest in the hotel. Hence the number of guest is 0" and that is perfectly fine theoretically.
it' s simple - if it' s theory then it is impossible to be two infinite numbers of something but there can be two infinities of something different for example infinite number of cats and infinite dogs. so together it will be infinite squared which is also infinity.
You probably saw the problem after you press the submit button. He can't kill ALL the guests and left ONE. That's a contradiction.
With 3 sets of 2 cards I can be sure that at least 2 sets are ordered in the same way. With an infinite number of people made of a finite number of atoms with a given order I can be sure that if I book into the hotel, an infinite number of me are already booked into the hotel into an infinite number of different rooms.
@StrikaAmaru Is infinity odd or even? positive or negative? infinity is not an integer, the moment you sum infinity it becomes integral, how integral can be infinite?
So what if a infinite number of busses loaded with infinite people pulled up? empty out all people f.e. and let the first ones go into rooms which are a multiple of 2 such that 2^n where n=1,2,3.. The next coach with 3^n the next with 5^n, the next with 7, and so forth with all prime numbers. This way there will even be an infinte amount of rooms left vacant :P
This is why it seems wrong. This hotel has an infinite amount of rooms, but not infinity rooms. Let me explain: It has all positive INTEGERS of rooms, which is an infinite amount of rooms, but not infinity rooms. If I would say "How many room does it have between 3 and 5" you see that part of the hotel has finite rooms. A TRULY infinite hotel would have a room for every single fraction. Then there are infinite rooms between 3 and 5. (just example numbers).
continues...
Doesn't that mean there are finite amount of rooms if every single room is occupied. If it was infinite, there would always be space for any amount.
'Doesn't that mean there are finite amount of rooms if every single room is occupied'
No. There is a one-to-one correspondence between rooms and guests.
'If it was infinite, there would always be space for any amount'
Incorrect. There is, for example, no bijection between the set of real numbers and the set of natural numbers, despite both sets being infinite.
And also I don't even understand why we are arguing about something so crazy ! I guess everyone litterally NEED to be right and prove his point. I'm perfectly ok with infinity - infinity is different from zero, but you need to demonstrate it and that, as I understand it, would mean that, given infinity is considered as an element, it can be "different" from itself which seem to me as just a new definition of equality which I don't agree on.
Actually the paradox is that infinite as a concept/numerical value/anything is paradoxical in nature
Bro, all of the hotel's rooms were occupied. If each guest only moved down one room to the right, how many rooms become vacant? Only one, which is the starting room, and one room definitely can't hold an infinite number of people. By moving to the room number that's twice bigger than yours, only the odd numbers are left, and with infinity there is an infinite number of odd and even numbers, which satisfies the two groups of infinite people.
Unfortunately this does not work. What room number do you send the guest to? You cannot ask them to go to room number infinity + 1. But you can easily ask them to go to room 1, after asking all your guests to go to room number n + 1 where n is equal to their old room number.
these are great
It doesn't necessarily destroy logic, it just forces you to use different logic.
Hilbert's Hotel deals with "actual infinity", rather than the type of infinity you are talking about when you say "x approaches to ∞". If you understand ∞ as the number that represents the class of all the sets of cardinality ℵ-null (e.g. the set ℕ of natural numbers), then you can define "∞ - n", for each natural number n, as the cardinal of ℕ \ {0,1,2,...,n}, i.e. ∞ (ℵ-null). But how would you define ∞ - ∞?
David Hilbert perguntou o q aconteceria se um novo hóspede chegasse procurando um quarto. R: é fazer com q cada hóspede se mude p/ o próximo quarto. O hóspede no quarto 1 se muda p/ o quarto 2, e assim por diante, de modo q o novo hóspede possa ficar no quarto 1. Mas se chegasse uma caravana com um número infinito de novos hóspedes?
It all depends on the set you're using ! Google "Infinity plus one" and you'll see that it make sense for hypereal numbers.
Not all infinities are equal. Also, ∞ - ∞ is known as an example of indetermination.
Well... when he fills the hotel in start... he already assumes that someone is placed in room "314.424" i wonder how old is that guest in first place.
@oriocoookie No. That isn't true. No set can be a proper subset of itself. However, infinite sets do have proper subsets which are equinumerous to them.
Isn't this just the paradox of adding infinity...or am I missing something greater or perhaps something exaggerated.
Why move to odd numbers? Why not just the next room?
Because that would only free up room number one.
I think it's a cool idea, although it also just comes down to, if he has enough rooms, he can fit it. :D
Which infinity does this conundrum best address?
the paradox is that infinite sets are proper subsets of themselves
@Swolf943 Not exactly pointless. It should be said that finite mathematics is possible and probably cultivated by a few mathematicians, but it is much less useful than mathematics with infinities. Ask any physicist. The universe they propose is finite, yet infinities turn out to be indispensable in the explanation of it.
So if the Hotel is Infinitely full, where the hell does Mr. Infinity Plus One come from?
An infinite number of buses is the same as an infinite number of passengers. An infinite number of passengers is already solved.
Philosophical question - Do you agree ?:
We cant grasp infinity. But if I watch this video a thousand times now, I have a better sense of infinity than one who doesent.
Maybe you can't grasp infinity, but I can and so can most people (with a little nudging). It just means endless - easy peasy. This topic is math not philosophy. It's about definitions and using deductive reasoning.
@@Chris-5318 yes but we are taught in math that when mathematical concepts lead to contradictions, we conclude that they are nonsense. If we still want to play mathematically with the idea for fun, no problem! But don’t expect to ever find an actual real world analog.
@@salman-11924 Do you have a point? Are you claiming that there is a contradiction? If so why didn't you give an example ('cos I don't know of any)?
Math is not constrained by the limitations of reality. If infinity led to contradictions then it wouldn't be a part of math. "Fun" - infinity is an extremely important part of mathematics. Mathematics would be seriously incomplete without it. It simplifies so many things.
OTOH mathematical infinity is used all the time by scientists and engineers. e.g. integrals with infinite end points and Taylor/Maclaurin series. The world would be a very different place without it.
Simple: they keep on switching rooms infinitly or untill they die.
What happends though if there is an infinite amount of Busses with each having infinite amounts of Guests? How do you fit them in? Is it possible?
Yes it is. You take the guests off the buses in a diagonal pattern:
passenger 1 bus 1
passenger 2 bus 1, passenger 1 bus 2
passenger 3 bus 1, passenger 2 bus 2, passenger 1 bus 3
passenger 4 bus 1, passenger 3 bus 2, passenger 2 bus 3, passenger 1 bus 4
etc.
This is commonly, but incorrectly said as ∞ * ∞ = ∞
Ah, well, infinity is an idea, not a material number. Using it as one destroys logic. The idea of an infinite hotel is a bit silly, but is a good way to explore the handling of the concept.
This isn't a paradox, and this specific "paradox" always makes me angry because of how absurd it is. There is an infinite amount of rooms, so it can never completely fill. There you go; The solution is right there. No need to complicate it and act like it's more complicated then it really is. The fact some people act like this is a paradox blows my mind.
'There is an infinite amount of rooms, so it can never completely fill'
Yes, they can all be filled. If every room is occupied by a guest, like in the problems' premise, the hotel is full.
'There you go; The solution is right there'
I mean, your 'solution' is incorrect. It is based on just ignoring the conditions set up in the problem.
If there are 2 guests in the hotel and 3 more arrive the 5th guest will have to walk to room 5 which is the closest available room. If there were 4 in the hotel and 5 arrive, the 9th guest will have to walk to room 9, 3B guests and 3B + 1 arrive, guest number 3B+1 will have to walk to room 3B+1, etc. What's my point? If there are N guests in the hotel and more than N guests arrive, the last arriving guest will have to get a key to room that was not previously occupied. Bottom line is, the hotel cannot contain an infinite number of people
All the rooms are occupied in Hilbert's hotel. Keys aren't needed, the guests are very trustworthy. Use some common sense.
Doesn't this imply that the infinite of rooms is greater than the infinite of guests though?