How An Infinite Hotel Ran Out Of Room

Publisert 10. mai. 2021
If there's a hotel with infinite rooms, could it ever be completely full? Could you run out of space to put everyone? The surprising answer is yes -- this is important to know if you're the manager of the Hilbert Hotel.


References: Ewald, W., \u0026 Sieg, W. (2013). David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917-1933. Springer Berlin Heidelberg. --

Gamow, G. (1988). One, two, three--infinity: facts and speculations of science. Courier Corporation. --


Special thanks to Patreon supporters: Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal


Animation by JD Pounds and Jonny Hyman
Thumbnail by Iván Tello
Music by Jonny Hyman and from Epidemic Sound and E's Jammy Jams (Hotel Lavish - Radio Nights, Steps in Time - Golden Age Radio, What Now - Golden Age Radio, Book Bag - E's Jammy Jams, Arabian Sand - E's Jammy Jams, Firefly in a Fairytale - Gareth Coker)
Written By Derek Muller and Alex Kontorovich
Sound Design by Jonny Hyman



  • There is more than one infinity, there is Aleph Null, Omega, Omega Squared….

  • i know this must be a stupid doubt but why dont the newcomers just move to the room next to last... why does the manager has to bother everyone

  • WTF

  • Veritasium: "Have you ever had a stroke?" Me: "No." Veritasium: "Would you like to?"

  • I am slime

  • This is why everyone hates mathematicians. This is so stupid. If there aren't enough rooms for infinite guests then obviously there are not "infinite" rooms. You're describing infinity and not infinity.

  • who cleans the rooms?

  • Scp-182737362821818273364646328929181827374664646467721819191928373746-hotel

  • 1:26 some people are still going to their room... literally it took years to the poor guy in the room 4673480886491. Poor Marvin

  • wow, that's a good cliffhanger! subscribed!

  • If the infinite names has just two letters, doesn't that mean that their is going to have the name that formed throught flipped diagonally picked letters? I mean their is no end to number of combinations so we can computer it, but no matter how you pick and flip the letter, its still going to be someones name

    • The new name is indeed present on the bus, as it follows the rules for name creation. However, it provably does not show up on the list that was supposed to have all the names. Such is the problem.

  • I want to show this video to my math teacher

  • Infinite Hilbert Hotel's

  • *my roblox character named infinite is satisfied.*

  • I like the part where he said infinite

  • make everyone (including the party bus) go up one

  • I don't get it, surely the diagonally formed naming would also eventually appear in the ordinary course of listing. Seems like they are all double booking to me. But if all the names never ends it's like an infinite number of possible irrational numbers which is kinda cheating, give them a room number of pi, they'll spend so long reading their room number they'll never leave the bus.

  • Simple solution: rename the rooms the same way that the people are named. Each person goes into the room with their name.

  • The problem with the uncountable infinite fitting with a countable infinite is simply a problem on paper. This is a theoretical problem for the person who wants to plan ahead and make this graph before they arrive… this is not a problem in real life, because in this scenario, this uncountable infinite is in fact countable one by one. We can’t plan ahead for every room, but if they all line up and keep coming in, we will always have another room for them and we will always have another row to fill out on the spreadsheet. You didn’t plan for the opposite letter person showing up, because you couldn’t count him? Well you demonstrated how this is not a problem as you explained it… you wrote it down. If it can be written down, it exists it is tangible it can be counted. The problem isn’t in getting them rooms, the problem is in counting the hypothetical.

  • Infinite is not a number

  • “Your mother arrives”

  • There is a solution for the person without a room to get a room; that’s the manager’s name! The hallway with infinite rooms is the room!

  • The problem I have with the "differently sized infinities" concept in math is that it doesn't make sense to me. You start with the premise that there are an infinite number of passengers with an infinitely long, unique name consisting on the letters A and B. and you write down all the rooms in this infinite hotel and all the names of the passengers, which is part of the premise so we assume that this impossibility is possible. if you were then to flip the letters along the diagonal to make a new name and don't find it on the list, what you have discovered is that you haven't written down all the names not that the list of passengers must exceed the list of rooms as both are, by premise and definition, without limit. The only way to have this problem is to impose a limit on one of the factors, in which case, it's no longer infinite and the whole thought experiment just kind of falls over. I know there was a whole math civil war over this very discussion but I've not yet had anyone explain how the problem doesn't violate it's own rules. I can see it as a way to illustrate how to deal with problems of incredibly large numbers that would be almost impossible to work with practically, but not as a proof of the logical existence of the childhood taunt "Yeah well... mine is infinity plus one!" which is how I've seen it presented.

    • @Hydros92 Time isn't part of this. You set the rule for how the new name is generated and there it is. Notably, we do not know what the new name is. Can't know, really. unless we contrive the list to take a certain shape. But the rule works. It takes in an input that makes sense and it has an output that we understand, so such a name must exist. We don't actually have to create the name for it to prove this reality. Notably, your assertion that an infinitely long list must contain all possibilities is trivial to disprove. I just have to give you an infinite list that doesn't do that. Check it: BAAAA... ABAAA... AABAAA... AAABAAA... and so on. Not only are we missing tons of possibilities, but we don't even have one with more than one B. As for differently sized infinities? There's no grand need for it. It's just true. If it weren't true then it wouldn't be true. Such is math.

    • @eggynack Thank you for taking the time to address my confusion on this issue. I think perhaps my problem with the logic of this thought experiment is that the premise states that there are infinite passengers, so the list will be infinitely long, all the names are infinitely long but also unique, so all possible permutations must then exist in the list and to carry out the task of flipping a character in each name would take an infinite amount of time and would never complete, so to prove the name wasn't on the list would be an impossibility as you could never generate the name to prove that it wasn't there, without introducing a finite value. Maybe I would be able to gloss over that if I knew the need to have one infinite to be larger than another infinite as it doesn't sit right in my head on why we would ever need to make such a logical loophole.

    • @Hydros92 Infinite does not mean all. An infinite list can be missing stuff. It can have limits. The only thing an infinite thing is not allowed to be is finite, and this list is definitely not finite.

    • @Tom Svoboda I appreciate you taking the time to try and reframe the problem. Maybe its my understanding of the usage of the word infinite in this context, as to prove something isn't in an infinite list you'd need to impose a limit, specifically the list would be infinity minus the entry missing, which would make it finite rather than infinite and thus not satisfying the original assumption of an infinite list rather than proving that there exists greater or lesser values for infinite.

    • It's a proof by contradiction. Assume that you can list all people -> find a name which is _provably_ not on the list -> the original assumption was wrong, so the listing is impossible. Which we wanted to prove to begin with. Also it's unnecessary to formulate it as a proof by contradiction. The diagonal argument proves this literal statement: "for any list of infinite binary strings indexed by natural numbers there exists a binary string not on the list". Clearly this implies that a list of _all_ binary strings cannot exist.

  • a infinte amount of guests is the same like a infinite amount of busses with infinite guests.

  • I see a problem with "an infinite number of buses with an infinite number of people"...isn't that equal to ONE buss with an infinite number of people? How can u use plural on infinite?

  • I feel bad for any developer who has to write the software for this hotel

  • Explain why this is irrelevant & dumb af, there's no hotel this big you donut

  • This is flawed. A hotel with infinite rooms, all occupied, by an infinite number of people. Then a new person is invoked that is not part of the infinite number of people. If invoking a person is allowed, why not invoke an empty room?

    • @Joop Meijer that depends. "infinity + 1" equals infinity if you're concerned only about size (adding an extra element to an infinite set doesn't change its size). it doesn't necessarily equal to infinity if you care also about an ordering. for example the natural numbers are implicitly ordered into an infinite line 1 < 2 < 3 < 4 < .. adding +1 corresponds to adding an extra element to the end of this line, which changes properties of the ordering: the new line now has a last element which wasn't true before. so "infinity" and "infinity+1" are different objects now. the amount of elements stays the same though, it's only the ordering that got change. look up cardinal numbers and ordinal numbers.

    • @Tom Svoboda Hi Tom, would you say that infinity + 1 is greater than infinity? Is this allowed?

    • Your argument doesn't make any sense. Consider a hotel for 20 people which is full. Then a new person is invoked that is not part of the 20 people. If invoking a person is allowed, why not invoke an empty room? Have I shown that the concept of hotel for 20 people is flawed?

  • To get this job your qualifications must be insane

  • Your favourite element is molybdenum since your profile is atomic no. 42 and molybdenum matches this atomic number so I predicted that your favourite element is molybdenum.

  • It's name is "the infinite hotel Paradox"

  • Think about this... There are an infinite set of positive integers. There are also an infinite set of positive even integers (and odd), but wouldn't, by their definition, the even numbers be a smaller set of infinite numbers? a 1/2 infinity? While both are infinite, if you had a set of them, the infinite positive integers would contain the infinite even and odd numbers set, so wouldn't the first (by definition) have to be bigger by at least a factor of two? It's a very fun thought experiment. The original problem that is referenced here is "are there more integers between one and infinity then there are numbers between 0 and 1, where instead of flipping the A's and B's, you just take the digit and tick it up one, creating a whole new number that wasn't on the list before. Different interpretation, but same principle, (and yes, there are, for the same reason.)

  • I am confusion

  • There is a great way of understanding different infinities Imagine the infinite amount of numbers in between 0 and 1, you can take an unlimited amount of decimal numbers. Now take every single number between 0 and 2, here it is near double of all numbers between 0 and 1, which is infinite, but there are still infinite numbers between 0 and 2.

  • literally just put the next guy to show up in the next room...

  • That's over thinking

  • This in no way makes sense to me. Infinite number of rooms=infinite amount of guests. There should be at least on room left

  • Some infinities are bigger than other infinities. Boom! Problem solved.

  • What if you tell ABBA and the rest of the party bus to treat the A’s and B’s in their names like 1’s and 0’s and convert from binary to decimal to get their room number? I think they would also need to put a 1 in front of every name so that names starting with 0’s are still unique

    • On second thought they don’t need to bother converting, just send ABBAAAAAA… into room 1100111111… send BABABABA… into room 10101010… by telling them to use the simple strategy of changing A’s to 1’s and B’s to 0’s and popping a 1 in front. If they have infinitely long names I’m sure they don’t mind having infinitely long room numbers, no?

  • Just imagine being in room 938749837413058145 and being told to go to a room with double that number :(

  • "infinite hotel infinite rooms infinite people infinite buses infinite spreadsheet" very realistic stuff you see.

  • This seems like something hotel managers have nightmares about and wake up in a cold sweat

  • Tricky but I find it cute topic haha

  • Ow, my brain

  • Or you know. Just tell them to walk down to the next vacant room.

  • How much people of the ABBA-people could get a room and how much would be left without? :D

    • @Tom Svoboda I'm not sure if you checked the link I was suggesting to use stack since unlike set you can not see the full stack, however since the cardinality is 2^alpaha_0 it won't make a different. If you have a stack based hotel where everyone travels from room 1 to their room then it will fit because you are just adding to a infinite stack

    • @gamecoolguy619 what exactly do you mean by "make it countable"? the problem is unambiguous, it's impossible to pair natural numbers with infinite binary strings. there's no way around it.

    • @Eins gleich Null nope since 2^alpha_0 is bigger than alpha_0 and any set with that cardinality is uncountable There is a way to make it countable (it's on computer science stack exchange): /questions/141522/hilberts-hotel-for-guests-with-infinite-string-name

    • in this case it becomes countable ;)

    • Alpha_0 can fit, 2^alpha_0 - Alpha_0 will be left over

  • The colossal scorpion unequivocally enter because sphere notably boil on a few fierce cone. materialistic, cut magazine

  • But infinity isn’t a number

  • My brain hurts

  • They can just go get a room and give the money to the manager while passing near him

  • My head hurt

  • What if you just assign each new guest to room 1 and tell everyone to move 1 room up. Repeat till infinity

  • so time does not exist in this experiment? even if it takes the smallest amount of time to give every guest a room. the manager would need infinite amount of time for that. so why bother about the guest after your first infinity. he waits an infinite amount of time. so the manager never has to deal with him

    • @Tom Svoboda but u will never reach the 2 second mark . 1 + 1/2 + 1/4 + 1/8 + 1/16 .. is near 2. but not 2. i was thinking. u have to "end the first infinity" to deal with the next one. and that sounds strange. how to end an infinty? to be honest i maybe only understood half of what u said. maybe i missed the point

    • @Stefan Paetrow If you absolutely must, math can model time in this experiment without a problem. For example you can say that it takes 1 second to accomodate the 1st guest, 1/2 second to accomodate the 2nd guest, 1/4 second to accomodate the 3rd guest etc. The hotel will be full after exactly 2 seconds. Or you can enlarge your timeline appropriately. If it takes 1 second for every guest to accomodate and they start accomodating at time t=0, then infinitely many guests take [0,∞) time. But you can picture two timelines stacked next to each other (connected at +∞ of the first timeline and -∞ of the second time line). Any point on the second half of the enlarged timeline now corresponds to a situation where infinitely many people have already walked into the hotel. Math is not supposed to do silly things like I've just done, but my point is that it can, easily. Math is the general framework, physics is just setting the parameters right. Getting rid of the constraints dictated by physics (and sometimes common sense) and treating the framework as a thing on its own turned out to be extremely effective. A lot of stuff possible in math is useless and unrealistic, but that's a completely logical thing. Just like a random string of symbols will almost surely not be an english word, a "random math scenario" will almost surely not resemble anything with a direct counterpart in the real world.

    • @Stefan Paetrow you have a nice time too buddy

    • @mar98co1 i think this kind of stuff hits the boundries of my knowledge and imagination at the moment. numbers like "i" in -2=i² .. or quantum theory. which i dont understand. but thats why i like this channel. it makes me think and also sometimes gives answers. (it just recently answered how to calculate pi. what i didnt know for a long time). thank you for your replys @mar98co1 and have a nice day ^.^

    • @Stefan Paetrow Well, immaginary numbers started as pure mathematical nonsense. Look at the role they have in physics now, they're absolutely central. This type of stuff is plenty useful for mathematics and has some niche applications in physics for now

  • Owner of the hotel must be infinitenaire

  • it must suck toget the last room

  • It's completely Illogical that ALL the rooms are full but if someone new shows up they MAKE a new empty room, so they weren't all full?. Who makes this stuff up?

    • They do not create a new room. It's the same set of rooms. One of the already existent rooms is rendered empty by moving the guests.

  • and there is an uncouble infinite amount of possible diagonals so you can never list the whole thing because there would still remain an uncountable infinite amount of ppl not on the list thats why its uncountable

  • Just put people from all busses in one infinite queue.

  • I would buy an infinite hotel for the infinite hotels which are housing intimate hotels in their infinite hotels.

  • The Hotel with aleph null rooms' manager: hold my number.

  • I think at the end there you meant something a little about 010001001011010101001111110101000100101001111000100100100111000101100100100101001001110011011100000010101111010101001010101-

  • My new favourite joke: Yo mama so fat when she came to the hilbert hotel they said, "sorry, no room".

  • Teacher: The test isn’t hard! The test:

  • .

  • wrong thete is infinite rooms and there are infinite people even 9999999999 infinities + infinity = infinity sheesh

  • I remember this old Vsauce video.

  • wow thanks i finally understood why its bigger, a really good way of explaining!

  • Why not the first client goes to room 1, second to room 2 and so on. The infinite clients from the bus could just take number 123433, 123434 and so on to infinity ??

    • @ArmaGhedoN That's not a mistake. It's just a thing that happens.

    • @eggynack You did a little mistake in the beginning. The bus never empties, nor the hotel gets full.

    • Well, say they do that. The infinite bus arrives with its infinite guests, and each guest goes to the room that matches their number. Once you've emptied the bus though, the hotel is full, yeah? No matter what room you name, I can state which guest is inside that room. So we've just kinda kicked the can down the road, because, y'know, what happens when a new guest arrives? Or a new bus? We're back at the start of the video with its hotel assumed full and need to start moving guests.

  • I would have gotten the fields medal had you been my maths teacher 🔥😭💝

  • Pov you saw this from tiktok

  • You can't just ask all the guests to switch rooms every time a new guest shows up. that kinda ruins the whole purpose of the Hotel.

  • I guess they just had a lot of people come in, that's why it's full...

  • This is a riddle from ted ed they are copying them

  • Is Hilbert Hotel up for Sale? 🤔

  • Yes.

  • You did this to my brain at 2am...

  • I hate all of you. Infinity is finite. fml

  • lol the infinite bus had me crackin up

  • "But there is a limit" Yes, the fire code

  • high quality production! watching this in 2160p!

  • i am infinitly confused

  • I'm not convinced. Simply unload the busses one person at a time. Their names are arbitrary.

    • I feel the video put this a bit poorly. It's not simply a bus like the previous buses except you named the passengers differently. There are necessarily more passengers in the bus if they can be named that way in the first place. If you tried to rename the passengers to match up with those earlier buses, you'd completely fail. And, while you can indeed get started unloading the bus one guest at a time, that's never gonna empty the bus, or even get meaningfully started. The big question we have to ask is where each guest is gonna end up when they leave the bus, but any attempt to answer will leave infinite guests unaccounted for.

  • i just learnt limits in maths lol

  • Infinite infinite busses aren't any more then just one infinite bus since it is still infinite

    • yes they are, it's a basic mathematical fact

  • Theres a lot of Stupidity on this video that i cant even write a comment on it

  • This video is really really stupid.

  • The about chart is actually a 1 and 0 and it how pics were invented.

  • 1:11 long bus

  • now I want to know how this led to the iPhone... Does anyone know the answer?

  • Words cannot express the feeling of dismay that welled up inside me when infinite infinite busses showed up

  • Counter to the whole point of the video, but the clerk monster just needs to change the way the rooms are identified to match the way the people on the bus are.

    • You can't change the rooms to match the names any more than you can match all the guests up with rooms. The same proof applies equivalently to both.

  • Another infinity teaser... is 1/0 positive or negative? If the divisor approaches zero from the positive side, it would be an increasing positive number, but what if you approached zero from the negative side?

  • Benzoate Ostylozene Bicarbonate

  • Surely first guest goes to room 1, second guest to room 2, third guest to room 3. Whenever someone comes you move them to the end, +1 every time.

  • Well… Assuming that there are no other way but to walk, there’s a limit to the number of rooms. Assuming a walking speed of 5km/h (3miles/h), if the guests arrive at 4PM and leave at 10AM, you can’t get much farther than 45kms long, since you have to get to your room and back. And that means no sleeping, only walking for 18 straight hours

  • An infinite number of infinite capacity buses showing up at a infinite capacity hotel. Amazing.

  • Amazingly insightful

  • Ted ed?

  • If you replace A with 1 and B with 2 and apply the same technique of creating different combinations, you can always keep generating new unique rooms (with its room number containing 1s and 2s) for every unique person (with a name containing As and Bs).

    • @Prabhdeep Singh Infinite stuff is usually static. Like the natural numbers. They're not perpetually generating themselves. There's rules for what constitutes a natural number, and then the set just exists, pristine and unchanging. If I come back to the set tomorrow it won't have grown a few numbers. They are definitely infinite though. And they are pertinent here, cause each hotel room is associated with one of the natural numbers. They don't self-generate, but there also aren't any new ones to find. The technique that works for the names does not work for the room numbers, as the outcome of the process will not be a natural number at all. The reason the names are greater in quantity than the rooms is because, while you can map the names to the rooms such that every room is accounted for, you cannot do so in such a way that every name is accounted for. Simple as that. Sure, it's perhaps an unintuitive result, but we're working with a different kind of infinity here. One with properties distinct from those of the hotel. It's an infinity infinitely greater in magnitude.

    • @eggynack Replace "generating" with "finding" and the logic still holds. And there can not be a static pile of rooms because it will imply thay they are limited in number and hence finite. I may be wrong but I am finding it hard to believe that some infinities are bigger than others. How can one claim that one thing is bigger than other if both things are capable of extending themselves endlessly with no limits?

    • Rooms are never generated. It's just a static pile of rooms.

  • Hilbert's Hotel? Naah. Cantor's Condominium.