Infinity Paradoxes - Numberphile
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- Опубликовано: 15 авг 2024
- Infinity can throw up some interesting paradoxes, from filling Hilbert's Hotel to painting Gabriel's Trumpet... Mark Jago is a philosophy lecturer with a background in computer science.
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The money game is known as St. Petersburg Paradox - it is quite famous!
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Hilbert's Hotel is impossible, because you would have infinitely many complaints.
Unless you have guests with infinite level of patience.
You would have an infinite number of arsonists.
But, then again you would have infinitely many serial killers, that are killing an infinite number of normal people..
Not if you had infinite lives.
but then, there are infinite possibilities that you DON'T have infinite lives
"The manager's clever." Well, there's a paradox right there.
which is?
@@yonishachar1887 Presumably the idea of a clever manager. Btw, you're commenting on a 4 year old comment, were you expecting a reply?
@@danielalorbi Actually yeah, even if it will arrive in a year. it will still be worth it.
RUclips is a site that you can't quit, so the chances of replying are bigger
@@danielalorbi Well he got one didn't he?
I’m here too :)
I was promised infinity paradoxes, and you only gave me four. I will send you an invoice for infinity dollars, with a payment deadline of infinity. The appeal process will take infinity years, so don't bother.
Clingfilm Productions FFFFUUUUUUUUUUUUUUUUUU
Hahaha this is funnier than it’s supposed to be xD
Infinitely don’t bother
Well, no, you were promised infinity paradoxes, not infinitely many paradoxes. Infinity is not a description that tells you how many, infinity is just the quality of some such description.
@@angelmendez-rivera351 woooosh
Gabriells trumpet problem is actually solved by pointing out that hypothetical paint is actually a layer of zero thickness, so you actually need zero hypothetical paint to paint the entire infinite suface of the trumpet.
Insisting on a physical thickness for the paint results in dividing the problem into two parts. Where the paint layer is thinner than the trumpet we simply use thickess times surface area, where the trumpet is thinner than this we simply use the volume of the trumpet. Both of which are bounded. Problem resolved.
@Philip Moseman that's nonsense.
That was my thought when I heard it. He tells us about intuition changes when talking about infinity, however he doesn’t when he talks about the paint. It must follow the same principle, as you said.
@Philip Moseman You may invent your own version of math if you so choose. But I must warn that nobody will listen.
It was a bad paradox imo, the paint will run out eventually. Paint doesn’t go on forever.
@@Memistical Why not?
By the way I love the way Brady keeps showing us a sideways "8" and thinks we won't notice it's not the infinity symbol.
???
@@MrDennis8169 In the sideways 8, one of the circles is smaller than the other. That's what he was showing. You can see it in the thumbnail.
@@EGarrett01I thought that was just a slightly different symbol for infinity? Unicode displays it ∞ like that...
This, Numberphile, is why people created limits.
Precisely. "Infinity is a process, not a number." - Richard W. Hamming
@@Qladstone where does Richard W. Hamming say that?
Actually this video explains why we created measure theory. Calculus dodges the question a bit too much.
@@hassanakhtar7874 Could you expatiate please?
Well, there are limits to what humans can understand. But the question is:
Do these limits converge?
My favorite fact about infinity is you can split it up into infinite infinities.
My favourite fact about infinity is that It's all impractical nonsense.
Skel Neldory that’s what you think
@@SkelNeldory its not fact its ur nonsense opinion
@@nolimitderrick4822 yes man ryt
@@SkelNeldory And that's precisely what makes it so fascinating
I have a 0% chance to hit any point on a dartboard,
only if there was no effort
no, you have a 100% of hitting A point on the dart board, but there is a 0% chance of hitting any GIVEN point.
I don't think that there's a 0% chance of hitting any given point. I think that there is an infinitesimal chance of hitting any given point.
The point has 0 area. 0/A, A being the area of the dartboard, is 0, therefore you have a 0% chance of hitting it that point. 0% chance of hitting it does not mean that it is impossible. Definitions become a bit wonky when you introduce infinite sets, especially uncountably infinity sets, but it truly is 0%.
What the Hectagon?!
I disagree with the idea that a point has 0 area. I think that a point has infinitesimal area.
I like how the thumbnails of these videos show the people during a moment where they have the weirdest expression of their faces XD
That casino must be at the Hilbert Hotel... where I hear they play amazing darts, and they have an absolutely angelic trumpet player named Gabriel whose horn is painted as beautifully as he plays it ...
Lol nice. 👌
Lol.
Underrated comment😂
Let’s say I have a infinitely deep v-neck, but I’m also wearing a sports coat. Infinitely awesome lol.
Driving a white Ferrari, itty bitty shorts, sloppy steaks at Truffani's....
5:42 "so its this kind of granular nature of our existence that gets us out of trouble".a really beautiful/funny sentence :)
quick thought, for the first paradox, wouldn't it take infinitely long to move the customers?
hornylink Well couldn't they move at the same time?
***** I don't know. the paradox very specifically moved them in sequence. I don't think it's possible to move them all at once because that would require you to move all of infinity
hornylink Brady's visualization moved them in sequence, he could be wrong, *or* he just simplified the process. They could very well shift positions at the same time to escape the problem you mentioned (they can take how ever much time one person needs to shift, instead of infinite time).
I don't see how it could be done differently without taking infinite time.
hornylink Whenever you take infinity into account, get rid of time all toghether (unless specified otherwise), it will make more sense that way
Whoops! yeah generally. but I figured it was appropriate in this case since it was a video about how infinity fails to mesh with reality
It is paradox that push mathematics forward and deepen the understanding of human being.
in the casino one, there is no possible loss, so... why would you bet anything less than all your money? I don't get the logic
+yovli porat You missed something: The pot always starts at 1$ and doubles from there, the
question is how much the casino could charge you to enter in a single
game before you think it's not beneficial to enter anymore because
you'll lose money rather than win it. 1? 2? 50? And the answer is, given
an infinite amount of times you play the game, no matter what you end
up paying, you'll win an infinite amount of money.
+BintonGaming . Thank you for explaining that, I was a bit confused about that one.
+BintonGaming Where is the loss/gain in this game? Why would you pay $1M to win $1 and/or possibly lose the stake. I just don't get what I'm risking against what I can expect to receive!?!?
+Brady Dill
No. The expected winnings would be the sum of all the possible amounts each multiplied by the possibility of getting the certain outcome. Since the game could in theory continue infinitely, then the expected value is infinite, despite in most cases you would not make it beyond a couple of rounds. That is why it's a paradox. On paper you should pay whatever the cost, but it would rarely be beneficial in real life.
+yovli porat I think the casino "charging" you to enter has not been clarified in the video. So it is like "it costs you 50 pounds to enter a game that starts with 1 pound in the pot". The question is "how much would you pay for an entry ticket given mathematically infinite amount of earnings?". At least this is the only way I could make sense of it. Otherwise if you can put anything in the pot and u either get it back or casino doubles it and gives it to you next turn there is no loss and anyone would bet anything they've got in there.
I’m living for his “can I speak to your manager” haircut. The video is pretty cool too
It's on a man, and he has an accent, it's perfectly acceptable
The vid's pretty neat too
@@soupisfornoobs4081 yep
who's the idiot who wouldn't put all the money he has in the box? the game looks pretty easy to understand and you are gonna win 100% of the times if you can flip the coin as many times as you want, there's only 2 flip possibilities and none in gonna make you lose...
I wouldn't. You don't get back your money when you win, you only get those from the pot. 1 pound pot that is doubling. That means that you buy-in for 1000 000 pounds and if you win in third round, you get 4 pounds (1x2x2) and the million is lost for you.
You are only guaranteed to win 100% of the time if both you and the casino already have an infinite amount of money (so that you can always try again and the casino can always pay). So in the thought experiment you should bet all of your money but no in real life.
First flip tails oh look you win £1 there you go.
It's that much per game. The game ends when you flip a tails and take home the money, then you'd have to pay for a second game. It does on the surface seem odd paying a million+ per game thinking you're probably going to lose well before the 21st flip and take home way less than the million you payed.
In practice you might run out of bank balance before you make enough money but take an average and your winnings per go will outdo any game price.
Infinity doesn't really apply to physical things. Atoms are small, but there is a certain number of them, not infinite. So the dart's chance of hitting the board is based on how many atoms there are.
I was thinking that too, and how it completely invalidates the second paradox (the trumpet one). There is a measurable smallest unit possible (the planck length), so infinitely small is theoretically impossible because it has a finite end.
The thing is, the trumpet "paradox" is based on the paint having to be made of atoms while the trumpet doesn`t, which kindof counteracts itself considering you can`t even fit a finite particle in the thinnest part of the trumpet, so there can be no paint there. the "paradox" can be solved by imaginig the usage of infnitely small paint particles.
I like to think of infinity as dimensional. Say you have a line, a square is composed of infinite lines, but we can still measure squares with a different unit by integrating.
If you apply that thinking to the dart board, you have infinitely many points, but there is no probability of hitting a single point, you have to integrate in order to calculate what the probability is of hitting a certain area.
I hope the way I said it is somewhat understandable, its difficult to explain.
Mathematical point since there is infinitive points in any area it is infinite
Either way, 1/infinity isn't TECHNICALLY zero in this situation, it's better described as Epsilon
His hair makes me mad.
....because I know mine will never look that fabulous.
That vneck tho
Theoretically, in the fourth paradox, if they flip heads an infinite amount of times than you would never get your money no matter how much the pot grew because it would never land on tails for you to be able to get the money.
True, but the probability of flipping all heads forever is zero, so that contingency does not affect the expected value.
It's so comforting learning about things like Gabriel's Horn and struggling with it just to watch a video like this and see that mathematicians struggled too. Struggled to the point that some just said to ban it haha
lol
yeah true tho
The Hilbert Hotel: If the hotel is infinite, and it is full, everyone must be in it. So NO ONE could walk up to check in. That is the paradox
Gabriel's Horn: If the surface area is infinite, the volume must be infinite. I don't know why he says otherwise.
Dartboard Puzzle: Wherever the dart does hit, it will be touching an infinite number of points (assuming the dart tip is finite). This is possible because a point occupies no space--it is in the 0 dimension.
Double Your Money: I may not be understanding this properly, but it sounds like no matter what your getting free money. I mean, they are putting a pound in the pot to begin with, so you get at least that no matter what side the coin lands on. Just pay a pound to play and you can't be any worse off than you were before, right?
-Feel free to respond and tell me if you think something different. I've been trying to comprehend infinity for a while now, and I may have misinterpreted some of the paradoxes.
LeinadONyt Hilbert: based on infinity equals infinity plus One, so a new room at infinity plus one opens up and you slide everyone up a room to let the new guest in
Horn: length = infinite
Area = length * width , length increasing width decreasing means you are about same and the shape of the horn means slight increase in area ( harmonic series) very slowly up to infinity.
Volume = length * width * height, length increasing but both width and height decreasing means volume going to zero increase and thus a finite value ( not infinite like length or area )
Dartboard: if it was really zero you would never hit. Need it to be an infinitesimal to solve the paradox.
Pot Game: the question is how much would yo pay to play. If $1 entry it is obvious you should play. If it is $25 you risk losing $24 if it goes tails the first time. What if entry is $1000? The math says pay any amount to play - that is the paradox.
Gabriel's horn volume is provably finite, numberphile recently made a video on that (same reason some fractals have infinite perimeter and finite area)
The question for the betting game should be: you're in a casino, a host offers one round of this game to the highest bidder (always starting from 1£), how much do you bid?
Infinite hotel: imagine that each time someone in the hotel gives birth they throw the baby out the window. now each moment an infinite amount if babies is asking for a room. at the reception.
Infinity _doesn't_ equal _everyone._ Infinity means there is an infinite amount of people... don't ask where they're coming from, but at no point does he claim that "all" the people are in the hotel. However, the manager is _not_ smart if he decides to move everybody one room down. Even if only half of them complain, that's still infinitely many bad Yelp reviews. What he should do is send every new customer to the first free room. They'll never reach it, because there's still an infinite number of rooms, but they'll also never get to a "last" room and conclude that there's no vacancies. So... IDK, it's a different kind of paradox.
*"Fractals:* You can see infinity with what they call fractals, see the Mandelbrot Set. A simple Formula or pattern can repeat itself an infinite amount of times without ever resulting in the same thing or outcome." *- from ~~The Present~~ at TruthContest♥Com*
Question Everything The true fractal is infinite although you can not literally zoom in infinitely or calculate the fractal's rule an infinite number of iterations ):
+Cooper Gates How about if you look in a mirror while holding a mirror. That would be a fractal and you would see yourself holding a mirror and within that mirror see yourself holding a mirror etc to infinity (zoom in or not does not matter)
+twee Weekes This relies on the speed of light. Super fast, but not infinitely fast.
Hilbert's Hotel: You can't fill an infinitely large hotel because there will always be a door available. You even said that yourself.
Gabriel's Trumpet: How the actual f*ck can an infinitely large trumpet have a finite surface area
Dart Paradox: Just like you said, the dart has surface area to it. If i throw a dart, multiple points will be hit (a infinite amount of points at that).
Betting: Defuq? Just bet like a buck, win. Bet all that. Win more. Bet that. Win more. Repeat for infinite cash. Problem?
The most interesting, (and troublesome), is the trumpet paradox.
Unfortunately it isn't properly explained in this video.
I first saw it excellently explained in Lancelot Hogben's book, Mathematics For the Millions. I advise you to not research it. I am doing my best to forget it
Dan Kelly Trust me, I had no plans on researching an obviously flawed "paradox" like that one
Paul Kelly He was a FRS and if you read his explanation you will be convinced.
Dan Kelly Sorry, but whats an FRS?
Paul Kelly Fellow of the Royal Society, something like that.
"We wouldn't bet everything, would we?"
There is no loss state. The risk is 0. You go home with, at a minimum, the amount you put in.
You don't put in the amount you pay. You put 1 pound. The question is how much would you pay to be able to start with 1 pound.
Alon Loewenstein Do you realize you just replied to a 3-year old comment whose question was already answered by a dozen other RUclipsrs?
I'd like to see who else answered it.
Why wouldn't we just pay 1 pound because that is a guaranteed win?
+Alon Loewenstein The other people who answered are currently moving between rooms at the infinite hotel.
In the first example it should not be called a full hotel. Because the infinite rooms are matched by the same number of guests, so the way it was phrased does not make sense to me.
It is full as there is a guest in every room.
The hotel IS full, though, because every room is paired exactly with one guest, or one set of guests, anyway. What this paradox proves is that the map n |-> n + 1 is bijective for the set of natural numbers. It proves the set {0, 1, 2, ...} and the set {1, 2, 3, ...} have the same number of elements despite the fact that the latter is a proper subset of the other.
that's what I'm saying. The most basic law of infinity is "infinity - infinity = infinity"
I've been watching this channel for years, and now I'm finally old enough that it's helpful for completing math homework. Thank your for your help, Numberphile
The problem with Hilbert's hotel is that it assumes the person in room 1 can move to room 2 before the person in room 2 has moved out. If you disallow that, the person in room 1 would never be able to move out, thus never creating a vacancy.
Well there's an infinite hallway they can use to do the move
all guests get notified simultaneously through the intercom. they move to the next room simultaneously done.
or room one knocks on room 2s door and tells him to move .takes an infinite amount of time to move everyone ,but immediate vacancy and each roomie is displaced for a max 10 mins
I'm not sure if I understand the scenario on the last one. How would the player ever lose money if he is either getting back his money or doubling it? What am I missing?
You bet 100 dollars, the first flip comes out tails, you get 1 dollar. You just lost 99 dollars.
the pot starts with 1 pound in it
darrenflips: exactly what I thought.
@@Shawn-hk1ud Even if you bet $1000. The worst that can happen to you is getting your $1000 back. There is no scenario where you lose money.
The question is about the entry fee for the game. The pot only doubles if you get Heads on the coin. You only get what is in the pot if you get Tails. If your entry fee payment was bigger than what is currently in the pot, then you have a net loss.
the thing that the video narrator and the commenters dont acknowledge is that infinity ONLY exits in the math NOT in reality.
Mathematics is a symbolic language used for a modeling tool it isnt the actual thing you're describing anymore than the words you use to describe a thing replace that thing.
Oni Raptor the only thing it COULD apply to is the length or expansion of the universe
Actually, if you believe infinity does not exist in reality, then you clearly do not know physics.
The first paradox seems intuitive to me. If there's infinite rooms, you can fit infinite guests.
yea i never understood why videos had to explain that
Theres already infinite guesta
Not necessarily, some infinities are bigger than others.
What is counterintuitive, is that even if the hotel is full, we can keep fitting guests, but also there is the fact that there are sets of guests that can not be fit inside the hotel.
Really frustrated with all the people claiming that these paradoxes are in some way 'false'. Take it from me; they do make sense. Just the one with the dartboard is frustrating if you know about infinitesimals... But mathematicians don't use them either.
+Anouk Fleur They're all equivocations of theoretical and physical
terms; except the hotel where the phrase "and for the moment there
is somebody in every single room" breaks the infinite paradigm
altogether.
+snetsjs yeah and in the Dart paradox you can't have an infinite ammount of points on a finite surface. That's the difference between mathematical and real life paradoxes.
The trumpet one is related to scale. Refer to Mandelbrot's paper on the coast of Britain. the same concept applies.
That's related to Koch curves and how you can use an endless line to enclose a finite area.
The casino paradox doesn't work (or wasn't explained well enough) as there would be no apparently point that the casino could ever win. Why would they offer the game? Did I miss something?
Say you pay a billion dollars and the first toss is a tails. You take $1, making you a net loss of $999,999,999. You pay another billion dollars - the first toss is a heads: $2 in the pot, the second toss is a heads: $4 in the pot, the third toss is a heads: $8 in the pot, the fourth is a tails - you take home $8. Net loss $999,999,992. It seems that a billion you are losing too much money, but the maths says that even if everyone paid a billion a go, the casino would still lose in the end.
Kristian Allin the pot starts with 1 pound in it
So why not just pay 1 unit of money if you get tails you not lose anything and if you get heads then tails youve doubled your money so why bet more the 1?
@@kimbapai1095 if you lose, the game ends and you take what is in the pot. So if the game ends, you can't toss the coin anymore, therefore can't gain anything more. So it makes sense to bet everything that you have on each toss, since you can't lose anything, and you don't know how many rounds you'll win before getting a tails. On the other hand, if you could replay any amount of time, you could start low...
Kimbap Ai the question is how much would you pay to play. It is agreed for sure you should play if the entry is only $1 but would you play if the entry is $25 , would you play if the entry was $1000? The math says you should play regardless of the entry price and regardless of if you only play one game.
Here's one I came across on a gambling forum, Wizard of Vegas.
Infinite agents are each given either a white or a black hat. They can all see each other's hats, but they can't communicate once the colors are assigned. An infinite subset of them have to guess their own hats without a single error (but no limit on abstentions). Is there a strategy that can get the chance of success over an arbitrary number?
After it was demonstrated to the OP that he had (ironically) committed the gambler's fallacy, the board came to three solutions:
1. Starting at some n, the agents have to separate into fl(2^n/n^2) groups of n agents each. Each agent looks around their own group, and if they all have the same color hat, that agent guesses the opposite; all others abstain.
2. (The OP's shot at redemption.) Starting at some n, the agents separate into groups of (2^n-1), and each group indexes themselves starting at 1. Each converts the indices of those wearing black hats into bitstrings and XORs them. If they get their own number, they guess white; if they get zero, they guess black.
3. (This solution was a variant of the OP's erroneous solution, which explains some of the sillier features, but it is, as far as I can tell, correct.) The agents stand in line, and starting at some n and at the beginning of the line, the agents look behind them for groups of n agents, bounded on either side by exactly three black hats and an interior white one, with between 0 and 2 black hats among them. If there's exactly 1, they know the game is lost and give up; if 0 or 2, they then keep going, looking for such a group of 2n, 4n, 8n, and so on. Then they look to see if they themselves are one of exactly kn agents surrounded on either side by three black hats with white cushions and either 0 or 1 black hat in between. If so, and there are no black hats, they guess white; if 1, they guess black.
The expected value will always be that half the guesses are wrong, but you'll notice that all of these manipulate the guessers such that each group will either have a few correct guesses or many wrong ones.
1. Each group will have a 2^(-n) chance of all having the same hat and everyone guessing wrong, and an n*2^(-n) chance of all but one having the same hat and that one guessing right. The math's a bit tricky, but it adds up such that the chance of there ever being a wrong guess converges to a nontrivial value, but that of there eventually being another right guess will always be 1.
2. Each group will have a (vanishingly less than) 2^(-n) chance of the sum coming to 0, and everyone wearing a white hat guessing wrong, but if that isn't the case, the one whose index is the sum will guess right, so there will certainly be infinite guesses. It's easy to see that the chance of a wrong guess converges (NB: for 0
My 5 year old grandson just made me draw a banknote with infinite francs written on it. So, I think he's gonna be a mathematician.
Very nice paradoxes from Prof Jago.
Let me try to solve these paradoxes here:
Hilbert's hotel:
Infinite rooms, infinite + 1 customer.
X+1>X for all natural number X, but INF+1 = INF.
This is just the way infinity is. its not a real number, its a concept like i. and its useful.
Gabriel's trumpet:
finite volume, infinite area.
Actually, in a mathematical world, all 3D shapes are comprised of infinite 2D layers. Just like you need infinite dots to fill a line.
Dart:
we can solve it by ... infinitesimal? And of course limit tends to zero can also do the job. The concept is that there is a small gap between any real number you can name and zero. The dart will always hit that narrow gap.
Infinite Gamble:
We are certain some day, money will come back, the question is, how fast and how risky?
Lets assume each game takes 10 seconds. How much can you safely(50%+ certainty) earn for one day? it means 8640 games in total, but for simplicity's sake, let round it down to 2^13 =8196
In the 8196 games, we can safely expect half of the times you get 1.
then one forth you get 2, one eighth you get 4, ... and one in 8196 you get 4320.
meaning 4320+4320+4320+4320+4320... (13 of them)
So after a whole day of tossing, you can safely expect to earn 56160. meaning 6.85 per game.
Its true that if you gamble longer and longer, the odds are on your side, and increasingly so. But even a bet of 10 make this game rather risky. at least on day one, you have to be prepared to lose 25800(or more)
So it is actually rather reasonable that people spend only 10 or 20 on it. you will be broke before you can see any big money coming in.
But yes, if any casino introduce that game, it will bankrupt before you can try, I'm pretty certain.
In the second paradox: if the smaller end of the trumpet indeed get smaller and smaller, it will not extend infinitely long because there's something called planck distance, its the closest distance objects can get.
well in physics maybe but not in maths. You know the horn is just a concept not a physical object so the diameter of the horn which is 2/x can get arbitrarily close to 0 without being limiting to whatever distance.
You lot are going to hate this, but I think this is what a physicist might do for the last one:
Value of bet V= 1/2+1/2+1/2 ~ 1/2ζ(0) = -1/4 < 0 => do not take the bet.
He only partially went in to the Hilbert's Hotel paradox. The second part to it is if an infinite number of guests come to the hotel how could he accomodate them. And the answer to that is to have everybody currently in rooms to take their room number multiply it by 2 and go to that room. Then you have an infinite number of odd rooms to accomodate the infinite number of guests.
For the third problem (the dartboard problem), it is possible to use the concept of infinitesimal numbers to solve it. After all, the odds of hitting a single point out of an infinite selection of points is infinitesimally small. If ε is an infinitely small number and ω is an infinitely large number, then 1/ε = ω/1, and the odds of hitting one of the ω points is 1/ε.
I don't really see how this helps because summing the probability over all the points still doesn't get you anything useful in this scenario. And moreover why would the probability be 1/ε and not 2/ε or any other infinitesimal?
Regarding the dart board, why not say the chance is infinitely small, but not zero?
So, if the hotel has infinitely many rooms - how can it be full in the first place?
With an infinite number of guests.
Fair enough. But that is more of a tautology, really. I think the main problem with the hotel analogy is that at some point when it was built it was first empty and then it was filled later.
But since filling inifinitely many rooms with infinitely many guest would take infinite amounts of time, you'd never be finished and therefore you could never get to a filled state.
If you accept the concept of infinitely many rooms why not to accept infinitely many guests in them? You can imagine that a room is filled iimediately after is it built. Both concepts go hand-in-hand, it makes no sense to accept one and deny the other.
Think of an infinite number line.
Now, think of another number line directly beneath it placed in such a way that the numbers on each line line up.
1,2,3,4,5,6,7,8,9,10,11,12,13,14 ... Infinity
1,2,3,4,5,6,7,8,9,10,11,12,13,14 ... Infinity
So you have something like this ^. Let the first line represent the room numbers and the second line represent the number assigned to each guest staying there.
You can see that the hotel has infinitely many rooms because line one extends to infinity. You can see that the hotel also has infinitely many guests because line two extends to infinity. Finally, you can see that the hotel is full, because there will never be a number on the first number line that doesn't have an equal corresponding number on the second number line.
You can Supertask the guests into their rooms in less than a minute
Why does this guy have the same haircut as a 13 year old boy.
+Jasper Southwell he likes it?
benjamin button
I was gonna leave a like but you had 69
The problems with infinity arise when you try to mix abstract concepts (infinitely long, infinitesimally small etc) with real objects (paint made of particles that cannot be infinitesimally small, a person's wealth that cannot be infinitely large).
"Although it's always crowded
You still can find some room"
-Elvis, Hilbert's Hotel
heres one if pinocchio said "my nose will grow now"....will it grow???
Infinity is the opposite of zero, yet so similar in many ways
When I think of paradoxes like this, the one that comes to mind is a question - how long does a ball take to fall distance X if we time how long it takes to fall the first 1/2, then time half of that, and so on. There are infinitely many 1/2s, so infinite amount of time - is the paradox. However, math has the idea of limits which solve this nicely, so the issues where infinity is paired with something real... isn't all that paradoxical.
For the dartboard thing, I don't think the explanation he gave is quite satisfactory. You don't necessarily have a problem if you say that the probability of hitting a single point is zero, because when you consider the dartboard, it is an uncountable union of points - which means you cannot expect the probability measure of the whole dartboard to be equal to the sum of the probabilities on all points (that makes no sense, as you have an uncountable number of points). You'll have to integrate instead on a probability density function - which is finite and very integrable...
I think I came up with a good explanation of the dartboard, however it's pretty philosophical. We could say that the area of the point is infinitely small, we'll call it dx. Since dx is infinitely small, we can say dx=1/(infinity). Now let's say we want to find the probability that the dart will even hit the board, since there are infinitely many points, and they each have the probability of being hit, dx, we could multiply infinity*dx, and then make the substitution infinity/infinity, which simply equals 1. That proves you are able to hit the dart board, mathematically, so I think the math works out. If I made any mistake, I apologize as I am rather tired while writing this.
The last one is called the St. Petersburg paradox. The expected payout is infinity only when you play this game infinitely long. In case of 1500 tosses of a coin (that is about 750 games) and a first payment of 1$ a fair price for 1 game would be just 2,8$, which means that on average after 1500 trials a player and a casino will have 50% chance to win. The general formula for this case (first payment - 1$) is y=0.25*log2(x/p)-0.5, where p=(1+1/ln2)/16≈0.15, x - total amount of tosses, y - a fair price for one game, log2(x/p) is the logarithm of x/p to base 2.
5:05 bold of you to assume I can hit the dartboard every time
I think once you realise infinity is not a number and therefore cannot be calculated with, a lot of the puzzle becomes easier to understand.
2:38 That only works because paint is 3D. If you had purely 2D paint, that wouldn’t work.
The simple reason that people won't bet a lot to play this game is because of diminishing value of a dollar: your first thousand dollars is worth a lot more to you than going from $1,000,000 to $1,001,000.
I know very few people who would place a $10,000 bet with a 5% chance of winning $1,000,000 regardless of expected outcome.
The knife that cuts through the gordian knot of the painting Gabriel's trumpet is that our intuition that the volume of paint needed to paint a surface scales with the area of the surface is based on the assumption that the coat is of uniform thickness. But for any finite nonzero thickness you choose for your paint, no matter how thin, the infinite trumpet will not only eventually get too narrow for a coat of that thickness to fit inside it but will be too narrow _for almost its entire length_ (that is to say, the part of its length that is thinner than your coat tends toward 100% of the length as the length approaches infinity). As such, you either have to give up the assumption of uniform thickness, in which case the intuitive relationship between volume and coated area no longer applies, or let it have infinitesimal or zero thickness, in which case the multiplication by infinite area becomes troublesome.
Just to test the casino paradox I wrote a script to run preform the scenario it 1,000,000 times and spit out the average money earned per try. What I found was strange. Although most of the averages were around 12, there were a few large outliers.
Funny thing is, as I lowered the amount of times the scenario was ran (1,000 instead of 1,000,000) the averages usually hung lower, more around 5, but it was harder to speculate which averages were outliers, as the results were far more spread apart.
Huh..
is it just me or do i find his hair very distracting..
Infinitely distracting
@Joe Exotic toché 😂
the dartboard paradox, does the infinitesimal monad or the omega from surreal numbers help us deal with this any more cleanly? It seems having a definable infinitesimal may get us out of trouble.
Hilberts hotel, you can check in any time you like, but you will never have to leave
Gambling Game - I must be missing something here. The question is, "How much would you pay to get in the game?" If you can get in for a penny, why would you pay more? You have a 100% chance of winning at least 99p. He is either not describing the game correctly or I don't understand him correctly.
You don't have free reign to decide how much you pay - the casino sets it. You only decide if the set price is worth it. If it's 1p then yeah sure, obviously you win no matter what. But if it costs 5 pounds to enter, are you willing to take the risk you just get Tails immediately and effectively paid 5 only to take home 1 pound? Mathematically, you should go for it even if the cost to take part is 100,000 pounds, because the place of infinity in the game makes the 'expected outcome' something that would never happen in reality.
@@TenArashi I would love to see mathematical proof of that. Because it does not make sense like that. Your chance of getting infinite amount of money is infinitely low as well. You have to keep rolling heads forever or a lot. So putting in all your money doesnt make sense. You are probably not gonna get 16 heads in a row ($65,536). Therefore putting in your house doesn't make sense. You are *definitely* not gonna get ∞ heads in a row, you are not even gonna get 80 heads in a row.
So you put all your money in, for an absurdly low chance (in other words impossible) of getting infinitely or just a lot more money. You said 100,000 pounds, to win that money back you need to roll 20 heads. Yeah so you get the point. It logically doesn't make sense, also mathematically doesn't make sense unless you just ignore the luck factor and say "i can roll a coin without getting any tails all day baby".
The chance of it hitting any one point on the dart board is infinitesimal. Problem solved.
Agreed.
Glad I'm not the only one who thinks so. This is pretty far outside my area of expertise.
+Daniel Bundrick what's that mean
+The Realist Infinitesimal means 0.000 ... 001 where the ellipses represents infinite zeros. It means the chance is as close to zero without being zero as is mathematically possible. I hope that helps.
+Daniel Bundrick You cannot place a final value at the end of a string of infinite zeros.
for the casino paradox, the answer is that money is not equal to happyness. Let me explain:
Let's take a more simple case. It's a free to play game, you have 99.99% of winning infinity dollars and 0.01% chance of loosing 500 000 $ and that's all the money you have. The expectation is infility dollars, but if you win infinity dollars you wouldn't be infinitly happy. You would be as happy as someone who wins 500 million dollars. Assuming you would win 500 million dollars, the expectation becomes a negative number and you shouldn't play this game.
The dart board you said has a 100% probability of getting hit by the dart
But the specific point is very unlikely to be hit because there are infinitely many other points that could also be hit
When I watched it a couple of years ago I understood about 10% of it.
Now, I understand about 30%.
See you all in 3 years.
... and 4 months
same lol
I'm not terribly impressed with the first paradox as it seems to me the infinite rooms would have to be infinitely full. Perhaps i just don't get it haha
I'd have liked to go more into the cone with infinite surface area. I can't wrap my head around Jago's statement that you'd need an infinite amount to coat the surface area though, you should only need the volume the shape would hold. Is he saying that by coating the surface you're removing paint from the volume? I found this paradox to be the most intriguing of them all.
I don't understand why the game paradox is all that interesting, and I don't think they can say you should get an infinite amount of money together to play it without more information.
At first I thought they were asking how much you would BET with those game rules (replacing the pound in the example with your bet) - which of course would be infinity because you'd be risking nothing on the game. There's no way to lose money. It kind of exposes the mechanics of the game when you look at it that way.
A very similar game would be if heads comes up, then you get a pound, but if tails comes up, you also get a pound. And the game never ends. You just steadily get one pound each flip. This game is much less confusing not because infinity was taken out, but because probability and multiplication were.
Of course, if you had a choice between playing those two games you'd still pick the example Jago gave because the money has the potential to grow exponentially if you miss the payout flip a few times. So it takes fewer turns to make more money.
But how much time does it take to flip the coin and get paid? You might not want to risk an infinite number of pounds (or even just a really, really large number of them) if each flip takes a day...
In one of the 'Hitch-hikers Guide' books I remember something about if the space outside the expanding universe is infinite and all the matter and energy within the universe is finite then as a ratio, technically we don't exist.
For the "infinite trumpet" paradox, I could say that you need another of those trumpets filled with paint to paint the second one :P .
Visual_Vexing Well, no. Not quite. The problem is that area is just an infinitesimally small region of finite volume. Therefore, finite volume can cover infinitely many infinitesimally small regions of volume, which means it can cover infinite area. It all boils down to understand the relationship between area and volume. It clashes with reality because infinitesimals are not physically real.
@@angelmendez-rivera351 yup exactly, thanks for the explanation I saw it earlier in these or the gabriel's horn video comments
Sorry but you can't have a full hotel if there's an infinite amount of rooms.
Guy Smith No you can't because an infinite amount will never fill up an infinite amount.
so infinity > infinity
is infinity < infinity?
robin van Sint Annaland No infinity is bigger than another. Nor are they equal.
+Guy Smith your wrong, deal with it...
+John Trollinski (SalvadorSTM) The set of all irrational numbers is bigger than the set of all natural numbers, so some infinite sets are larger than other infinite sets (which means some infinities can be larger than others)
For the last paradox, I think the problem comes down to the value of money vs. what we really want, happiness and/or well-being. Consider a middle class homeowner playing the game, and suppose they really do put up their house as the buy in. Pretend you succeed in doubling your buy-in. Are you going to be twice as happy? Better off by a factor of 2? No, you won't. Once your basic needs are met, there are diminishing returns on having more resources. But look at the flip side. Suppose you succeed in earning back only half your buy-in. Now you lose your home, or your car and can't get to work, or you struggle to feed yourself if you get laid off, or you can't send your kids to college, or whatever. Are you half as happy, or half as well off? No, I think you'd be much less than that. And the worse you do in the game, the greater the disparity becomes between your pay-out and your resulting happiness/well-being.
Here’s a better version of Hilbert’s hotel:
And even cleverer manager says “ no don’t move everyone down. We can leave the guy in room one where he is and just move everyone else and the new guy can go into room two.”
Then a still slightly cleverer manager says “no no no only do that for everyone in room three and after. We can leave the guys in rooms one and two alone and put the new guy in room three.”
Suddenly a queue of infinitely many managers forms, each slightly clever than the last…
In what room does the new guy end up?
Depending on whose advice one would take. And if you try to take all managers' advices, then you won't move anybody, and the new guest won't get a free room.
Did you hear that explosion? That was my mind.
Sound like a semantic error than a paradox. Full is one concept infinite is another. if you say its full then its full. if you say its infinite then youre saying it cant be filled. its just a bad sentence. A square triangle.
No, "full" and "infinite" are not incompatible concepts.
Let's look at a particular example. Suppose you have the rooms in your hotel labeled as 1, 2, 3, 4, etc. - one room for every positive integer. Now, suppose that you have guests in your hotel with very strange names; they are named 1/1, 1/2, 1/3, 1/4, etc. - one person for every positive integer. Now, suppose that for every positive integer n, person 1/n is in room n.
Clearly, there are infinitely many rooms and infinitely many people. Yet there are no empty rooms. Is room 3 empty? No, the person named 1/3 is in room 3. Is room googol empty? No, the person named 1/googol is in room googol. etc. You cannot find an empty room, making the infinite hotel full, no?
but the paradox is that you can always find an empty room for another guest even though you've said there are the same amount of rooms as guests, showing that it isn't full.
Infinity has no end, so if you ever tried to fill it, you couldn't. Things being full is a completely realistic and possible goal, but infinity is 100% abstract and doesn't exist in real life.
PirateOfPlayTime "but the paradox is that you can always find an empty room for another guest even though you've said there are the same amount of rooms as guests, showing that it isn't full."
No, this does not show that the hotel wasn't full. The "paradox" (it is a paradox in a more literary sense, not a logical one) is that for infinite sets, the part can be equal to the whole, and that rearranging an infinite set can make it appear bigger or smaller than it previously did.
"Infinity has no end, so if you ever tried to fill it, you couldn't."
Sure, unless you allow for supertasks. But that's not the point. It's irrelevant how the hotel came about to be full. The problem starts with the assumption that it is full. And being full is not in contradiction with being infinite.
"Things being full is a completely realistic and possible goal, but infinity is 100% abstract and doesn't exist in real life."
So? Things being bigger than other things is a completely realistic and possible goal, but Graham's number is bigger than the number of particles in the universe, meaning that you can't have Graham's number amount of things in real life. Does that mean it's complete nonsense to say that Graham's number is bigger than 1?
The flaw in your thinking is that you can actually prove such concepts when talking about infinity.
I agree. This example falls down at the first step. He say the hotel is full, and has an infinite number of rooms. But if the rooms are infinite, the hotel can never be full.
Solution of Hilbert's Hotel: First the problem: We are mixing ordinals with cardinals. The problem's statement is a representation of the ordinal omega+1 - countably infinitely many guests, and one newcomer after all of them. We use a non-order-preserving mapping from omega+1 to omega. We show that cardinality of omega and omega+1 are the same but the hotel, as presented in the problem, is an ordered set and we destroy it's order during the process.
Wrong with the trumpet... You had mathematical paint you could paint the entire infinite surface with any finite volume of paint. Q. How many 2D surfaces can you fit in a finite volume? A. Infinite.
I was not understanding how a horn with infinite length could have finite volume, but I looked up Gabriel's Horn on Wikipedia and I think I was able to understand it. Tell me if this is the correct way to understand it:
It's like taking a cube's volume, let's say 1 cubed foot, then adding half of that volume, 1/2 cubed foot, then adding half of that, and so on. So the number of cubes (analogous to the surface area) would be infinite, but the volume of the cubes would forever approach 2. Is this the correct way to think about this?
The last math class I took was calculus 1 a couple years ago in college, so I probably would not be able to understand the proof :[ Thanks for the offer though! I really like math and I can understand it if given enough time and the right explanations, but I regretfully skipped a math class my senior year of high school, so, when I took math in college, it felt like it just went a little too fast for me. If I had a personal math teacher who would teach me at my own pace, then I'd learn all the math I could XD but I don't T^T
***** No matter what pace you go, always continue to learn, explore, and have fun with math. :)
ExtremeMagneticPower
I intend to! ^-^
If you take a piece of paper and divide it in 2 pieces, then take those two pieces and divide them again and so on you will end up having an infinite amout of fragments that add up to the original piece of paper.
Simone Noli
Wow, that is a super simple way to think about it XD Thanks!
that haircut though....couldnt take him seriously if he was my teacher
Whoa! Shallow you.
Wow. That is really unfortunate. Being unique is important and should have more value.
I expect if he were your teacher when he'd mark your maths test your score would be the inverse of your coolness.
Psychology: If they are different than us, they are weird and should be banned. Huh, I just gave a lesson in psychology, history and politics!
That's how an intelligent person looks.
The dart board and trumpet one don't make sense because it defies simple laws. There is something called a Planck length- the smallest possible volume to ever exist. There is no such thing as any space smaller than that, so technically you are able to calculate the surface area of the trumpet and the probability of two points touching
About paradox #1, wound't it be more convenient to direct the new customer directly to the last occupied room +1? (Or the first unoccupied room if you prefer.) OK : it may take a while to walk to there but certainly less than moving customer #1 to room 2, customer #2 to room 3 and so on a few zillion times.
There is no unoccupied room. That's the whole paradox
I think the problem is that it's a silly exercise to begin with. Infinity is not a number, but a concept. Once you reach an answer of "infinity" for anything, that's where your exercise should stop. You can't use an answer of "infinity" as a new starting point to continue your evaluation of anything. When you start adding 1 or talking about shifting an infinite number of people around - you're just talking nonsense because you started out with nonsense.
My paradox that I think I came up with is that it is impossible to generate a random number. Image you could generate any number at all, even decimals. What are the chances o you getting a one? Well there are infinite other possibilities, so your chances must be zero. So the same could be said for two, three, four... ect So you can't get any number. And now I realize that is the same as the dart board thing
I like it :)
Giant Grass The chances are infinitesimally small, not zero. In real problems, the choices are finite, but your idea still works in principle.
Cooper Gates Like they said there is the exact same paradox here as in the dartboard thing: if you're generating, say, a real number between 0 and 1, and if the probability of generating a specific real number is greater than 0, then the sum of the probabilities for the (infinitely many) real numbers in that range is infinite. Similar to the dartboard, the paradox goes away if you consider “areas” rather than points - in case of random numbers it could be, e.g., only those numbers that can be represented by a given data type on a computer (or with a given amount of decimals).
Arkku Multiplying an infinitesimal by an infinitely large number is simply greater than zero, it is not necessarily infinitely large because an infinitesimal is infinitely small.
An infinitesimal is greater than zero but infinitely close to zero - still infinitely smaller than something like 10^(-(10^800)).
Cooper Gates It seems to me that any sum of probabilities other than 1 is still a paradox. =)
Since the smallest physical space possible is a Planck Length, there aren't an infinite number of points on a dartboard. Since there is a finite number of points, there is some real number probability of hitting one of those Planck Length areas
The coin game paradox seems like it might relate to the difference between 'expected gain' and 'expected happiness from that gain'.
Exactly this. Money has a decreasing marginal utility. Risk aversion is a natural outcome of decreasing marginal utility.
What happens when it's smaller than an atom?
I am thinking about it
death
A quark.
@@methatis3013 yes but smaller than a Quark is Space Time Fabric, smaller than that is death 💀
@@retroboyo4238 but is it worth talking about if it has no mass. No thicc, no point
The St. Petersberg Paradox (the version of the casino game example used in Economics and Psychology) has always irked me because it really isn't a paradox in any logical sense, the possible world in which the "Paradox" can be adequately demonstrated is so far removed from the actual world that it has no relevance to it, even if you tamper the "Paradox" such that it could be demonstrated in the actual world, agents remain rational despite not choosing to play the game, and even in that possible world in which the "Paradox" can be adequately demonstrated, agents remain rational even though they choose to not play the game.
It is not a real paradox
About the casino: I think the main point is that for humans, the utility of money doesn't rise linearly. There is a small chance of winning really big, and because that goes on to infinity, mathematically any amount of money to bet makes sense. But to humans, whether you get a trillion dollars or 10 trillion dollars doesn't make a huge difference. Any sensible person would take a 1/10 chance for a trillion dollars over a 1/100 chance for 10 trillion dollars even though the expected outcome is the same.
That's actually an interesting point about the human psyche. Let's get rid of the infinity and propose this game: You can choose between either getting 500$ or a 1/2 chance of getting 1000$, 1/2 chance of getting nothing. Most people will take the 500$. Iirc there was a study about this and the amount where both options were equally popular were at about 400$.
So in reality, we would have to multiply the chance with our actual utility for that money (though that is of course hard to pinpoint), but it would definitely a row that converges to some finite sum.
Sorry, Dr. Philosophy, as an engineer, you never have to deal with these paradoxes in what I like to call "reality." They're fun to think about, though. :)
Daric Soldar That's not only fun, entire parts of mathematics have been built because mathematicians tried to tame infinity and dealt with these paradoxes. Some of this parts of mathematics are used by engineers.
From the top of my head I can think of the theory of distributions (Dirac and whatnot), Fourier analysis,... and so on.
If you're dealing with integral & differential calculus (and as an engineer I presume you do it all the time) then yes - you're dealing with these things all the time. You've just forgotten the foundations of what you do every day ;-)
Lol I was exaggerating for comedic effect.
Daric Soldar Whats even funnier is that almost all current sciences are direct derivatives of philosophy, so arent you one yourself ;) ?
Whoops! Yep. A PhD stands for "Doctor of Philosophy". So, technically, any scientist or engineer with a PhD is a philosopher.
The first one isn't strictly true, the hotel has an infinite number of rooms so can never be full to start off with.
But it also has an infinite number of guests already checked in. The problem here is that ALL countable infinite sets have the same cardinality (number of objects) as each other, so you could theoretical map them one-to-one, even though intuitively some countably infinite sets SEEM larger. So for example, the set of all even numbers and the set of all integers have the same size, both being countably infinite, even though not all integers are even.
I did put that last paradox to the test playing gta San Andreas, I went to the casino and played roullette with Carl Johnson. I started betting small sums of money to colour red, if I loose, I did double the bet, and so forth. Won a lot of money though
It's known at the Martingale gambler's fallacy. It only workd (as noted in this video) if the gambler and casino have unlimited wealth and place no cap on the betting.
OneEyedJack01 yeah but he’s playing gta
+Brendan McCabe Yea, that's precisely the context he's explaining in the comment, lol.
For the last example you should have done the one where a frog wants to cross a lake, but every jump it does is half the distance of it's last jump, so it'll never cross the lake even though it's allways moving forward.
6:10
Ok, WTF is this game? It makes no sense from the casino’s standpoint, not only because they don’t have infinite money, but mainly beacause from what was said in the video there is no way that the casino makes money of of this.
There are two ways to interpreting the terms of the game, each with the same problem.
1. You pay some x amount of money, then the casino does the dobbling if tales and you get the money if it is heads. So basically, the only way the casino makes money is by the initial amount of money you put in to play. This makes no sense, as from what you said you can put any amount of money, so if you put basically nothing to play you will end up the same as if you put everything.
2. You put some x amount of money which is dobbled if heads and you get the money if it is tails. Soo basically there is no way to loose any money, so you should put literally everything cause there is 0% chance of loosing any.
Both of this hav the same problem, the casino will make no money of this and there is no way to loose in this.
The money that you are paying to play the game does not go into the pot. Its just a fee to play the game. So regardless of what you pay to play the game the pot starts at 1 and doubles every time a head is flipped.
beau Haegele So the game makes no sense from the casino’s point of view
Sorry but your explanation of Gabriel's trumpet was so bad.....
+avril14thlove you cant even solve a 1x1
+Ian Thomas
It's not about physics. You brought up that anything can be divided in half, so think about this. What happens if a series continually divides in half? It approaches a number. Not infinity, but a real number. That's what Gabriel's trumpet is doing, so the volume is not, in fact, infinite, but the surface area is infinite.
+avril14thlove Elaborate
+avril14thlove I think it was perfectly fine, everything he said made a great deal of sense to me. He explained it better than most people can. Ever stop to think maybe you just haven't grasped the concept?
+avril14thlove Look up surfaces of revolution. This is just taking the graph of y = 1/x for x = 1 and beyond and rotating it around the horizontal (x) axis to create a two-dimensional figure.
This guys fringe tends to infinity
Cantor's fringe...
Is the last one really a paradox? I think most people wouldn't pay more than a few pounds (even if they expected to win a lot) because they know the chance of getting to a point where your winnings outweighs your buy-in gets exponentially smaller with each iteration in the sequence...
Any volume of paint will cover an infinite surface if you can spread the paint infinitely thin.
That's just the way things work when moving between dimensions in mathematically perfect geometry.
A line segment in math has a length that's a positive number and a width of zero. You can cram an infinite number of parallel line segments into a square. Likewise that square is 0 mm thick so you can stack and infinite number of them in a cube and it will never get any more full.
Since surface area is measured in square cm and a drop of paint would be measured in cubic cm you wouldn't even need to fill the trumpet. Just use a single drop of paint and make the layer of paint (1/infinity) mm thick.
So with the hotel, if there's room a room to move to, why does everybody have to move? Why can't the new guest just walk and go to the room that the person in the room next to it would have moved to.
The guest doesn't have to move to room 1, although there would be a wave of 2 people in 1 room that advances along forever, everyone is only sharing their room for a moment. Either that or everybody moves exactly in sync.
Wow! I remember I formed the dartboard problem by myself couple of years ago :-) I used another example but the idea was the same. That was a math lesson about cartesian coordinate system. I was wandering what is the probability of targetting a chosen point on the table by piece of chalk. What's surprised me - I solved the problem exactely in the same way like presented here :-)
Infinity is a fun mathematical concept. I wonder if it exists in reality, same with imaginary numbers.
I really don't understand the fascination with the Hilbert's Hotel Paradox. Excluding people standing in the corridors whilst moving from room to room and room sharing etc i don't believe you could fit any more people in. There would always be a 'next' room but it would always be occupied. I don't think the scenario should be looked at as 'x' (in this case infinite) amount of people and 'x' amount of rooms but instead as having 'x' amount of occupied rooms. So the room automatically comes with a person in it regardless of how many rooms there are. Just my logic I'm no expert.