Dark

How An Infinite Hotel Ran Out Of Room

Veritasium
Subscribe
Views 5 621 827
79% 325 000 85 000

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

Published on

 

May 10, 2021

Share:

Link:

Download:

Loading link...

Add to:

My playlist
Watch later
Comments 0
Waail EL Aourabi
Waail EL Aourabi 3 hours ago
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 !!!!!
Tom Svoboda
Tom Svoboda 2 hours ago
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
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.
Daniel Stefanov
Daniel Stefanov 4 hours ago
"HOW AN INFINITE HOTEL RAN OUT OF ROOMS" 0:35 "Let's say all rooms are full" Well ya done it, congrats :\
Phlypour
Phlypour 5 hours ago
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 23 minutes ago
@Kazper TeH_OnE The hotel has infinitely many rooms and every room has a guest inside. No room is free.
Kazper TeH_OnE
@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 5 hours ago
Why can't an infinite hotel be full?
Rubia, Giechris P.
Rubia, Giechris P. 5 hours ago
This is probably a dumb question to ask, but how does a hotel with infinite rooms get full?
Iza Kast
Iza Kast 5 hours ago
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 24 minutes ago
@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 4 hours ago
​ @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 5 hours ago
Where did you see "a number assigned to infinity"?
Vishi Karthik
Vishi Karthik 5 hours ago
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
FUN 4US 6 hours ago
ABABAABABABABABABA×♾️ 😂really his parents should must be proud of him.
oddlynick
oddlynick 6 hours ago
Me: ok My brain: error
Austin Yu
Austin Yu 7 hours ago
Reminds me of the comparison between the size of set of rational numbers and real numbers between 0 and 1
Bobv177
Bobv177 7 hours ago
This feels like vsauce
Soda Milk
Soda Milk 8 hours ago
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 9 hours ago
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
Timberwolfe 10 hours ago
So Stupid. Like a little kid saying INIFINITY PLUS ONE!!! And Let us remember that this is ALLLLLL Theroretical!!! NO REAL PROOF WHATSOEVER
Timberwolfe
Timberwolfe 9 hours ago
Why doesn't he just add the Infinite Bus to the next Infinite Rooms? No need for the chart
Timberwolfe
Timberwolfe 10 hours ago
I think the content maker RAN OUT OF IDEAS. Answer: The Manager sends them to Infinity Rooms. Done
hena kumari
hena kumari 10 hours ago
Well the definitely makes me eligible to be a manager at the plaza hotel I guess 😂😂
Chul Yeom
Chul Yeom 10 hours ago
The godly burst computationally surround because brother-in-law whitely thank but a humdrum specialist. mute, free tongue
tory carlozzi
tory carlozzi 10 hours ago
The medical mini-skirt selectively charge because tomato suprisingly suspend around a nervous taurus. complete, addicted calculus
Mrdog011 -
Mrdog011 - 11 hours ago
Me rn: 👁👄👁 huh?
Steel Soldier75
Steel Soldier75 13 hours ago
So what is the highest integer that you can't add one to? Answer that and I'll agree
Tom Svoboda
Tom Svoboda 5 hours ago
Why should there be a highest integer?
FortNikitaBullion
FortNikitaBullion 15 hours ago
What if there's a COVID-19 outbreak?
OffBrandStudio
OffBrandStudio 16 hours ago
0:14 “You are the manager” Me:ok… hold up. Why am I blue?
Ron Mascarenhas
Ron Mascarenhas 16 hours ago
Everything is infinite, except the poor lone receptionist. 1:Several Infinites. Not fair math!
LittleMopeHead
LittleMopeHead 17 hours ago
I believe The trebliH Hotel might be able to handle this.
Petet Xul
Petet Xul 18 hours ago
*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 18 hours ago
Go to every room until you find a room with noone, thats yours
Tom Svoboda
Tom Svoboda 17 hours ago
You will not find a room with noone. All rooms are occupied.
Mads Baunbæk
Mads Baunbæk 18 hours ago
3:40 and thats how the ABBA band started...
Nebula
Nebula 18 hours ago
A mathematicians wet dream
WolffLandGamez
WolffLandGamez 19 hours ago
The guy with room number infinite that was just told he has to move down infinite spaces 😑
Artur Styszyński
Artur Styszyński 20 hours ago
1:40 Isn't 7x2 = 14?
Unknown
Unknown 20 hours ago
So dumb, and utterly pointless this is.
Mahabir Neogy
Mahabir Neogy 21 hour ago
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 8 hours ago
@A B oh now i can imagine it.
A B
A B 16 hours ago
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 22 hours ago
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
Preetam Priyadarshi 23 hours ago
My brain stopped working sorry for that....and thank you
Sunrit Roy Karmakar
this is just stupid
Romarain
Romarain Day ago
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
A B 16 hours ago
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 17 hours ago
You're confusing the existence of the name with our ability to write it down.
Dark Soul
Dark Soul Day ago
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
Nicholas Colombo 11 hours ago
But infinity could be a number.
FalloutJack
FalloutJack Day ago
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
FalloutJack 15 hours ago
@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
A B 16 hours ago
@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
FalloutJack 16 hours ago
@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
A B 16 hours ago
@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
FalloutJack 16 hours ago
@A B So, you're saying my infinite Hilbert Hotels can be stored inside my Hilbert Hotel? Neat.
Precious Baldon
Who even thinks of this
JCCyC
JCCyC Day ago
Sorry, if you want to dance, jive, and have the time of your life, you gotta stay in the bus.
Vargub Lahkar
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
A B 16 hours ago
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
Some infinites are bigger than other infinites ..well it made sense in TFIOS !
Bethany Tjaden
???
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
A B 16 hours ago
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
Start learning math
Sumit Kumar
Sumit Kumar Day ago
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 Day ago
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 Day ago
카운터에 한글이 있네
Eleanor Conway 2!
Ababbabaabaababababbababababababababaabbababababababababbababababababababababaabba
Vung Muan Ching
wtf ;_;
Jaime L
Jaime L Day ago
Welcome to The Hotel California!🤪
Speed Junkie
Speed Junkie Day ago
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
eggynack Day ago
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
Wait will someone tell me what purpose this vdo serves..... P.S- I'm here for the frst tym
Stifler2277
Stifler2277 Day ago
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
Brien831 Day ago
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 Day ago
infinite can not be "complete"
Andrew Ye
Andrew Ye Day ago
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
eggynack Day ago
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
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
After watching this video, you made me believe that infinity is finite ☹️
MONEYBOY512
MONEYBOY512 Day ago
This is not even a realistic situation
MONEYBOY512
MONEYBOY512 Day ago
How does this help me
Hm
Hm Day ago
"Just go in an empty room"
Steve Newcombe
Steve Newcombe 2 days ago
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
A S 2 days ago
We all know "Uvuvwevwevwe Onyetenyevwe Ugwemuhwem OSAS" aint getting a chance at this hotel.
Balaji MD
Balaji MD 2 days ago
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 days ago
"You pull out an infinite spreadsheet of course" - of course 😅
olgierd ogden
olgierd ogden 2 days ago
I’d like to comment but my answer is to long)))..
Archie Areopex
Archie Areopex 2 days ago
This is... mind blowing..
Derek Nereida
Derek Nereida 2 days ago
The good meat byerly scold because list demographically obey till a equal work. accidental, last encyclopedia
Rob Gibson
Rob Gibson 2 days ago
When you have infinity + 2 guests. Duh.
Zander Leslie
Zander Leslie 2 days ago
Its infinite cause they break there legs on the way there
Chloe Chong
Chloe Chong 2 days ago
I lost all my braincells when watching this entire video
Nico Pauly
Nico Pauly 2 days ago
I’m glad I now know what to do if I’m ever put in this situation. Thanks!
Jada Lydia
Jada Lydia 2 days ago
I think people aren’t going to like to be moved
Global warming is hot
Could they not just accept everyone and have them walk until they find an empty room?
A B
A B 16 hours ago
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. 2 days ago
I know the RUvid 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
eggynack 2 days ago
@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. 2 days ago
@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
eggynack 2 days ago
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 2 days ago
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
A B 16 hours ago
It’s a famous thought experiment thought up before both of them
Daniel Röder
Daniel Röder 2 days ago
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 days ago
which room does BBBBB... get?
Mechros
Mechros 2 days ago
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
Mechros Day ago
@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
Brien831 Day ago
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 2 days ago
Anybody else's brain full after 2 min
TheRealRB3902
TheRealRB3902 2 days ago
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
@Red Panda ohhh yeah that looks amazing ty :D
Red Panda
Red Panda 2 days ago
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 - 2 days ago
I just lost thousands of brain cells by watching this one video 👁👄👁
Petru-Mihai Petrenchi
I think I saw this first on vsauce a couple of years ago
Rexanious
Rexanious 2 days ago
the problem isnt infinity or sth its how they will walk to their room lmao, "yes sir your room is 7252846 please walk $down the hall"
Puar XI
Puar XI 2 days ago
Instead of making everyone goes to the next room. Why don't just let the new guest taking the room that the last person would take anyway?
Just a HKer
Just a HKer 18 hours ago
Because, when dealing with infinity, there is no "last" person.
flare
flare 2 days ago
This is called The Banach-Tarski Paradox. It was the topic of a VSAUSE Video a while back which used the Hilbert Hotel example.
flare
flare 2 days ago
@Tom Svoboda I KNOW
Tom Svoboda
Tom Svoboda 2 days ago
@flare banach tarski paradox is the theorem about reassembling a ball into two balls. this video is way more basic stuff (different sizes of infinite sets), but it's a needed ingredient for banach tarski, so it was covered in the vsauce video. but the banach tarski paradox itself is something different. pythagorean theorem is about triangles, but not everything involving triangles is the pythagorean theorem.
flare
flare 2 days ago
@Tom Svoboda oh wait it isn’t?
Tom Svoboda
Tom Svoboda 2 days ago
this is unrelated to the banach tarski paradox (even though it was mentioned in that vsauce video)
Ayush Patil
Ayush Patil 2 days ago
This video is already there on Ted channel uploaded 2014.he just told the same
Lordlouckster
Lordlouckster 2 days ago
Ted-Ed used powers of primes. bus-th prime ^ seat
eggynack
eggynack 2 days ago
This way is better, cause it actually fills all the rooms. The power of primes thing leaves infinite rooms empty.
Flávio
Flávio 2 days ago
Me watching: Hey, that's not allowed ✋🏼
Atapd
Atapd 2 days ago
COPY FROM TED-ED
Muazzam Hazmi
Muazzam Hazmi 2 days ago
I don't understand this but I agree
HyperBlade 93
HyperBlade 93 3 days ago
the most interesting part of this whole thing is wondering how much money is being raked in from all those guests.
Bitfire31337
Bitfire31337 2 days ago
That's easy: An infinite amount. Depending on the net profit per guest and night it might be a bigger or lesser infinity though 😅.
MegaMrblackguy
MegaMrblackguy 3 days ago
This is dumb.
Jiri Valek
Jiri Valek 3 days ago
you can compare the amount of members of an infinite group only if you simplify it by using a finite group...but than you are comparing our number systems rather than the actual number of members.
Tomor Inferno
Tomor Inferno 3 days ago
wouldn't infinity mean that no matter how many Infinity are filling that infinity it would never be full? Due to Infinity not being like 1+1 but more like everything+everything+everything=everything
D. Dillon Duffield
Surreal! No I mean, do a long video on Surreal Numbers! 👌😸
Guhan SEN
Guhan SEN 3 days ago
Thats y i took Biology over maths
MisterSaur
MisterSaur 3 days ago
She: How much you love me? He: Infinite.. She: Will you buy that dress for me? He: Infinite also has limits.
schotte
schotte 3 days ago
Infinite number of rooms, maybe - but I’d hate to imagine the breakfast buffet.
Barty
Barty 3 days ago
What did I just watch
Surekha Veer
Surekha Veer 3 days ago
This is hard as dark season 3😂😂got it
Next
The Secret of Synchronization
20:58
The Infinite Money Paradox
10:32
The Banach-Tarski Paradox
24:14
3 Perplexing Physics Problems
14:00