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

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

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

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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)

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# How An Infinite Hotel Ran Out Of Room

324 000 Көрүүлөр 6 млн

I call no on this one. Convert all As to 0s, and Bs to 1s, then shift them from binary to decimal base. You now have an infinite number of unique positive integers, EXACTLY like when you had a single, infinite bus show up. Each person's binary-converted name equates to their seat assignment in the bus. Then you just use the same solution you used on the bus. If that can't work because of the unique diagonal, THEN THE BUS COULDN'T HAVE WORKED EITHER, FOR THE SAME EXACT REASON. This is a logical error, ultimately. You see, the string of characters formed by the diagonal is itself infinitely long. Which means you can never resolve it in the first place. It's meaningless. The name/number will never be on the list, because it is, itself, one of the forms of infinity. And infinity is not a number. It's a concept. This fails to prove that you can't house all the guests, because there will never be any particular guest with an infinitely long name.

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" 😂😂

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

카운터에 한글이 있네

Ababbabaabaababababbababababababababaabbababababababababbababababababababababaabba

wtf ;_;

Welcome to The Hotel California!🤪

I understood the logic, but I still don't understand the logic

U could again assign the name of all the people with the seats infinitly and give them the room

Wait will someone tell me what purpose this vdo serves..... P.S- I'm here for the frst tym

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.

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.

infinite can not be "complete"

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.

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.

After watching this video, you made me believe that infinity is finite ☹️

This is not even a realistic situation

How does this help me

"Just go in an empty room"

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?!

We all know "Uvuvwevwevwe Onyetenyevwe Ugwemuhwem OSAS" aint getting a chance at this hotel.

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'

"You pull out an infinite spreadsheet of course" - of course 😅

I’d like to comment but my answer is to long)))..

This is... mind blowing..

The good meat byerly scold because list demographically obey till a equal work. accidental, last encyclopedia

When you have infinity + 2 guests. Duh.

Its infinite cause they break there legs on the way there

I lost all my braincells when watching this entire video

I’m glad I now know what to do if I’m ever put in this situation. Thanks!

I think people aren’t going to like to be moved

Could they not just accept everyone and have them walk until they find an empty room?

I know the KGshows 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.

@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.

@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

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.

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

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.

which room does BBBBB... get?

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.

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.

Anybody else's brain full after 2 min

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

@Red Panda ohhh yeah that looks amazing ty :D

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.

I just lost thousands of brain cells by watching this one video 👁👄👁

I think I saw this first on vsauce a couple of years 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"

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?

This is called The Banach-Tarski Paradox. It was the topic of a VSAUSE Video a while back which used the Hilbert Hotel example.

@Tom Svoboda I KNOW

@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.

@Tom Svoboda oh wait it isn’t?

this is unrelated to the banach tarski paradox (even though it was mentioned in that vsauce video)

This video is already there on Ted channel uploaded 2014.he just told the same

Ted-Ed used powers of primes. bus-th prime ^ seat

This way is better, cause it actually fills all the rooms. The power of primes thing leaves infinite rooms empty.

Me watching: Hey, that's not allowed ✋🏼

COPY FROM TED-ED

I don't understand this but I agree

the most interesting part of this whole thing is wondering how much money is being raked in from all those guests.

That's easy: An infinite amount. Depending on the net profit per guest and night it might be a bigger or lesser infinity though 😅.

This is dumb.

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.

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

Surreal! No I mean, do a long video on Surreal Numbers! 👌😸

Thats y i took Biology over maths

She: How much you love me? He: Infinite.. She: Will you buy that dress for me? He: Infinite also has limits.

Infinite number of rooms, maybe - but I’d hate to imagine the breakfast buffet.

What did I just watch

This is hard as dark season 3😂😂got it

Just go to the next open room lol

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

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

WTF

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

Underrated comment 🤣

I am slime

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

@Tom Svoboda You don't. Because the bus doesn't exist and neither does the hotel.

how do you sort the people of the last bus into the hotel then?

who cleans the rooms?

Scp-182737362821818273364646328929181827374664646467721819191928373746-hotel

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

wow, that's a good cliffhanger! subscribed!

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

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

I want to show this video to my math teacher

Infinite Hilbert Hotel's

*my roblox character named infinite is satisfied.*

I like the part where he said infinite

make everyone (including the party bus) go up one

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

No infact, if you only had all the irrational numbers as guests, you could fit them into you hotel. The rationals and the natural numbers have the same magnitude.

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

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

Infinite is not a number

“Your mother arrives”

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

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

@Tom Svoboda @eggynack Thank you both for helping me understand this, I can see where I was going wrong in my logic with it now. I'm not a smart man, I'm more of a nuts and bolts type thinker than a theoretical one but you both genuinely helped get the concept through my thick skull so again, thank you for that. Also my apologies if the second @ doesn't work

@Hydros92 The exact mathematical statement we're dealing with goes as follows: there doesn't exist a 1-1 correspondence (a pairing) between natural numbers and infinite binary strings. It's the same thing as trying to pair a 3-element set with a 5-element set, it's just not possible. The finite case is easily understood: the naive reason is that 5 is larger than 3. The latter set has more elements, so there will always be leftovers. We use this intuition to _interpret_ the infinite case as the set of binary strings being larger than the set of natural numbers. But it's just semantics. We're not arguing that it's "truly larger" in some metaphysical sense. It's more like giving a meaning to the word "larger" in this context. It doesn't matter what you call it, we don't need to use the words "larger" or "size" at all. The point is that the _fact itself_ holds, it won't go away, and it demonstrates itself throughout mathematics.

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

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

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

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

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

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

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

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

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

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

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

To get this job your qualifications must be insane

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

It's name is "the infinite hotel Paradox"

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

I am confusion

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

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

That's over thinking

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

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

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

@TheMadHatt no, the rooms are explicitly indexed by natural numbers, and natural numbers have only finitely many digits.

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

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

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

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