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.


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


Publicerades den


10 maj 2021



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joels5150 28 minuter sedan
Oops, I saw the Fatal Flaw episode before this episode, and they covered this concept extensively.
23 Garibay
23 Garibay 3 timmar sedan
Илья Левицкий
In the last problem, I, as a manager, can competently forget about their names so they will become a countable infinity. It's better to be shown another way: The infinite bus is filled with pregnant women. As soon as a woman moves into a room, she gives birth to another woman. So when all the "original" women got into their rooms, they spawned another infinity of women. And those could not be fitted.
Cobosa 6 timmar sedan
This was exhausting just to watch.
Mint Wolf
Mint Wolf 6 timmar sedan
no 4 is my name
no 4 is my name 7 timmar sedan
Funny how no one talks about the complete infinite list. It's like saying empty full bottle, or dead live rat. If you can have a complete list, it's not infinite. Simple as that..
Tom Svoboda
Tom Svoboda 5 timmar sedan
Firstly, an "infinite list" in this context is simply a collection of names indexed by natural numbers. First name, second name, hundreth name etc. A name for every natural number. One such list can be defined e.g. by saying that the n-th element is the name which has all A's except for a single B at the n-th position. The beginning of the list would look like BAAAAA ABAAAA AABAAA etc. We can't _physically write_ all the elements down, but that doesn't matter. The defining condition determines the list unambiguously, we know what the n-th element is for every index n. In this sense the list is completed object, there are no more free numbers we can assign names to. Secondly, when he mentioned "complete list" in the video, what he meant was an infinite list (in the sense explained above) which would contain every possible name from the bus. Such a "complete list" cannot exist, but not for the reason you've described. The proof is the diagonal argument. If the bus contained e.g. people indexed by rational numbers, then a complete list of all those people would exist: there exists an assignment of natural numbers to rational numbers such that every fraction is covered. No such assignment exists for the infinite binary strings, there will always be leftovers. It's a mathematical fact.
Dobbs Mill
Dobbs Mill 7 timmar sedan
More importantly, why is this receptionist looking at his computer?
Dobbs Mill
Dobbs Mill 7 timmar sedan
Infinite roomed hotel, full already. All you are doing is creating infinite waves along the corridor
Roman YouTube
Roman YouTube 7 timmar sedan
I mean why don't we name the rooms ABABBABABABA... etc.
eggynack Timme sedan
Just as impossible as moving those guests into the rooms. The same exact proof applies.
__ PATHANIA __ 8 timmar sedan
I am so eagerly waiting for this to become a part of class 10 textbooks.... poor children 😂
Suddenly Pineapples
Suddenly Pineapples 8 timmar sedan
This hurt my brain…
Haitham Gamer kiddo
Haitham Gamer kiddo 8 timmar sedan
Frederik Qu
Frederik Qu 9 timmar sedan
I hear Hilbert, I know things are getting goooood
Nee Mann
Nee Mann 9 timmar sedan
This is dumb and makes no sense
RedOrav 11 timmar sedan
I was counting the rooms 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17... wait what?
EccentricGentelman 12 timmar sedan
As I understand it, the difficulty of getting people into this hotel is not a limit of rooms and space, because it has an infinite amount of both. The only difficulty is finding a way to organise it. The manager couldn't fit the ABs into the hotel, not because there were weren't enough rooms for them but because he had no system to do assign them rooms.
EccentricGentelman 11 timmar sedan
@Tom Svoboda I don't actually know, I'm not a mathematician and I never claimed to be. And I only just learned of the Infinite Hotel paradox today. I imagine that if this were real, I could just point to the hotel's infinite rooms and say "go there and grab a room." That would fix the problem in my mind. But apparently that's not good enough, in this hypothetical you need to give each of them a specific room yourself. And working out a system to assign them rooms is where this breaks down and not the number of rooms themselves. At least that's how I understand it. Just before I saw this, I watched Ted Ed's video on the same subject. According to them, the paradox reminds us our minds can have difficulty coping with infinity and so can our orderly numeric systems.
Tom Svoboda
Tom Svoboda 11 timmar sedan
No matter what amount of people from the last bus you fit into the hotel, there will always be leftover names (actually the vast majority of the bus will be unassigned, literally no more than 0% of the bus can be accomodated at once). How does this not say that the hotel doesn't have enough rooms for all of them?
Christopher Jokes
Christopher Jokes 12 timmar sedan
Must've suck to switch room every day
Jeremiah Harris
Jeremiah Harris 13 timmar sedan
Isn’t this a paradox? The end of the video makes it sound as if it’s not. “Some infinites are larger than others” doesn’t that by definition make it not infinite? If the infinite rooms can’t occupy the named infinite bus, that’s a paradox right?
Tom Svoboda
Tom Svoboda 12 timmar sedan
It's a paradox in the sense that it's counter intuitive, but logically sound. There _are_ differently sized infinite sets in the sense that two sets (finite or infinite) are defined to have the same size if it's possible to pair their elements one to one. The diagonal argument shows that it's not possible to pair the natural numbers (the rooms) with the infinite binary strings (the people in the last bus). It's impossible in the exact same way it's impossible to pair a 3-element set with a 5-element set. There will simply always be leftovers.
Waail EL Aourabi
Waail EL Aourabi 17 timmar sedan
I've got question here 👋 ! Any room number can be written in binary system, 0s and 1s ! Isn't that equivalent to the As and Bs in the names of the guests ? Like if the names of the guest were made of 0s and 1s instead of As and Bs, couldn't we assign each guest to a room number equivalent to the decimal writing of his name ?? Somebody answer me please !!!!!
Waail EL Aourabi
Waail EL Aourabi 5 timmar sedan
@Tom Svoboda couldn't we say the same thing for names ? Like names have finetly many letters ? I didn't catch the diference between the names and the numbers ! For me for any name random ABABABA... The is a (room) number n which can be written as 0101010... In binary system ! Which means that the normal numbers and the infinite bus are infinitely equal
Tom Svoboda
Tom Svoboda 16 timmar sedan
numbers have only finitely many digits. for example the name BBBB... would be 1111... but that's not a number, just an infinite string of digits.
Levan Katsadze - LeoKac
Levan Katsadze - LeoKac 18 timmar sedan
4:34 - This logic is wrong, you can't change all of the names, because you will never change all of the names, because they are infinit, never ending, you will never reach the last name, because there IS NO last name. It is never ending.
Tom Svoboda
Tom Svoboda 12 timmar sedan
It doesn't matter that you can't write the new name down in its entirety. Its _existence_ doesn't depend on it. Clearly this name is included in the bus (because the bus contains _all_ possible names), and it provably isn't on the list by construction. Therefore it's not possible to organize all names into the rooms.
Daniel Stefanov
Daniel Stefanov 18 timmar sedan
"HOW AN INFINITE HOTEL RAN OUT OF ROOMS" 0:35 "Let's say all rooms are full" Well ya done it, congrats :\
Phlypour 19 timmar sedan
It appears mind-blowing, but every time I hear a theory fighting theory... it only shows the limitations of all those theories. You cannot assume that there is an infinity hotel that is full. Infinity can't be full because it's infitit. Whatever's inside is already there making the infinity look full, but you can always expand. Just add a second floor for uncountables XD
Tom Svoboda
Tom Svoboda 13 timmar sedan
@Kazper TeH_OnE Incorrect in what sense? What I've written is a literal premise of the thought experiment.
Kazper TeH_OnE
Kazper TeH_OnE 13 timmar sedan
@Tom Svoboda This is incorrect. The problem boils down to trying to throw infinity into infinity and saying it's not possible because there's a +1. The only way this is true is if there were never an infinite amount of rooms to begin with. If there are an infinite amount of rooms that means there are an infinite amount of empty rooms (place holders) that will be able to hold any and all guests as well as everything and anything else that can be measured. It is never full, it is continuously filling an empty room that is then replaced with another empty room. The place holder Zero.
Tom Svoboda
Tom Svoboda 14 timmar sedan
@Kazper TeH_OnE The hotel has infinitely many rooms and every room has a guest inside. No room is free.
Kazper TeH_OnE
Kazper TeH_OnE 15 timmar sedan
@Tom Svoboda because... infinite... However many guests that show up, regardless if they can be assigned a number or not will have a room. Number's are simply a way for us to measure and communicate the things that are being counted. There will always be a placeholder to continue that measurement if needed.
Tom Svoboda
Tom Svoboda 19 timmar sedan
Why can't an infinite hotel be full?
Rubia, Giechris P.
Rubia, Giechris P. 19 timmar sedan
This is probably a dumb question to ask, but how does a hotel with infinite rooms get full?
JGB 9 timmar sedan
just a single infinite bus is enough to fill it initially
Iza Kast
Iza Kast 19 timmar sedan
I could never wrap my head around this problem. Isnt infinity defined by it's inability to be quantified? By assigning a number to an infinity, doesnt that mean you've quantified it? and by that metric, it should no longer qualify as an infinity. No matter how many numbers, letters, or any point of quantification you use, infinity should be a greater amount than it.
Tom Svoboda
Tom Svoboda 14 timmar sedan
@Iza Kast *Isnt infinity defined by it's inability to be quantified?* Yes and no and it depends on what exactly you mean by "quantify". From the Cambridge disctionary: quantifiable - possible to measure and express *as a number* Numbers (natural, integers, rational, reals, complex) are finite by definition. Infinite literally means "not finite". So with this definition of quantifiable as measruable by a number (i.e. by something finite), it's true that anything that is infinite cannot be quantifiable. It's basically a tautology. That however doesn't mean that infinite implies impossible to grasp, describe, and work with. The set of all natural numbers N = {1,2,3,4,..} is infinite. That doesn't mean anything else than that its amount of elements is larger than any finite number. In other words, if we pick any natural number n, we can find a subset of the natural numbers with exactly n elements, for example {1,2,3,..,n}. This says that the amount of elements of N is at least n. Since this is true for all numbers n, the amount is larger than any number. This is a perfectly unambiguous condition that can be reasoned about. 0:41 The hotel has rooms indexed by natural numbers. For every number n there's a hotel room. And for every number n there's also a guest sitting in the room n. No finiteness of the hotel is implied by this statement. 2:15 Same thing. There's a bus for every natural number n, and there's a person inside for every natural number m. So all people in the buses are indexed by pairs of natural numbers (n,m), first index is the bus second index is the seat. The example is supposed to show that a set indexed by nat. numbers (the hotel rooms) and a set indexed by pairs of natural numbers (the people in the buses) actually have an equal amount of elements in the sense that the elements can be paired one to one. The pattern of the pairing is indicated by the spreadsheet. Again, nothing about this implies a limit, an end etc. 5:17 Yes, there are different sizes of infinite sets in the sense described above: infinite sets are compared by pairing their elements. The sets have the same size if a pairing between them exists. First set is smaller than the second set if any attempt at the pairing results in leftovers in the second set. In the video we saw that it's not possible to pair a set indexed by the natural numbers (the hotel rooms) with a set indexed by the infinite binary strings (the last bus). This is a combinatorial statement, it's impossible in the exact same fashion as it's impossible to pair a 3-element set with a 5-element set. There will always be leftovers in the bus. 4:48 The list is an attempt at sorting the people into the hotel rooms, i.e. the pairing described above. The diagonal argument shows that for any such attempt there will be someone unaccounted for. The diagonally created name is not a name which doesn't appear on the _bus_ , it's a name which doesn't appear on the _list_ giving the pairing. It's a general construction which for any hypothetical list produces a name which provably doesn't appear on the list. This is how you prove that the bus is truly a larger set than the hotel.
Iza Kast
Iza Kast 19 timmar sedan
​ @Tom Svoboda Isnt this assigning a number to an infinity? 2:15 Isnt it implying that within any of those 'infinite' buses filled with 'infinite' people, there's a finite number/metric that can be written out on a supposedly 'infinte' spreadsheet? If infinite by definition means unending, wouldn't 0:41 "but all the rooms are occupied", imply a end, a countable number/metric of people/buses? 5:17 even states "Some infinities are bigger than others" isnt that quantifying it? To imply it is bigger means it's is comparatively larger than something else. The purpose of numbers/letters/words/etc is to denominate an X object compared to a Y object. If only X object exists, there would be no need for defining Y. Since you're here im just gonna ask the other questions plaguing me. 3:42 states that "there's a person with every possible infinite sequence" So why would 4:48 's combination "..guaranteed to appear nowhere else on that list" if the owner is assigning a different room to each new guest?
Tom Svoboda
Tom Svoboda 19 timmar sedan
Where did you see "a number assigned to infinity"?
Vishi Karthik
Vishi Karthik 20 timmar sedan
Okay I don’t wanna freak anyone but imagine this so your in the hotel all alone, you look all the way down the hall and since it’s infinite it’s super long but you look down the hall and see somebody there and there running towards you.
FUN 4US 21 timme sedan
ABABAABABABABABABA×♾️ 😂really his parents should must be proud of him.
oddlynick 21 timme sedan
Me: ok My brain: error
Austin Yu
Austin Yu 21 timme sedan
Reminds me of the comparison between the size of set of rational numbers and real numbers between 0 and 1
Bobv177 22 timmar sedan
This feels like vsauce
Soda Milk
Soda Milk 22 timmar sedan
Damn, this my gf make this problem? Because there's a 0% chance that it ever happened or will ever happen.
Gourob Kundu
Gourob Kundu 23 timmar sedan
Isn't this the same way we can prove that there are more real numbers between 0 and 1 compared to all the Natural numbers?
Timberwolfe Dag sedan
So Stupid. Like a little kid saying INIFINITY PLUS ONE!!! And Let us remember that this is ALLLLLL Theroretical!!! NO REAL PROOF WHATSOEVER
Timberwolfe Dag sedan
Why doesn't he just add the Infinite Bus to the next Infinite Rooms? No need for the chart
Timberwolfe Dag sedan
I think the content maker RAN OUT OF IDEAS. Answer: The Manager sends them to Infinity Rooms. Done
hena kumari
hena kumari Dag sedan
Well the definitely makes me eligible to be a manager at the plaza hotel I guess 😂😂
Chul Yeom
Chul Yeom Dag sedan
The godly burst computationally surround because brother-in-law whitely thank but a humdrum specialist. mute, free tongue
tory carlozzi
tory carlozzi Dag sedan
The medical mini-skirt selectively charge because tomato suprisingly suspend around a nervous taurus. complete, addicted calculus
Mrdog011 -
Mrdog011 - Dag sedan
Me rn: 👁👄👁 huh?
Steel Soldier75
Steel Soldier75 Dag sedan
So what is the highest integer that you can't add one to? Answer that and I'll agree
Tom Svoboda
Tom Svoboda 19 timmar sedan
Why should there be a highest integer?
What if there's a COVID-19 outbreak?
OffBrandStudio Dag sedan
0:14 “You are the manager” Me:ok… hold up. Why am I blue?
Ron Mascarenhas
Ron Mascarenhas Dag sedan
Everything is infinite, except the poor lone receptionist. 1:Several Infinites. Not fair math!
LittleMopeHead Dag sedan
I believe The trebliH Hotel might be able to handle this.
Petet Xul
Petet Xul Dag sedan
*Reminds me of the pullover “philanthropist” criminal Billy Gates. Invents a problem that doesn’t exist and “sells” a solution to it. For the money that exist.*
Mads Baunbæk
Mads Baunbæk Dag sedan
Go to every room until you find a room with noone, thats yours
Tom Svoboda
Tom Svoboda Dag sedan
You will not find a room with noone. All rooms are occupied.
Mads Baunbæk
Mads Baunbæk Dag sedan
3:40 and thats how the ABBA band started...
Nebula Dag sedan
A mathematicians wet dream
WolffLandGamez Dag sedan
The guy with room number infinite that was just told he has to move down infinite spaces 😑
Artur Styszyński
1:40 Isn't 7x2 = 14?
Unknown Dag sedan
So dumb, and utterly pointless this is.
Mahabir Neogy
Mahabir Neogy Dag sedan
what is the problem? i didn't understand.If he just tell them to knock every room and whenever they find an empty room it will be their's.
Mahabir Neogy
Mahabir Neogy 22 timmar sedan
@A B oh now i can imagine it.
A B Dag sedan
Then individuals have to move an infinite distance. They’ll never get there. Whereas with the other strategies each individual only has to move a finite distance. Each number is uniquely paired with one person
Tushar gola
Tushar gola Dag sedan
The new name we generated at the last... we can just say him to move to the last room of the hotel......(infinite dead body)
Preetam Priyadarshi
My brain stopped working sorry for that....and thank you
Sunrit Roy Karmakar
this is just stupid
Romarain Dag sedan
Well, no : for a guy to have a name composed of the diagonal you mentioned, it would have to reach the infinity of names, wich CAN'T be reached because it's infinite. So, that supposed guy doesn't exist, it's an intellectual ghost. Infinite numbers are enough stupid¹ for us to imagine something geometrically based on it... (¹: Infinity doesn't exist, it's just a concept and an hypothesis, and it has never been verified, nor could be. However, the absurdity of the diagonal is directly comprehensible and therefore the silliness of its author verified).
A B Dag sedan
The limit as x approaches infinity of 1/x is zero. Infinity applied, a useful concept. It’s defined much more rigorously than it is in this video in pure mathematics. So it has been proven (to use your language)
Tom Svoboda
Tom Svoboda Dag sedan
You're confusing the existence of the name with our ability to write it down.
Dark Soul
Dark Soul Dag sedan
Fun fact: Infinity is not a number but rather a Definitive Constant that's why we don't include Infinities in any Domain/Range
Nicholas Colombo
But infinity could be a number.
FalloutJack Dag sedan
You can't have a complete infinite set. You can hand me a big box marked "Everything", but you can't tell me you know the limits of what Everything is, regardless of whether or not the box filled with Everything actually includes its own box, the box of a different universe's Everything, whether it's a multiversal Everything, or even if it's just the *concept* of Everything, as in to say that anything that can be conceived exists in that box. The imagination has no limits, and numbers themselves don't really have an upper-limit. Why should conceptual infinity? Besides...in your theory of many infinite buses of people, you didn't count on my having infinite Hilbert Hotels.
FalloutJack Dag sedan
@A B Math doesn't really limit itself. In fact, if anything should be less limited than potentially the universe (which we can't measure at this time, but know that it's constantly expanding, so it's generally regarded as effectively infinite until better answers come in), it's the math. Put simply, you may run out of *names* for numbers, but you won't actually be able to stop counting up if your goal is to reach the end. If you ask a computer to start counting, it will never stop until deprived of the ability to do so. And that doesn't mean that it stopped, only that it *was* stopped by an outside force. The set of infinity is just a placeholder for 'An ongoing forever number'. You can assign a meaning and a category to it, but it's still a forever number, much the same way a letter-assaigned variable - like X - is a stand-in for an unknown number. Basically, if you're trying to define a finite infinity, you're working too hard and not getting paid enough for it. The argument will go on forever because you're trying to say that the concept of no limit has a limit. Eliminating the concept would destroy the meaning of the question, and therefore render the whole thing moot, so you can't have it that way.
A B Dag sedan
@FalloutJack nothing in this universe is infinite (at least not in the observable universe). Not in a mathematical sense. There are different sizes of infinities, countable, uncountable, and other orders
FalloutJack Dag sedan
@A B Oh no, that bit was just an aside. The main point is that arguing infinities literally has no end. I was removing whether it can self-reference itself from argument there because it wouldn't be relevent.
A B Dag sedan
@FalloutJack no, I’m saying you define that a set cannot contain itself. So a set of all sets does not contain itself. An axiom of modern set theory.
FalloutJack Dag sedan
@A B So, you're saying my infinite Hilbert Hotels can be stored inside my Hilbert Hotel? Neat.
Precious Baldon
Precious Baldon Dag sedan
Who even thinks of this
JCCyC Dag sedan
Sorry, if you want to dance, jive, and have the time of your life, you gotta stay in the bus.
Vargub Lahkar
Vargub Lahkar Dag sedan
I think the question itself is inherently wrong, since we know the some infinitys are bigger then other infinity, but the hotel owner nor us have a way to calculate is the number of infinite people is bigger then the infinite number of rooms, if the room infinity is bigger then people infinity, the lack of room does not exist
A B Dag sedan
We know how many rooms there are, a countable infinite of them since they’re numbered with all the positive integers. And the different groups of people have different infinity sizes assigned to them, culminating in an uncountably infinite group
Himadri Barad
Himadri Barad 2 dagar sedan
Some infinites are bigger than other infinites ..well it made sense in TFIOS !
Bethany Tjaden
Bethany Tjaden 2 dagar sedan
Stefany Del Toro - Paz
im so confused, if the hotel is infinite but you run out out of rooms, a new person shows up and you tell everyone to move down one, then you obviously had more rooms ?? Can someone explain am I just dumb?
A B Dag sedan
There’s always another number. To put it in a mathematical sense, you can match every whole number to another whole number, and you can list them. 1, 2, 3, … on the other hand you can’t list the decimals between zero and 1. 0, 0.0000…. What’s the second term?
Knightly Ishaan
Knightly Ishaan 2 dagar sedan
Start learning math
Sumit Kumar
Sumit Kumar 2 dagar sedan
You would need an infinite amount of money funded by infinite number of owners and an infinite number of workers to make a hotel with infinite number of rooms and an infinite number of staff to run the hotel and infinite amount of mathematicians to deal with infinite amount of money . Find the mistake and you'll get the infinite amount of profit made by the hotel with..... "repeats again" 😂😂
yan wu
yan wu 2 dagar sedan
Bruh, I spent 6 minutes of my life watching a video, hoping I would understand so I can teach my friends but I didn’t
Jaewon Cheon
Jaewon Cheon 2 dagar sedan
카운터에 한글이 있네
Eleanor Conway 2!
Eleanor Conway 2! 2 dagar sedan
Vung Muan Ching
Vung Muan Ching 2 dagar sedan
wtf ;_;
Jaime L
Jaime L 2 dagar sedan
Welcome to The Hotel California!🤪
Speed Junkie
Speed Junkie 2 dagar sedan
I understood the logic, but I still don't understand the logic
Jyoti Prakash Das - 08
U could again assign the name of all the people with the seats infinitly and give them the room
eggynack 2 dagar sedan
Changing all the names to match the rooms is just as impossible as putting all the people into rooms. The same proof functions for both.
Harshita gupta
Harshita gupta 2 dagar sedan
Wait will someone tell me what purpose this vdo serves..... P.S- I'm here for the frst tym
Stifler2277 2 dagar sedan
Look "infinity" is a concept and not a number. For instance, how many numbers is between 1 and 2, you get 1.0000000......infinity...1 up to the number 2 ( 1.0000001, 1.0000002, 1.0000003.........) So are there more numbers between 1 and 3, than between 1 and 2? It makes sense that there must be more numbers between 1 and 3 than between 1 and 2, but theoretically there is not, since infinity is a concept and not a number since: n+1> 1 So in conclusion infinity + infinity is not > than infinity, you cant count infinity and you can neither assign a character (x) in it.
Brien831 2 dagar sedan
We are not talking about numbers, but sets and their sizes. You can determine, if two sets are the same size if you can find a bijection between them. A bijection is a map, that maps one element of one set to exactly one element of the other set and vice versa. For example there is a bijection between the natural numbers, and the rational numbers, wich means they have the same size. However for every open interval in the real numbers, for example (0,1) there exists no bijection to the natural numbers. In mathematics we say the set of all real numbers in (0,1) is uncountable, and for the rational numbers we say they are countable. Now in fact the set of all numbers in (0,1) has the same size as the real numbers as there exists a bijection even though the real numbers cover (0,1). If you want a more intuitive definition about set sizes, you may want to look into measure theory.
Andrew Ye
Andrew Ye 2 dagar sedan
infinite can not be "complete"
Andrew Ye
Andrew Ye 2 dagar sedan
One thing I am confused on. When describing infinity, we are still giving it a finite set amount, not an infinite. This still doesn't work. Infinite isn't something we can do/explain.
eggynack 2 dagar sedan
It's a set amount, but it's not finite. The natural numbers are always just the natural numbers. Unchanging, solid, very much infinite.
Rishabh Raj XB 39
Rishabh Raj XB 39 2 dagar sedan
Our mathematics teacher once tried to explain the concept of infinity to us, but he ended confused himself.😂 I'm gonna share this video to him.
Anakin Skywalker
Anakin Skywalker 2 dagar sedan
After watching this video, you made me believe that infinity is finite ☹️
MONEYBOY512 2 dagar sedan
This is not even a realistic situation
MONEYBOY512 2 dagar sedan
How does this help me
Hm 2 dagar sedan
"Just go in an empty room"
Steve Newcombe
Steve Newcombe 2 dagar sedan
I just got my head around the hotel then these freekin huge busses start rocking up; loads of em. Just how big is the car park?!
A S 2 dagar sedan
We all know "Uvuvwevwevwe Onyetenyevwe Ugwemuhwem OSAS" aint getting a chance at this hotel.
Balaji MD
Balaji MD 2 dagar sedan
This is really entertaining but conceptually wrong.. We can better understand the concept of infinity by the Indian word 'Purnam'.. Below is a famous verse which can make you realize the absolute.. Term 'Infinity' is a cheap version and has lost the value overtime.. FYI: 'OM Purnamadah Purnamidam Purnat Purnamudachyate Purnasya Purnamadaya Purnamevavashishyate'
Chip Vos
Chip Vos 2 dagar sedan
"You pull out an infinite spreadsheet of course" - of course 😅
olgierd ogden
olgierd ogden 2 dagar sedan
I’d like to comment but my answer is to long)))..
Archie Areopex
Archie Areopex 2 dagar sedan
This is... mind blowing..
Derek Nereida
Derek Nereida 2 dagar sedan
The good meat byerly scold because list demographically obey till a equal work. accidental, last encyclopedia
Rob Gibson
Rob Gibson 2 dagar sedan
When you have infinity + 2 guests. Duh.
Zander Leslie
Zander Leslie 3 dagar sedan
Its infinite cause they break there legs on the way there
Chloe Chong
Chloe Chong 3 dagar sedan
I lost all my braincells when watching this entire video
Nico Pauly
Nico Pauly 3 dagar sedan
I’m glad I now know what to do if I’m ever put in this situation. Thanks!
Jada Lydia
Jada Lydia 3 dagar sedan
I think people aren’t going to like to be moved
Global warming is hot
Global warming is hot 3 dagar sedan
Could they not just accept everyone and have them walk until they find an empty room?
A B Dag sedan
They’d need to walk an infinite distance, not one would find a room over an infinite amount of time.
Hannah K. V.
Hannah K. V. 3 dagar sedan
I know the SVdown comments section is a stupid place to ask a question like this, but I'm going to ask it here in case anybody knows: At the very end, there is 1 creature that has a different sequence of A's and B's than the infinite amount of other creatures. It can't be more than 1, since there are only 2 letters to choose from, and you've already switched 1 of those in each other creatures' to come up with this one that is different. But since this is only infinity + 1 + infinity (those already at the hotel), couldn't you put the 1 first, and then arrange the rest according to even and odd numbers? I've seen the ABBA thing done with decimals in other videos, so let's do it with those as well. For numbers, you could add 1, or 2, or 3, or subtract 1, 2, 3, etc. as well. Let's say you could subtract/add/whatever to these numbers to produce an infinite amount of decimals that are different from the original infinity. Well, in this case, you'd have infinity (original) + infinity (those you just found that are different from the original) + infinity (those at the hotel). So couldn't you just put each person in a room in multiples of 3? There are infinite number of them, just like the multiples of 2. Wouldn't that mean that all infinities are the same, and countable/uncountable infinities don't exist? Obviously there are different infinities, so why wouldn't these work? Please help, I've had this question for months now and I still haven't found an answer.
eggynack 2 dagar sedan
@Hannah K. V. It's not that they're infinite together. There's no need to combine anything with anything else. The set is simply uncountably infinite unto itself. The point of the proof isn't that you have this infinite list and then you can add one to it, or add infinity to it, or even add uncountable infinity to it. The point is that, no matter what list you produce, it will always be incomplete. It could be some specially designed list, or a completely arbitrary list, or anywhere in between, but it will always be missing something. A list of the evens, the odds, the naturals, the integers, all of these are very possible. Nothing missing. The reason why what I did above was interesting was because it demonstrates something we already knew, that an attempt to list an uncountably infinite set will always, always, be missing an uncountably infinite set's worth of elements.
Hannah K. V.
Hannah K. V. 3 dagar sedan
@eggynack What I don't understand is why the second infinite list of numbers is the same as the first, but together they're uncountably infinite. Why doesn't infinity x 2 = infinity? Earlier in the video it was mentioned that when you have 2 infinite sets, you can match 1 with the evens and 2 with the odds. So why can't you do that with the original set of infinity and the new, different set of infinity? I hope this makes sense it's difficult to explain lol
eggynack 3 dagar sedan
The amount of numbers you're missing isn't simply one, or even infinite. It's uncountably infinite. What's left over is exactly as numerous as the set you started with. There's actually a pretty cool way to do create such a set of missing numbers via the diagonal argument, though I dunno how to do it offhand with the binary set presented here. Just with the base 10 representation. But check it. Consider a list of the reals between 0 and 1. .4987891234... .1972304802... .7171717171... .9999199999... .3141592653... and so on. So you do the standard diagonal trick, adding one to each of the digits along the diagonal, getting you .50806... But, as you note, you can also add two to each digit, getting .61917... Or, y'know, you could alternate between 1's and 2's, getting .60907... Or, and here's the cool part, you could use literally any combination of 1's and 2's and you'd get a new number. And we can represent these as decimal numbers, with, for the sake of argument, the 1's replaced with 0's and the 2's replaced with 1's. So the ones so far are like .0000... .1111... .1010... and so on. Except what do ya got when you have literally any combination of 0's and 1's after a decimal point? You got the set of all real numbers between 0 and 1 in binary. Which, y'know, that's just the set of all real numbers. In other words, in attempting to list the set of all real numbers, you missed literally all of them. 100%. And you can't do any better than that. Any attempted list will miss as many numbers as were in the original set. It is a far greater infinity than the infinity of the list, or the hotel.
Trexy Lemur
Trexy Lemur 3 dagar sedan
is anybody gonna talk about how ted-ed did a video similar to this also im not saying he copied their video but it was a really good ted-ed video
A B Dag sedan
It’s a famous thought experiment thought up before both of them
Daniel Röder
Daniel Röder 3 dagar sedan
But there is a way to fit all guests into unique rooms. Just tell them to treat their name as binary instructions to find their room, A = 0 and B = 1. For example, if their name starts ABBA... then they add 1 ( as offset because first room is called 1) + 0*1 + 1*2 + 1*4 + 0*8 ..... so after the first 4 letters that guest would stand before room 7 and then calculate his next step. Everyone is guranteed to have a unique room this way. The guest at room 1 would have all As in his name, the guest in room 2 would be named B followed by infinite As. Every guest would have a unique room to himself after infinite calucations for the room number (which might be slightly annoying for the guests but gives you enough time to write the guestlist...). If there is another person at that room, that guest would have the exact name so that can't happen as we know. Your room list would look something like this: AAAAAAA.... BAAAAAA.... ABAAAAA.... BBAAAAA.... AABAAAA.... and so on. The argument with the name that doesn't appear on the list: If you start writing out the first letters of the first names of the list you will see that the technique of switching the letters on the diagonal will generate a string consisting out of all Bs. Because on the diagonal there will only be As going down the list. Otherwise it would mean that the two graphs Y=f(2 to the power of X) and Y=f(X) )intersect for a number X > 0. "All Bs" happens to be in the last room and at the end of that infinite list, because by definition it is on the opposite end of "All A", the guest who has to add every 2 to the power of X term.
Tom Svoboda
Tom Svoboda 2 dagar sedan
which room does BBBBB... get?
Mechros 3 dagar sedan
There's so much that doesn't make sense. For example there are an infinite amount of people on an infinite amount of buses. It could also mean you could fit an infinite amount of people in one infinitely long bus. This would create another "Hilbert's hotel" but on a bus. If all values are not finitely set but are infinite (including the rooms), it would quite literally never end. Though I do understand that this video's point was to try and prove different sized infinities.
Mechros 2 dagar sedan
@Brien831 yeah I had a look into it after watching this video. I read something similar but with numbers. It made a lot more sense than the hotel. Comparing sets of natural numbers and decimals, and when you get a number from each decimal in each decimal place it creates an entirely new number that didn't exist on the list. It was done in this video when they sorted the names but it makes more sense with plain decimals.
Brien831 2 dagar sedan
Yes his explanation is a bit sloppy there. If you want some further explanation look into countable and uncountable sets. Sometimes abstraction is easier to understand than simplification.
Isaac Rock
Isaac Rock 3 dagar sedan
Anybody else's brain full after 2 min
TheRealRB3902 3 dagar sedan
5:56 anyone know where I can get a wallpaper like this? This looks so nice. Even if I can't get a premade one, could anyone at least tell me how to make something like this if they know? thanks so much in advance edit: by wallpaper i mean like a wallpaper for my macbook
TheRealRB3902 2 dagar sedan
@Red Panda ohhh yeah that looks amazing ty :D
Red Panda
Red Panda 3 dagar sedan
I don't know where to find a wallpaper that look exactly like that but maybe you will like vaporwave grid wallpaper. You can search it on google.
Liam -
Liam - 3 dagar sedan
I just lost thousands of brain cells by watching this one video 👁👄👁
Petru-Mihai Petrenchi
Petru-Mihai Petrenchi 3 dagar sedan
I think I saw this first on vsauce a couple of years ago
Ant Wan - Marbella (Audio)
Ant Wan - Marbella (Audio)
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