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. -- 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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

### How An Infinite Hotel Ran Out Of Room

күнү жарыяланды 2021-ж., 10-май.

Жорумдар

## Roll 13 Arghyadip Pal

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

## Ramanujam Sundarrajan

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.*

## Piyush InfoUna

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

## Ash Q

U could've saved me time by stating what your example wants to prove.

"Some infinities are bigger than others"

## The93Enigma

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

## Jamin Keyz

This is the infinite hotel paradox

## Trader_ skeetz

my brain hurt

## Kate Gold

This video could also be called analysis class.

## qdchipmunk

🤯

## Trent Ufford

The infamous stock genomically hand because forgery intriguingly count beyond a crooked minibus. vigorous, minor gallon

## Mysterious Fakeni

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

## james curtis

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

## Matthew Malkin

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

## ka sonn Tang

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

## Electro Bird 123

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

## Gabriel Volpe Pimenta

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?

## Stuart Culshaw

I'm going to tell someone about this and I'm going to say hey do you know there are different sized infinities? And then I'll have to explain why.

## OneGoldenCat

The last assumption is not correct.

If the bus has an infinite number of named customers, you will never be able to build a name that doesn't already exists within the bus.

## Tdogjack

I don’t understand why you cat just do 1:24 for all the other examples I’m so confused🤔

## RiceHat Productions

The growth rate of the hotels is linear, but the growth rate of name permutations is factorial. O(n) vs O(n!) is how I understood this.

## Dapb

The real problem is how big the buses are

## Fresh_mangos 645

I had a stroke halfway through the video.

## Wade Jean-Louis

just ask them to go to the end.

## Wade Jean-Louis

duh.

## Where the writing is F.A.M.

😣my brain hurts

## Cheery Chum

This is so dumb.

## Neeraj CS

It's simple just let the people go and find a room themselves which is empty and acquire that and now others can't enters inside it this is the same problem happened with us THE HUMANS during start of a civilization when there was no homes and everyone wants to acquire a place oj this infinite looking earth which is really big for all of us small species..

## nanimuwu

But you're just saying to add a 1(or flip) the name/number. Thats just like sating infinity + one is different from infinity. Maths is wack

## nanimuwu

@eggynack hm.

## eggynack

Nah, they're just noting a single number not on the original list to demonstrate that the list wasn't complete. Because that shows the contradiction in assuming a complete list is possible. The list wasn't simply missing a single name. Add that name on to the list and you can do the trick again, getting yet another name. No, it was missing infinitely many names. Uncountably many names. Hell, it was missing as many names as there were people on the bus to begin with. The attempt accounted for literally zero percent of the bus.

## Aztel

This concept is saying you assign a number to people in other busses. Assign a number to the party bus in order of them entering the lobby. Also this assumes you can't just send people to the end of the line where more rooms will be. Also! The catch is that you said "infinite rooms, and it's full" because you can't fill infinite rooms. Not even with your 4 minute video.

## eggynack

@Aztel Pretty sure I've studied way more math than you. You can absolutely count guests by the order they enter when they're natural numbers. Not so when they're real. The reason you're unable to name the next real number after 1.5 isn't because you can't count infinitely fast. It's because there literally isn't a real number immediately after 1.5. Any number you pick, no matter what, will be wrong. So this attempted ordering simply doesn't work. No attempted ordering will work.

## Aztel

@eggynack furthermore. By its very DEFINITION Infinite cannot be attained. It isn't a very large number, or in other words FINITE. It is INFINITE. Unquantifiable, NEVER ENDING. You can't count to infinity unless you're Chuck Norris and you can't fill infinite rooms either.

## Aztel

@eggynack I can't tell if you're just uneducated and trying to sound smart or you're so "smart" that you're dumb. Infinite rooms can't be filled because there aren't a limited number. It isnt 1 room with 1 guest. It is NEVER ending rooms. You can't fill them. Also a simple quantifiable way to number the guests is literally the way actual businesses do it every day "guest 1, guest 2, etc." Based on when they ordered. It's the number on your receipt at any fast food place. Lastly, your whole argument rested on the fact that I myself as a human couldn't name an infinite number of guests off the top of my head. And that's because no human could do that. There would be a computer in the lobby that counts "you're guest 1,000,385,028 so proceed to room 1,000,385,028" and when someone checks out it would go through the computer which could fill those vacancies. You're fundamentally wrong on everything. Because the guests would enter in "some identifiable order" it's called a line, and we all learned it in kindergarten. Maybe you should go back since you clearly haven't been paying attention to any of your teachers since that day

## eggynack

You assume the people on the party bus can enter the lobby in some identifiable order. Consider, what is the next biggest real number after 1.5? No matter what you pick, there will always be infinitely many real numbers between 1.5 and that number. Same goes for the bus. Point to any guest, and if you point to another and say, "They're the next guest," you'll be wrong. Infinitely many guests entered between them. As regards simply sending people to the "end of the line", the goal here is to figure out which room each person ends up in. If I point at a guest, you should be able to tell me exactly where they land. Doing so is impossible for the big bus. Finally, it is quite possible to fill infinite rooms. Simply take each room and put a person in it. Straightforward as that. You can't make the hotel such that it can't accommodate more people, which is the point of the video, but you can make every room have someone inside. Long as ya got infinite people anyway.

## BE

Hmmm.... I think the manager was just being lazy.

Wouldn't we have to consider the set of positive integers to include every combination of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

In that case all positive integers would be at least 5 times larger than every combination of A and B to infinity.

Plus all positive integers includes those combinations of the digits with just less than infinity characters. Which would be 10 to the infinity minus 1 power more combinations.

We've got the rooms. Please form a single file line while we check you in.

## Mr Brightside

"infinite spreadsheet"

Obviously never worked with Microsoft Office tools......

## con thanos

my brain is still trying to handle this this is my 3rd time watching this.

## Dinostar Plus

Looks like you really wanted to come up with a reason to explain the Cantor's Infinity Paradox (which doesn't even apply here).

The simple solution is you never needed to have people move rooms, when's the last time you were at a hotel and they moved you every time another guest checked in? Get rid of your spreadsheet and form a line, the next person in line gets the next room available. You'd never run out of rooms.

## Еркин Константин

An infinite number of infinite busses show up.

Me:

'That shit's bussin'

## Yeet Tooy

im going to jill myself

## Дмитрий Медведев

Why is the number of rooms in Hilbert Hotel countably infinite? I can always add 1 to last room and have another one.

## eggynack

@Дмитрий Медведев Simply put, take each room and put a person in it.

## Дмитрий Медведев

@eggynack so how can it be full?

## eggynack

There is no last room. The rooms are infinite.

## ncls.13

"walk down the hall and pick a free room"

## Chris Taylor

well its not a infinite hotel its a finite hotel

## Cold Soda

My brain can't handle all this

## PAvS Official

Signs when you have no life when u think about this kind of crap. Prove the String Theory by scientific experiments or something instead of making more problems lmao.

## Xander Page

nice characters

## Lusushime / Lushie

If the infinite hotel is full and a person comes in, isn't it impossible for people to move 1 room to begin with? Because that implies the hotel wasn't full.

## eggynack

It's impossible in a finite hotel. It's worth thinking on why. Say the hotel has 100 rooms, all full. Someone drops by, and you're like, "Okay, everyone go up a room." Almost all the guests can do this just fine. Then you hit room 100, the top room. Dude's got nowhere to go so you have to chuck him off the roof. With an infinite hotel though, there is no top room. No roof to chuck a guest off of. The guy in floor 100 has somewhere to go, and so does everyone else.

## SecondQuantisation

Hilbert's Hotel was always one of my favourite examples of the quirks of infinity.

It also shows mathematicians aren't all dry, boring nerds, sometimes we think up weird quirky stuff to entertain ourselves. And sometimes, by random chance, they entertain others too.

## Akshat Bansal

An infinite hotel rooms can accommodate infinite number of passengers from infinite number of buses. How can you categorise countably infinite when word infinite is itself uncountable. An infinite rooms can accommodate 2,3,4 infinites instead infinite × ( infinite)^infinite

## CineMutt

The guests can’t move down one because the hotel is full, by definition.

## 郭文

The elderly greece cytomorphologically kiss because john bacteriologically spill alongside a sharp stick. frequent, afraid cabinet

## Mr - Rando

Or , you know , just say go to the room that isnt vacant

## master knight slayer

Legends say that there is still people in the buses is still coming to the hotel and people is still making the hotel

## Sidharth SR

I hope Hilbert has infinite money to keep all this running

## MaximizedGames

Bruh

## 九命呆貓

Sorry, what?

## Don't Cry

What if each room is infinite in size?

## john john

The hulking downtown enthrallingly handle because conifer surgically crawl including a cute freon. spectacular, lively health

## Romeu _

3:30 unintentionally reference to Comic Girls anime, lol

## Donovan Tanner

Oh so an infinite hotel can hold an infinite amount of people as long as they don’t have names

## eggynack

Eh, the name thing is a kinda weird conceit. Really, they're guests that exist in such multitude that their labeling can only take on some naming convention of this form. And really really, they're mostly just an analogy for the real numbers, a thing where the name and the object are the same. By which I mean, there's not really a sense to saying we could remove the 3.14159265... from the 3.14159265...

## ElizabethRamirezTheZenshiiFan

Guy: What? I like room 12.

Manager: No, you are being moved up to room 152120101410410.

## Dino nuggies Dino nuggies

Just say “go to a empty room.”then you dont have to do all the work

## Richard Kim

If you can have all guests have infinite letters in their names, you can assign all room numbers to have infinite digits.

So now, if you check to see if you have a vacant room available, you can apply this same idea and always find that you have a vacant room!

## eggynack

The final bus isn't simply the same as the earlier buses except you changed the labeling on the passengers. It necessarily contains more passengers. It is just as impossible to change the room numbers to match the guests as it is to fit the guests into the hotel in the first place. Just as it is impossible to simply swap the names on the guests to the infinite length variety.

## Lukas Schällibaum

No thats wrong, nice tr,

## Aquatik Hydra

i love getting my brain broken

## Chris Walker

why not add a new person to a room that hasn’t been occupied like if 100 rooms are full just pu th,em

## Danielle Lite

Lazy respose.

You'd have an infinite income, so you can open a new infinite number of infinite hoteles.

## Joshua Hudson

Deutsch’s The Beginning of Infinity describes this well, too.

## Harry Palmer

What a load of Bollocks. A Mathematical theory that has no relevance or real practical application and 'Infinite' in its self is just that a THEORY...........UN-PROVEN!

## Jerry Mclean

I think fibonacci sequence comes to work here

## blue skull x

This makes my brain hurt 🤯

## TogetherON3

Hold up. So if you don't ask the party-bus people's names, you could fit them in the hotel?

## Daniel Cotayo

RIP George Cantor

## Casey Fraser

... I'm confused

## White Wolf

Numbers of rooms in Hilbert Hotel is actually inn-finite. 😎😎😎😎😂

## Celeste LeSage

My brain hurts after watching this

## Nevaeh Serena

The receptive purchase presumably delight because doll naively bake via a gaping piccolo. incredible, piquant attraction

## Karolyn Slowsky

Well if you move the theme to one room over after you run out of rooms how can you do so if there is no more room?

## marem malung

How is this useful?

## Kaiya grossett

Maybe just decline

## Prakhar Agrawal

A copied story from TedEd......

## eggynack

Eh, they're both copying from David Hilbert. This one is also copying from Gregor Cantor.

## redplays2030's channel!

Best flex: your on the first room

## Jose Ignacio Penz

I saw the same video a few years ago from a channel called quantum fracture, i don't know is this is a inspiration or a copy, cuz they have almost the same situations

## eggynack

Probably neither. This video is based primarily on Hilbert's hotel, a thought experiment from the early 20th century, with the part at the end about uncountable sets drawn from Gregor Cantor's diagonal proof in the late 19th. Except, y'know, modified to fit the hotel conceit. Quantum Fracture and this are both pulling from that same source material (though I dunno whether the other video does the Cantor stuff as I haven't checked it out), so that probably resulted in the similarities you're seeing.

## Cyrus Naredo

When you got 1000 and you have to walk all the way to the room

## Benjamin guess

I never a video could physically harm me

## [GD] Speeder

The Backrooms Hotel

## LILAVATI PATEL

Imagine coming In the hotel and being told again and again to go from one room to another contiously

As u reach to another room u have to go to another room again for infinity 😂😂😂

Hotel will have people in negative no.s cause of this

## Fahad Binsiddique

My question is: how can a person walk till room 1mil

## Joey Outdoors

4:00 "Why won't this work?" At that point I would be like, "sorry, it will take me an infinite amount of time to sort this out so why don't you have a seat in the infinite number of chairs in our infinite sized lobby for an infinite amount of time.

## CommieBastard

Just make more rooms lmao

## Nathania Benita

Look I know this is about maths, but

THIS SQUIGGLY LINE RIGHT HERE- 2:41

*Molecular chemistry flashbacks.* If you know, you know.

## Kunsh Anand

Amazing

## Chocodrago

Infinites can have differant values? Wut

## Vivian Su

Hhh

## EthanHasntBrain

Why dont they just tell the guest to go to the rooms double there room numbers again

## The Savage

This is the second video that is flawed, the variance presented exists if you do the logic correctly. You are just looking to closely to small sample to see exactly where the match is.

## Stefan Schade

Of course I know the maths, but to come up with the analogy is still neat

## Junting Chen

this video is also made on ted ed, so not sure if this is pirated video or not

## Alex K

LOL, 8.3K people are really mad at math... 🤣

## Audiophile ꪜ

I somewhat understood the video, Untill BIg Bus showed up.

## Laura B.

Hä

## Dubo III

Cantor's diagonal argument, last one

## wefinishthis now

This all depends on your mathematical system and definition of "infinity". By the definition where infinity cancels out infinity then sure this video is correct, but by another definition, if there's infinite rooms, you can still fit an infinite number of people in them and it doesn't matter if there's an infinite number of infinite buses (ie infinity times infinity still equals infinity and not infinity squared).

## eggynack

@wefinishthis now I'm not really sure what it even means for you to have a different definition. The term kinda just has the one definition, and it's not whatever you're saying here.

## wefinishthis now

@eggynack Yep, that's the part where I have a different definition of 'uncountable infinity'. By definition, if an 'uncountable infinity' is not enough then it ceases to be an uncountable infinity by definition.

## eggynack

@wefinishthis now No uncountably infinity is a distinct idea from the infinitely many infinite buses. All those buses, for all their splendor, are countable, You can accommodate every person in every one of those buses if you so choose. Simultaneously. Uncountable infinity is simply a larger type of infinity, however. A greater magnitude. You can never accommodate all the people.

## wefinishthis now

@eggynack There will always be more infinite buses of infinite people, but there will also always be infinite rooms and so on. Adding more infinite buses won't change the fact that it's an infinite cycle of infinite rooms for infinite people. Hence the 'uncountable infinity'.

## eggynack

I dunno what definition you're talking about, but they already have the infinite rooms successfully fitting infinitely many people and also infinitely many infinite buses. You're still not gonna be able to fit the big bus at the end though.

## mahadi alam

Which means Hilbert has an infinite amount of money but he can't spend it for himself.