Hilbert’s Infinite Hotel – 60-Second Adventures in Thought (4/6)

A Grand Hotel with an infinite number of rooms
– and an infinite number of guests in those rooms.
That was the idea of German mathematician David Hilbert, friend of Albert Einstein and
enemy of chambermaids the world over. To challenge our ideas about infinity he asked
what happens if someone new comes along looking for a place to stay?
Hilbert’s answer is to make each guest shift along one room – the guest in room one moves
to room two and so on – so the new guest would have a space in room one. And the guest
book would have an infinite number of complaints. But what about when a coach containing an
infinite number of new guests pulls up – surely he
can’t accommodate all of them? Hilbert frees up an infinite number of rooms
by asking the guests to move to the room number, which is double their current one,
leaving the infinitely many odd numbers free. Easy for the guest in room one. Not so easy
for the man in room eight million, six hundred thousand, five hundred and ninety-seven. Hilbert’s paradox has fascinated mathematicians,
physicists and philosophers – even theologians.
And they all agree you should get down early for breakfast.

100 Replies to “Hilbert’s Infinite Hotel – 60-Second Adventures in Thought (4/6)”

  1. kaviramyead says:

    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 " = ".

  2. kaviramyead says:

    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.

  3. kaviramyead says:

    [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.

  4. Nicolás Tassone says:

    I'll say it again. He kills all the guests BUT the guy in room 1729. (The comma was a mistake, I'm sorry). Is it clear now?

  5. kaviramyead says:

    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.

  6. kaviramyead says:

    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.

  7. Brien Anderson says:

    Infinity is not a number so adding or anything else such as that will not make sense

  8. Nicolás Tassone says:

    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 ∞ – ∞?

  9. teltri says:

    GREAT! Exactly! I´m going to be a philosopher 🙂

  10. damatrix1001 says:

    Isn't this just the paradox of adding infinity…or am I missing something greater or perhaps something exaggerated.

  11. nasosmor says:

    i repeat its not bad to admit maths isnt your subject. infinity – infinity = 0 is false in any university maths book you ll open.

  12. nasosmor says:

    only this should stop your imagination . but i ll try to help you more although i am not sure i can! you inticipate infinity is not a number. but the point is that it isnt! clear and stable. if it was it would accept multiply or make it less. but it doesnt. even a high school student understands number like a quantity which u can add 1 and make it bigger. infinity cant get this option. there isnt any number bigger than infinity which is limit a ''dam'' of all numbers.

  13. nasosmor says:

    infinity cant get bigger or less so it isnt a value. you also say there is no thing as infinity + 1. so how do you take infinity as a number if u cant make a simple mathematical act by adding a number in it. it deosnt follow so the rules of numbers in R.

  14. lordofpengz Robert says:

    I liked the joke at the end…

  15. kaviramyead says:

    You're talking about math book like they were some kinf of bible. You still haven't understand that everything is possible in mathematics given the proper axiom and definition, even a division by zero. If what you mean is that in the R set we use generally infinity is not considered as an element, fine. But in a broader sense "equal", abreviated "=" means that "the left and right argument can be interchanged without breaking anything in all expression" and "minus" abreviated "-" means that ..

  16. kaviramyead says:

    [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.

  17. kaviramyead says:

    You're right, that's why we use another set : google Extended real number line., read the Arithmetic operations in wikipedia.

  18. orcodrilo says:

    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.

  19. nasosmor says:

    no i totally disagree. i talk about all math books in every university in the world so about something with huge validity . not about the bible!! a fairytale book. i proved you somehow in my other 2 comments why infinity can be a quaninty but it seems u didnt read them by mistake. the upper wasnt my only post 🙂 . anyway keep going .

  20. Mashroom Clear says:

    You know you don't need to make former guests shift one room, right? Just fill in the other infinite rooms. Doesn't that make things simpler?

  21. almirCCR says:

    4. El hotel infinito de Hilbert. Otra vez jugando con el infinito

  22. banana man says:

    Hopefully it wasn't your English teacher

  23. Paul Nong says:

    Statement A: If something has reached maximum capacity, it is full.

    Statement B: If something is full, absolutely nothing more can be added.

    Statement C: If the hotel has an infinite number of rooms and an infinite amount of people, then it has filled maximum capacity.

    Statement D: If the hotel has filled maximum capacity, then it is full.

    Statement E: if the hotel is full, then no more people can be added.

    The moment Hilbert suggested adding one more person this paradox became bullshit.

  24. LuminaryGames says:

    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.

  25. 淑仪 says:


  26. Crab says:

    these are great

  27. G says:

    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".

  28. ActiveReplay AR says:

    I thought of a paradox. Let's say you went to the past to shake hands with your younger self. Then you go back to your present day. What would happen is that when your past self grows up and finally ends up in your time period, he will go back to the past, and shake hands with his past self. This would happen over and over in a never ending time loop.

  29. ActiveReplay AR says:

    Because you didn't capitalize "M" in my, and you didn't end your sentence with a punctuation.

  30. ActiveReplay AR says:

    Gee bro, calm down O_O

  31. ActiveReplay AR says:

    I can tell you are having a bad day. LOL 😀

  32. ActiveReplay AR says:

    Maybe a cup of nice hot tea will make you feel better.

  33. ActiveReplay AR says:

    I'm 12, and whatever you say to me doesn't phase me in any way. So go ahead, keep throwing insults at me. It's the only way you can waste your time, after all. Plus, I didn't even want to start a fight. You just rage because I pulled a little joke on you. I hope you don't act like this to other people.

  34. ActiveReplay AR says:

    1) I'm a guy. 2) What the fuck is wrong with you?

  35. Spitz541 says:

    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

  36. Hua JING says:

    Love the video! It makes me smile 🙂

  37. Doom Fish says:

    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.

  38. Pieter says:

    " even theologians" haha fucking wankers 😀 religion is not a valuable profession!

  39. James Wong says:

    have fun hahahhahha

  40. Snooby66 says:

    Not sure the the fuss is all about.
    The hotel has an infinite number of rooms which are ALL FULL.
    So, there is no way to move the guests.
    Now, what to do with the remaining 55 seconds of this 60-second thought adventure ….

  41. martin shervington says:

    Hilbert's Infinite Hotel for you. Get down early for breakfast…(1 min)

  42. ForevermoreNevermore says:

    I don't understand the paradox… Infinity + infinity = infinity, right?

  43. Louis Becker says:

    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.

  44. Lightn0x says:

    I don't get where the paradox is.. we already know that one infinity can be different than another, so why is the fact that 1+inf=inf or inf+inf=inf so interesting?.. I guess it could've bothered people decades ago when there was no real concept of infinity present in mathematics but now?.. why do we still call it a "paradox"?

  45. Sina S says:

    But what if an infinity number of busses come along?

  46. 2019inuyasha says:

    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.

  47. Vishva Kumara says:

    Such a lovely place (Such a lovely place)
    Such a lovely face

  48. TheGrandBrand says:

    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).

  49. Richard Spence says:

    next step: uncountably infinite guests check in the hotel.

  50. Tom Adams says:

    Imagine being at the top and falling out of your window, lol. 

  51. CookinginRussia says:

    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.

  52. Magnemite Thames says:

    Simple: they keep on switching rooms infinitly or untill they die.

  53. Josh Solomon says:

    define your infinity 

  54. Wu Weiming says:

    Some infinities are greater than others…

  55. Eddie Temple says:

    Da…David Mitchell?

  56. Keith Fife says:

    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.

  57. Offshoreorganbuilder says:

    But why is this sort of thing important?  Am I missing something, or do the people who spend time pondering this sort of thing need to get a life, and do something useful?

  58. Paul Gray says:

    Does Hilbert's Hotel have enough space to contain the universe in which it exists?

  59. Paul Gray says:

    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!

  60. Neil Wiltshire says:

    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)

  61. ralphinator2 says:

    Why make any of the present guests move?  Just make the new arrivals go to the top. Since the hotel has infinite rooms, it also has infinite vacancies.

  62. doesn't matter says:

    What a bullshit omg

  63. mccowc0w says:

    But infinity is not a number.

  64. Nawaf The FearBraker says:

    Well, THAT'S why you can NOT use infinity as a number…But as a concept.
    Imagine it as an x factor, that's the only way you can solve equations with it, though it still makes almost no sense.
    i.e x+1=?
    Which again makes no sense, since ∞= INFINITY.
    It's like saying water +1=water+1 but water + water= 2 Water.
    See how that sounds stupid?
    I did not say gallons,Oz, Liters or anything so you can't calculate it…
    Wanna think of it as an x Factor too?
    Well (∞-1) + (∞+1) how much does that equal?
    If you said 2∞ you're wrong.
    (∞-1) + 1=? If you said ∞ you're right.
    Because it's already infinite again, you can't say water-1…
    You can calculate things (from what I tough myself and though about) as this:
    so 0+1+1+1+1+1+1+1+1+1 (gaining +1 every second to infinity).
    In my understanding of my awesum conclusions 😉 You can solve equations to that like:
    so in the first second it'll be 1
    second second it'll be 3
    3rd second it'll be 6
    so you can say ∞ (in the 3rd second) – 5=?
    you say ∞^3 (or whatever new something symbol that you can use to indicate what second it's in)-5=1
    Which is I.M.O 100%
    Please correct me on my theory or whatever that shit's called.
    Stupid yet correct answer to ANY ∞ equation = ∞…
    Just like 1^anything = 1

  65. DragonAurora says:

    The solution is that infinity continues moving and never ends. The guests would be constantly switching rooms forever.

  66. 藍狗 says:

    @Tabna Shahid

  67. pdxrailtransit says:

    But wouldn't it take an infinite amount of time to carry out the instructions to the guests?

  68. Kyle Jones says:

    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. 🙂

  69. pdxrailtransit says:

    And woudn't every porter expect a tip!

  70. ubiquity says:

    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

  71. 1m2a3t4t5 says:

    I always fail to understand this one. Where is the paradox? Infinite people can be accommodated in a hotel with infinite rooms…

  72. nicomp1 says:

    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?"

  73. Lazy Nomad says:

    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).

  74. Allen Spurgeon says:

    I dont understand how this is a paradox though. You can shuffle an 0/0 amount of people into an 0/0 rooms… Since 0/0 is just hypothetical, where is the problem?

  75. Robert Hernandez says:

    I guess it's only a fancy way that when you obtain infinity in your calculations, your theory crashed. It becomes a non-theory.

  76. BlackProphet says:

    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?

  77. kILLA ' says:

    Hilbert's hotel…Still too small for your mum.

  78. PinkJunkie 89 says:

    Why do the people have to move rooms? If the guests can move to a new room, then surely the new guests could just go there without moving everyone?

  79. Joshua Mwaniki says:

    Why move to odd numbers? Why not just the next room?

  80. Luca Schmidt says:

    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.

  81. ubiquity says:

    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

  82. Brendan Woods says:

    God damn infinity; you beautiful mindfucker you. The idea that in the space of 0 to 1 it would be impossible to count the decimals between the two makes me happy that humans have devise math well enough that that idea exists and infuriated that it exists at the same time.
    Not happy with the sentance structure there, but you (Reading this) know what I mean.

  83. Cohcky says:

    Can some one explain to me. How in an infinite hotel, where every room is infinitely occupied by a set of infinite people. Can each person move one room to the right? .. There is never going to be a space available for them to shift!? It's infinitely occupied.. They would keep asking each person to the right to shift and no ones ever going to get the opportunity to actually move.

    You're assuming there's an infinite amount of space available which is contradicting. How can you have a hotel full with an infinite amount of people. Suddenly vacant with an infinite amount of space available.

  84. Philip B says:

    Which infinity does this conundrum best address?

  85. Debi K Baughman says:

    I can see this being a problem in regards to the hotel, but in regards to life itself, and time it isn't taking into account death. death itself, in this account, is not a non-existance, but the dead being also doesn't use space in the same way as does the living being. The reason for the mistake here is a non-understanding of space, which is what he is basing his theory and conjecture on; see space as both empty and full at the same time (that's infinity) ; and then, well, then we get go into the understanding of Time which is both space and no space equally.

  86. Jonas Venture Jr. says:

    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.

  87. fauxmosmexual says:

    is that david mitchell?

  88. hamza mezbo says:

    Why david choose the hotel to explain his theory
    Why not a train or something ????

  89. skilliyay says:

    What is being described here is the difference between a one dimensional and two dimensional infinity.

    Everybody can grasp the idea of a one dimensional infinity. It is the endless string of numbers or in the this case the linear string of rooms in the infinite hotel.

    The infinite hotel described is like a row of chalets with infinite length. We don't even need to consider how many storeys the hotel has or it's width. The hotel is infinitely long in one dimension and thus can accommodate infinite guests. The supposed paradox arises when the new guests arrive………

    When we introduce the new guest(s) we add a second dimension to the concept. It is helpful to think of this second dimension as time, but we could just as easily be adding infinite storeys to the hotel.

    It would take time for each guest to move to the next room. We cannot fully describe this movement of guests using only one dimension, because it has now become a 2D concept. If you watch Carl Sagan's video on the 4th dimension, you can see the limitations of comprehending shapes from a higher dimensional space.

    In this 2D infinity, we can accommodate an infinite number of new guests in the second infinite plane.

  90. Loyal Honda Fan says:

    That germans should better find a way to fix their cheating emissions on their cars… booooy

  91. ed bell says:

    Sorry their is no vacancy at The Grand Hotel. No finite object can be infinite in number, because anything infinite would require an infinite amount space to exist in, thus leaving no room for anything else. The only thing that can be infinite is infinity itself. The all of the all is the only thing infinite. Everything else is a sub division of it. And that's it. There is nothing more because that is everything. No extra anything can exist because everything that can exist is contained in the all of the all and it is never ending and ever increasing in size if you were going to try and quantify it. You can't. Numbers are a way in which we compare one thing to another to determine length, width, height, volume, speed, velocity, etc, etc & etc. What could you possibly compare infinity too?

  92. iannickCZ says:

    If the infinite does not have "end", so the "start" neither, there is no room #1

  93. Roiben Blitz says:

    Infinity*2 = Infinity = Case closed Next please.

  94. Damsel Elysium says:

    I made a short film about this! The trailer is here! https://youtu.be/L1y4a4W_RVY

  95. GNM says:

    hi im in room tau of the real number infinite hotel
    and the guy in room root 42 is so annoying

  96. Silus says:

    Why does anyone need to move if there are infinite rooms. The new people just move into a room, irrespective of what the current occupants of other rooms are doing.

  97. Paul Perlova says:

    I actually applied this paradox to my job in a stockroom in HomeGoods in order to keep up with the trash and it actually works really well.

  98. Peter Anderson says:

    Why bother with maths? If the manager took the infinite people out of the hotel at the same time as the infinite number of people got out of each bus as it arrived, he would have only one set of infinite people? He could then fit the infinite number of people outside the hotel comfortably into the infinite hotel. Check out http://www.psawa.com/The_7_Dimensions_of_Everything_and_Nothing.html

  99. rianma 123 says:


  100. Michael Sebastian says:

    Have to inform the infinite people on the bus that they are already in the infinitely large hotel with infinite people.

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