How An Infinite Hotel Ran Out Of Room

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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.

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References: Ewald, W., \u0026 Sieg, W. (2013). David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917-1933. Springer Berlin Heidelberg. -- ve42.co/Ewald2013

Gamow, G. (1988). One, two, three--infinity: facts and speculations of science. Courier Corporation. -- ve42.co/Gamow1947

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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

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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

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References: Ewald, W., \u0026 Sieg, W. (2013). David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917-1933. Springer Berlin Heidelberg. -- ve42.co/Ewald2013

Gamow, G. (1988). One, two, three--infinity: facts and speculations of science. Courier Corporation. -- ve42.co/Gamow1947

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

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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

@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 ??

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...

There was too many people.

This is a riddle from ted ed they are copying them

Ted Ed isn't the one that created this problem so

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.

Yes

This Seems a very profitable business to me 🙂

To me, this is a problem of definition not of infinity. (Please help me make people see this comment.) 1) Solution for the Last Bus: Well... I make a new name, using the diagonal, as in the video. Then i give this New Name the room -1. Then I start the diagonal again and give the New Name room -2, and so on... That's it ! ^^ Sure, the task will take an infinite number of time, but like the other task on the first speed sheet. So, if i can complete the task, like the first one. All it remains to do is to give every positive numbers on my sheet an even number room, and every negative numbers on my sheet an odd number room. 2) Not convince ? Here is other an solution : Ok, let's say I decide to write all the names... There is an infinite number of names, which means I could go like this: AAAAAAAAAAAAAAAA ... BAAAAAAAAAAAAAAA ... ABAAAAAAAAAAAAAA ... (...) If i decide that A equal 0, and B equal 1. Then I have a list with ALL people write in binary notation. 000000000000000000 ... 100000000000000000 ... 010000000000000000 ... 110000000000000000 ... 001000000000000000 ... 011000000000000000 ... 101000000000000000 ... 111000000000000000 ... etc... You can even replace this with a ten digit notation if I want ! Conclusion : In fact... It's just a difference in definition / notation. One notation is countable, so your mind can pretend not having to do a never ending task... The other is uncountable, so your seems compel to face a never ending task... But, in fact, the difference is just that letters are not numbers. So, is it really a variation in 'infinity size' ? Or rather a gap in the founding of what we call 'mathematics' due to a narrow definition ? To me, the 'size problem' is only showing up due to a poor conception. The first task is a so called : Cartesian analytical approach. The second require a so called : Systematic approach ! That's all ! (You can also call that a philosophical paradox, a conflict beetween Ordinal and Cardinal sets ! Set theory ! )

No your mind is stuck by the fact you don't want to complete an infinite task. But there is already one, earlier in the video ! You are stuck in 'cartesian analysis'. ABABABAAAAA ... It's just 010101, it will soon appear at one of the 64 ways to have 6 digits in binary notation. 64, less of course the previous steps already wrote earlier. (The ones with 1 digit, with 2, with 3, with 4 and with 5.) Keep in filling the spread sheet, there is no name you can't write that way ! You just refuse to 'bypass' the uncountable by achieving a never ending task, as you have nevertheless done many time in the video. The only thing you have proven, are the limitation of the way you conceive things. I suggest you learn more about what is called a 'super-tasks' and the paradigm shift underway in mathematics, including all the work to redefine what is considered to be of the order of logic in the context of modern math, such as new set theories.

@Kévin Kozlowski Neither of your approaches works. The first one fails because you haven't proved that all missing names will eventually appear as the diagonal name (they won't), the second one fails because it misses e.g. the name I've given ABABABA... (I haven't found any correction in your post). You're trying to refute the irrefutable.

@Tom Svoboda Oh, you are right. Many thanks, i corrected the thing. ;)

you've missed e.g. ABABABA...

POV:This was in your recommended

can't you just assign each guest a number when they come through the door? seems more of a problem with how you organize your guests. Do we all agree that infinity is infinite but that on paper it's sometimes not the case depending on how you organize your data? I'm just trying to grasp what is being explained in this video. Is this to illustrate countable vs uncountable infinities?

@SoyKaf Right, and what I'm saying is that it is literally state the order the guests come through the door. Like, expanding on my previous example, say all the real numbers walk through the door, with their order based on size. The first one through is, I dunno, zero. That's guest one. So who's guest two? Pick anything. Say .1 for the sake of argument. Infinitely many guests go through the door between zero and .1. In point of fact, literally no matter what guest you pick, there will always be infinitely many guests between them and the first guest. Which gets you to the weird truth. There is no guest two. Can be no guest two. If you pick two arbitrary guests, it may be possible to say which one entered first, but setting them in this ordered list is impossible. This is what uncountability means in the first place, that it is not possible to set them in any kind of order. It just won't work.

@eggynack but what I'm saying is, you don't care who the guest are, all you care about is the order they came through the door. meaning guest ABBA comes in, he's now guest 1, guest BAAB comes in, he's now guest 2, etc... to inifinity. all you need to do is assign them a number when they come through the door and you now have room for them in the hotel. Am I missing something here?

You cannot, for the same reason you can't determine what the next real number is after one. You number the first guest to come through the door, but figuring out who the second guest is is impossible. And, yeah, the last part is demonstrating uncountable infinity stuff. You may note that, if you pop a decimal point on the left and swap the A's for 0's and the B's for 1's, then you wind up with the set of all real numbers between 0 and 1 in binary. A decidedly uncountably infinite set.

Saw a video about this a few days earlier but didn't really understand now I do thanks

just imagine how much money you would get

Infinite Money From Infinite People To Buy Infinite Amount Of Candy

Anyone know what the connection to smartphones is?

If everyone is identified by a unique name then you just tell everyone to take their name, put the letter B in front of it and then swap each point of their name so that the A is now a 0 and the B is now a 1 (putting a B at the front will prevent casses where AAAA and AAA are just 0) after this everyone now has a unique binary number assigned to them and you can give them rooms

@Jaime the problem is with cantor's diagonal not being suitable for this, I provided a solution on computer science stack exchange (not sure how it translates to maths) /questions/141522/hilberts-hotel-for-guests-with-infinite-string-name

Infinite binary sequences have the same problem, cantor diagonal's argument works the same way. you don't need to worry about the 00000... becoming 0 tho, sequences aren't reduced like that but they arent numbers

math problems be like:

( Infinity + 1 ) you

Surely at some point the solution becomes: Build a second infinite hotel.

@Craftypup453 of course, they would bith have to be built on infinite land

But where would you Bill on the second the large infinite hotel is the first hotel is covering the infinite space that is already infinite

I was watching this on my pc

You haven’t looked at the other side

Mathematician question be like How many chocolate cars can fit in my ford mondeo? How many bread crumbs can fit under my keyboard? How many snickers do you have to eat to get diabetes?

4:26 i looked at it wrong and thought verbatim was being held hostage an screaming through the spreadsheet oops

Or do none of that and just send them to the next empty room counting up one every new arrival

Amen

It’s 3 am

You forgot the most important part With infinite rooms comes infinite bills

Couldn’t they change the As and Bs into 1’s and 0’s then line them up in binary order then allocate a number based on the customers binary number

no because there would still be a number that is left out.

Wait. If there were an infinite number of matches between ababe etc. There would always be that combination... Infinite is infinite. It is an undetermined quantity but is infinite, so wtf

Right

Just have everyone on the bus take a number.

If the roomnumbers had decimals, then they would have fit. Basically roomnumbers now used are integer numbers, while the ab-people on the bus are real numbers. What a boring problem. i also wonder if the ab-people are simply binary numbers. Every binary number is an integer, so they should be equal amount. So much wrong with this problem.

Every binary number is certainly not an integer. You can do real numbers in binary just as you can do integers. .010101... is a binary real number, for example.

Yes

Welp... There goes my 50 brain cells...

Popular hotel

If we could move the person to another new room, then why can't we just move the newest customer to the end of the hotel?

Because there is no end. It's infinite.

Because we needed a problem to solve

Some infinities are larger than other infinities. TFIOS fans were hit hard.

There was a Professor who’s lectures used to be very interesting since he had an interesting way to teach and explain the concepts. One day, in the class, he asked the following questions, 1. What is ZERO? 2.What is INFINITY? 3. Can ZERO and INFINITY be same? We all thought that we knew the answers and we replied as following, ZERO means *nothing* INFINITY means *a number greater than any countable number* ZERO and INFINITY *are opposite and they can never be same* He countered us by first talking about infinity and asked, ‘How can there be any number which is greater than any countable number?’ We had no answers. He then explained the concept of infinity in a very interesting way. He said that imagine that there is an illiterate shepherd who can count only upto 20. Now, if the number of sheep he has less than 20 and you ask him how many sheep he has, he can tell you the precise number (like 3, 5 14 etc.). However, if the number is more than 20, he is likely to say “TOO MANY”. He then explained that in science infinity means ‘too many’ (and not uncountable) and in the same way zero means ‘too few’ (and not nothing) As an example, he said that if we take the diameter of the Earth as compared to distance between Earth and Sun, the diameter of earth can be said to zero since it is too small. However, when we compare the same diameter of earth with the size of a grain, diameter of earth can be said to be infinite. Hence, he concluded that the same thing can be ZERO and INFINITE at the same time, depending on the context, or your matrix of *comparison.*

I didn't understand the last part? ☹☹️🤔🤔

U could've saved me time by stating what your example wants to prove. "Some infinities are bigger than others"

But that's not how math, nor SVsoft works. Why would he make such a boring video?

If the infinity hotel can't hold all the AB people, how did the bus that carried them there do it?

@Cptn_n3m0 he is actually saying that (Infinity) < 5x (Infinity) But infinity is infinity

@Cptn_n3m0 yes.

Because the hotel is countable and the bus isn't

Bro what he did is called overthinking

This is the infinite hotel paradox

my brain hurt

This video could also be called analysis class.

🤯

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bro why don’t you have a big circle with lots of rooms and everyone goes in a hallway and then they just go to a number they got and then another set of number will pop up and they got a sequence of numbers that match this but the sequence of numbers are infinite bc this is fiction and I can think of logic spthat can do this but one large hallway is too inconvenient for a hotel

What happened to this place during lockdown? " we have infinite rooms, plenty close to the pool, just pick one"

Get an infinite amount of construction workers to build an infinite amount of more rooms.

Could it be described as there are infinite numbers between 0 to 1 but never a 2?

About here 1:58 I said "ok this is getting ridiculous"

Amazing video, as always. But… what happens if every room in the hotel also has an Infinite number associated to it, like room 01010000… etc. Then would it be possible to fit everyone?

Sure, though it would have to be a very different sort of hotel.