## Hilbert’s Infinite Hotel – 60-Second Adventures in Thought (4/6)

A Grand Hotel with an infinite number of rooms

– and an infinite number of guests in those rooms.

That was the idea of German mathematician David Hilbert, friend of Albert Einstein and

enemy of chambermaids the world over. To challenge our ideas about infinity he asked

what happens if someone new comes along looking for a place to stay?

Hilbert’s answer is to make each guest shift along one room – the guest in room one moves

to room two and so on – so the new guest would have a space in room one. And the guest

book would have an infinite number of complaints. But what about when a coach containing an

infinite number of new guests pulls up – surely he

can’t accommodate all of them? Hilbert frees up an infinite number of rooms

by asking the guests to move to the room number, which is double their current one,

leaving the infinitely many odd numbers free. Easy for the guest in room one. Not so easy

for the man in room eight million, six hundred thousand, five hundred and ninety-seven. Hilbert’s paradox has fascinated mathematicians,

physicists and philosophers – even theologians.

And they all agree you should get down early for breakfast.

What "indetermination" ? In function analysis it is of course, as you can't say that x^2 – x = 0 when x becomes arbitrarily big which is another way of saying that x goes to infinity. Because you never actually calculate "infinity^2 – infinity" but always a real arbitrarily big x. But, what we mean by "infinity – infinity = 0" is "x – x = 0" which is true for any element of R. In closed R, infinity is an element and of course infinity – infinity = 0 because of the definition of equality " = ".

The hotel can't be full in the begining as it is always in the process of putting people inside room since you can't reach a state when you say "there's an infinite number of people in the hotel". You'll barely say that there's an infinite line of people entering the hotel.

[2] If you can say "We have an infinite number of people in the hotel and it is full" it means you've been able to process an infinite number of things in a finite amount of time. The paradox only exists because we assume something much more extraordinary that simply an infinite amount of things.

I'll say it again. He kills all the guests BUT the guy in room 1729. (The comma was a mistake, I'm sorry). Is it clear now?

You're saying that infinity (how the hell do you put that sign anyway XD ?), infinity – 1 = infinity which is "true" except that "infinity – 1" has no meaning. All I was saying in the first place is that infinity – infinity = 0 otherwise infinity is different from itself. This is just a traduction of "a mad mathematician with a gun kills ALL the guest in the hotel. Hence the number of guest is 0" and that is perfectly fine theoretically.

And also I don't even understand why we are arguing about something so crazy ! I guess everyone litterally NEED to be right and prove his point. I'm perfectly ok with infinity – infinity is different from zero, but you need to demonstrate it and that, as I understand it, would mean that, given infinity is considered as an element, it can be "different" from itself which seem to me as just a new definition of equality which I don't agree on.

Infinity is not a number so adding or anything else such as that will not make sense

Hilbert's Hotel deals with "actual infinity", rather than the type of infinity you are talking about when you say "x approaches to ∞". If you understand ∞ as the number that represents the class of all the sets of cardinality ℵ-null (e.g. the set ℕ of natural numbers), then you can define "∞ – n", for each natural number n, as the cardinal of ℕ {0,1,2,…,n}, i.e. ∞ (ℵ-null). But how would you define ∞ – ∞?

GREAT! Exactly! I´m going to be a philosopher 🙂

Isn't this just the paradox of adding infinity…or am I missing something greater or perhaps something exaggerated.

i repeat its not bad to admit maths isnt your subject. infinity – infinity = 0 is false in any university maths book you ll open.

only this should stop your imagination . but i ll try to help you more although i am not sure i can! you inticipate infinity is not a number. but the point is that it isnt! clear and stable. if it was it would accept multiply or make it less. but it doesnt. even a high school student understands number like a quantity which u can add 1 and make it bigger. infinity cant get this option. there isnt any number bigger than infinity which is limit a ''dam'' of all numbers.

infinity cant get bigger or less so it isnt a value. you also say there is no thing as infinity + 1. so how do you take infinity as a number if u cant make a simple mathematical act by adding a number in it. it deosnt follow so the rules of numbers in R.

I liked the joke at the end…

You're talking about math book like they were some kinf of bible. You still haven't understand that everything is possible in mathematics given the proper axiom and definition, even a division by zero. If what you mean is that in the R set we use generally infinity is not considered as an element, fine. But in a broader sense "equal", abreviated "=" means that "the left and right argument can be interchanged without breaking anything in all expression" and "minus" abreviated "-" means that ..

[2] .. "in an ordered set, x – y means the predecessor of x by y unit". Now, if I accept the definition of equality and if we agree that infinity is equal to all its successor and predecessor then infinity = infinity, infinity – 1 = infinity and infinity – infinity = the null element, which is abreviated in all set as "0". Matrix A and B can be of infinite order and yes, A – B = O where O is the null element. Moreover, google that : Extended real number line.

You're right, that's why we use another set : google Extended real number line., read the Arithmetic operations in wikipedia.

The equation "x-x=0" works since x represents a fixed arbitrary number. But infinity is not a fixed arbitrary amount…….When we have the eqn "infinity – infinity = 0 " , the first infinity symbol is bounded to be no larger or smaller than the second, contradicting the unboundedness of infinity. Hence the equation does not work.

no i totally disagree. i talk about all math books in every university in the world so about something with huge validity . not about the bible!! a fairytale book. i proved you somehow in my other 2 comments why infinity can be a quaninty but it seems u didnt read them by mistake. the upper wasnt my only post 🙂 . anyway keep going .

You know you don't need to make former guests shift one room, right? Just fill in the other infinite rooms. Doesn't that make things simpler?

4. El hotel infinito de Hilbert. Otra vez jugando con el infinito

Hopefully it wasn't your English teacher

Statement A: If something has reached maximum capacity, it is full.

Statement B: If something is full, absolutely nothing more can be added.

Statement C: If the hotel has an infinite number of rooms and an infinite amount of people, then it has filled maximum capacity.

Statement D: If the hotel has filled maximum capacity, then it is full.

Statement E: if the hotel is full, then no more people can be added.

The moment Hilbert suggested adding one more person this paradox became bullshit.

Unfortunately this does not work. What room number do you send the guest to? You cannot ask them to go to room number infinity + 1. But you can easily ask them to go to room 1, after asking all your guests to go to room number n + 1 where n is equal to their old room number.

Hahahaha!!

these are great

Have anyone ask, do infinity even exist? considering it not a number, but an idea.

The only reason infinity exist is because no one or machine is capable of comprehending & obtain the finite limit of the universal scale. Infinity is an assumption if infinity exist at all, just like "monkey theorem" or "fractal".

I thought of a paradox. Let's say you went to the past to shake hands with your younger self. Then you go back to your present day. What would happen is that when your past self grows up and finally ends up in your time period, he will go back to the past, and shake hands with his past self. This would happen over and over in a never ending time loop.

Because you didn't capitalize "M" in my, and you didn't end your sentence with a punctuation.

Gee bro, calm down O_O

I can tell you are having a bad day. LOL 😀

Maybe a cup of nice hot tea will make you feel better.

I'm 12, and whatever you say to me doesn't phase me in any way. So go ahead, keep throwing insults at me. It's the only way you can waste your time, after all. Plus, I didn't even want to start a fight. You just rage because I pulled a little joke on you. I hope you don't act like this to other people.

1) I'm a guy. 2) What the fuck is wrong with you?

I tottaly agree with this idea.I allso want to add , that if the qifsha is one with the motren then and only then the infinity will be quaninty.Only the qifsha motren matters…be aware of that

Love the video! It makes me smile 🙂

Well… when he fills the hotel in start… he already assumes that someone is placed in room "314.424" i wonder how old is that guest in first place.

" even theologians" haha fucking wankers 😀 religion is not a valuable profession!

have fun hahahhahha

Not sure the the fuss is all about.

The hotel has an infinite number of rooms which are ALL FULL.

So, there is no way to move the guests.

Now, what to do with the remaining 55 seconds of this 60-second thought adventure ….

Hilbert's Infinite Hotel for you. Get down early for breakfast…(1 min)

I don't understand the paradox… Infinity + infinity = infinity, right?

This paradox is only sound if you believe it is possible to have an infinite amount of objects. This is almost certainly untrue, while infinity can easily apply to concepts (like mathematics) and possibly even space and time, it is hard to see how you could have an infinite amount of physical objects.

I don't get where the paradox is.. we already know that one infinity can be different than another, so why is the fact that 1+inf=inf or inf+inf=inf so interesting?.. I guess it could've bothered people decades ago when there was no real concept of infinity present in mathematics but now?.. why do we still call it a "paradox"?

But what if an infinity number of busses come along?

the amount of space available in the universe is finite and can't ever reach infinity. space itself is stretched toward infinity thus always getting larger. also the argument that each object is half the size of the previous is bogus because one plank length is as small of a unit that is possible.

Such a lovely place (Such a lovely place)

Such a lovely face

Hilbert's Infinite Hotel paradox with adding a guest (alternatively adding a finite number of guests) is clearly a play on the mathematical concept of "almost all".

Adding an infinite number of guests is in turn a play on the intuition that e.g. there is "more" natural numbers than there is even numbers, while according to the definition of two sets being equal in element count, there is just as many even numbers as there is natural numbers (this breaks only from rational numbers onward).

next step: uncountably infinite guests check in the hotel.

Imagine being at the top and falling out of your window, lol.

Another paradox if you are smoking a lot of pot, I suppose. Infinity is not a number. Attempting to treat it as one is the source of the problem.

Simple: they keep on switching rooms infinitly or untill they die.

define your infinity

Some infinities are greater than others…

Da…David Mitchell?

The number of points between 4 and 5 is infinite, the number of points between 5 and six are equally infinite and the number of points between 4 and 6 are likewise infinite. But since the difference between 6 and 4 is two, and the difference between 5 and 4 and 6 and 5 are both 1, the infinite set of points between 6 and 4 is double the number of infinite points between 5 and 4 and 6 and 5. However they are both still infinite.

But why is this sort of thing important? Am I missing something, or do the people who spend time pondering this sort of thing need to get a life, and do something useful?

Does Hilbert's Hotel have enough space to contain the universe in which it exists?

It seems logical that an infinitely large object cannot be contained.

It seems logical that an infinitely large container can contain anything and everything.

So we have a paradox.

Hilbert's argument begins after he has taken the liberty of containing an infinitely large object, the aggregate body of the guests, within an infinitely large container, the hotel.

Herr Hilbert is a very naughty boy!

This example misses the point by failing to continue the thought experiment with infinite buses containing infinite guests (who can be accommodated in Infinite Hotel) and then showing that the hotel cannot accommodate infinite guests denoted by decimal numbers. Thus showing that some infinities are larger than others. Hilbert would turn in his grave if he watched this (possibly continuously)

Why make any of the present guests move? Just make the new arrivals go to the top. Since the hotel has infinite rooms, it also has infinite vacancies.

What a bullshit omg

But infinity is not a number.

Well, THAT'S why you can

NOTuse infinity as a number…But as a concept.Imagine it as an x factor, that's the only way you can solve equations with it, though it still makes almost no sense.

i.e x+1=?

x+1=

x+1x+x=

2x∞+1=

∞+1∞+∞=

∞Which again makes no sense, since ∞= INFINITY.

It's like saying water +1=

water+1but water + water=2 Water.See how that sounds stupid?

I did not say gallons,Oz, Liters or anything so you can't calculate it…

Wanna think of it as an x Factor too?

Well (∞-1) + (∞+1) how much does that equal?

If you said 2∞ you're wrong.

(∞-1) + 1=? If you said ∞ you're right.

Why?

Because it's already infinite again, you can't say water-1…

You can calculate things (from what I tough myself and though about) as this:

∞=0+1/second/second.

so 0+1+1+1+1+1+1+1+1+1 (gaining +1 every second to infinity).

In my understanding of my awesum conclusions 😉 You can solve equations to that like:

∞=0+1/sec/sec

so in the first

secondit'll be 1second

secondit'll be 33rd

secondit'll be 6so you can say ∞ (in the 3rd second) – 5=?

you say ∞^3 (or whatever new something symbol that you can use to indicate what second it's in)-5=1

Which is I.M.O 100%

Please correct me on my theory or whatever that shit's called.

Stupid yet correct answer to ANY ∞ equation = ∞…

Just like 1^anything = 1

1^∞=1

The solution is that infinity continues moving and never ends. The guests would be constantly switching rooms forever.

@Tabna Shahid

But wouldn't it take an infinite amount of time to carry out the instructions to the guests?

So tell me if this argument holds water.

Lets say we follow the same procedure, but instead of moving the guest from room 1 to start, we instead have the guest from room 1 wait until the guest from room 2 moves first. That guest has to wait for the next guest and so on so that we only free room N when room N+1 has be vacated. Since there are infinite rooms this process will never be completed, as each guest will always be waiting on the next guest to move. Therefore the new guest never actually gets a room.

It's always seemed to me that this only works because we start at room one which is suddenly empty, but infinite room shuffling happens afterwards. If we require the infinite shuffling to happen FIRST, then it becomes impossible to complete, and the hotel indeed has no vacancies.

I'm sure I will be proven wrong somehow, but I'm curious to see the reasoning. 🙂

And woudn't every porter expect a tip!

We cannot say that the hotel has infinitely many rooms unless there were infinitely many builders building it. But we cannot say that infinitely many builders built it because that would imply that the hotel was built at all. We cannot say it was built at all because that would imply "completion". Completeness implies wholeness or in the mathematical sense, whole numbers. Since we can count whole numbers, this contradicts the idea of infinity. Therefore such a hotel cannot be built.

"`A` grand hotel with an `infinite` number of rooms" -> contradiction

I always fail to understand this one. Where is the paradox? Infinite people can be accommodated in a hotel with infinite rooms…

New Guest : "Hi, I need a room."

Clerk: "We have infinite rooms: here is your key for room [Infinity + 1]"

New Guest: "Where is that room? I can't find it."

Clerk: "Never mind. I will shift the other guests to the next highest room number and give you room #1."

New Guest: "Do you have HBO?"

The paradox is that one infinity can be contained in another infinity, or in other words some infinities are larger than the others.

There is an infinity of prime numbers within an infinity of natural numbers. So when the guests move to their new room numbers which is double their current one, there is created a smaller infinity of odd prime numbers (vacated rooms).

I dont understand how this is a paradox though. You can shuffle an 0/0 amount of people into an 0/0 rooms… Since 0/0 is just hypothetical, where is the problem?

I guess it's only a fancy way that when you obtain infinity in your calculations, your theory crashed. It becomes a non-theory.

What happends though if there is an infinite amount of Busses with each having infinite amounts of Guests? How do you fit them in? Is it possible?

Hilbert's hotel…Still too small for your mum.

Why do the people have to move rooms? If the guests can move to a new room, then surely the new guests could just go there without moving everyone?

Why move to odd numbers? Why not just the next room?

Philosophical question – Do you agree ?:

We cant grasp infinity. But if I watch this video a thousand times now, I have a better sense of infinity than one who doesent.

If there are 2 guests in the hotel and 3 more arrive the 5th guest will have to walk to room 5 which is the closest available room. If there were 4 in the hotel and 5 arrive, the 9th guest will have to walk to room 9, 3B guests and 3B + 1 arrive, guest number 3B+1 will have to walk to room 3B+1, etc. What's my point? If there are N guests in the hotel and more than N guests arrive, the last arriving guest will have to get a key to room that was not previously occupied. Bottom line is, the hotel cannot contain an infinite number of people

God damn infinity; you beautiful mindfucker you. The idea that in the space of 0 to 1 it would be impossible to count the decimals between the two makes me happy that humans have devise math well enough that that idea exists and infuriated that it exists at the same time.

Not happy with the sentance structure there, but you (Reading this) know what I mean.

Can some one explain to me. How in an infinite hotel, where every room is infinitely occupied by a set of infinite people. Can each person move one room to the right? .. There is never going to be a space available for them to shift!? It's infinitely occupied.. They would keep asking each person to the right to shift and no ones ever going to get the opportunity to actually move.

You're assuming there's an infinite amount of space available which is contradicting. How can you have a hotel full with an infinite amount of people. Suddenly vacant with an infinite amount of space available.

Which infinity does this conundrum best address?

I can see this being a problem in regards to the hotel, but in regards to life itself, and time it isn't taking into account death. death itself, in this account, is not a non-existance, but the dead being also doesn't use space in the same way as does the living being. The reason for the mistake here is a non-understanding of space, which is what he is basing his theory and conjecture on; see space as both empty and full at the same time (that's infinity) ; and then, well, then we get go into the understanding of Time which is both space and no space equally.

This one gets an instant subscription. TedED had an animation about this but they made a really shitty boring video. This channel did it even better and they did it way faster.

is that david mitchell?

Hi

Why david choose the hotel to explain his theory

Why not a train or something ????

What is being described here is the difference between a one dimensional and two dimensional infinity.

Everybody can grasp the idea of a one dimensional infinity. It is the endless string of numbers or in the this case the linear string of rooms in the infinite hotel.

The infinite hotel described is like a row of chalets with infinite length. We don't even need to consider how many storeys the hotel has or it's width. The hotel is infinitely long in one dimension and thus can accommodate infinite guests. The supposed paradox arises when the new guests arrive………

When we introduce the new guest(s) we add a second dimension to the concept. It is helpful to think of this second dimension as time, but we could just as easily be adding infinite storeys to the hotel.

It would take time for each guest to move to the next room. We cannot fully describe this movement of guests using only one dimension, because it has now become a 2D concept. If you watch Carl Sagan's video on the 4th dimension, you can see the limitations of comprehending shapes from a higher dimensional space.

In this 2D infinity, we can accommodate an infinite number of new guests in the second infinite plane.

That germans should better find a way to fix their cheating emissions on their cars… booooy

Sorry their is no vacancy at The Grand Hotel. No finite object can be infinite in number, because anything infinite would require an infinite amount space to exist in, thus leaving no room for anything else. The only thing that can be infinite is infinity itself. The all of the all is the only thing infinite. Everything else is a sub division of it. And that's it. There is nothing more because that is everything. No extra anything can exist because everything that can exist is contained in the all of the all and it is never ending and ever increasing in size if you were going to try and quantify it. You can't. Numbers are a way in which we compare one thing to another to determine length, width, height, volume, speed, velocity, etc, etc & etc. What could you possibly compare infinity too?

If the infinite does not have "end", so the "start" neither, there is no room #1

Infinity*2 = Infinity = Case closed Next please.

I made a short film about this! The trailer is here! https://youtu.be/L1y4a4W_RVY

hi im in room tau of the real number infinite hotel

and the guy in room root 42 is so annoying

Why does anyone need to move if there are infinite rooms. The new people just move into a room, irrespective of what the current occupants of other rooms are doing.

I actually applied this paradox to my job in a stockroom in HomeGoods in order to keep up with the trash and it actually works really well.

Why bother with maths? If the manager took the infinite people out of the hotel at the same time as the infinite number of people got out of each bus as it arrived, he would have only one set of infinite people? He could then fit the infinite number of people outside the hotel comfortably into the infinite hotel. Check out http://www.psawa.com/The_7_Dimensions_of_Everything_and_Nothing.html

2019?

Have to inform the infinite people on the bus that they are already in the infinitely large hotel with infinite people.