Infinity Paradoxes - Numberphile
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- เผยแพร่เมื่อ 23 มิ.ย. 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!
Extra interview footage from this video: • Infinity Paradoxes (ex...
Counting infinities: • Infinity is bigger tha...
Dividing by zero: • Problems with Zero - N...
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"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.
TH-cam 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 :)
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
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?
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?
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...
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
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....
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😂
5:42 "so its this kind of granular nature of our existence that gets us out of trouble".a really beautiful/funny sentence :)
It is paradox that push mathematics forward and deepen the understanding of human being.
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
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'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
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
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.
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.
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
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
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.
Regarding the dart board, why not say the chance is infinitely small, but not zero?
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.
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.
*"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.
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"
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.
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?
"Although it's always crowded
You still can find some room"
-Elvis, Hilbert's Hotel
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
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
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.
Hilberts hotel, you can check in any time you like, but you will never have to leave
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.
Did you hear that explosion? That was my mind.
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.
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.
Infinity is the opposite of zero, yet so similar in many ways
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.
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.
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..
5:05 bold of you to assume I can hit the dartboard every time
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 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).
is it just me or do i find his hair very distracting..
Infinitely distracting
@Joe Exotic toché 😂
2:38 That only works because paint is 3D. If you had purely 2D paint, that wouldn’t work.
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.
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
This is beautiful.
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.
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
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
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
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.
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.
I think it's ENTIRELY satisfying.
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
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.
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!
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.
Infinity is a state which defines the current position in a line of numbers, it could be a high number or a small number, it depends how old infinity is, so time is really important in the universe of infinity. Inifnity ends when it stops getting bigger, so when there is no time anymore.
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.
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. =)
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.
My favorite infinity paradox is the Banarch-Tarski paradox.
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 :-)
How do you exactly lose in the casino paradox game?
You don’t, but you have to pay to play
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
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.
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)
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.
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.
You cannot fit in even one more guest unless they have an infinite amount of time to wait for all the existing guest to move to the next room.
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.
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".
I'm a big fan! I think this video alloud me to ask you if you can do the Continuum Hypothesis of Cantor? pleeeeeeease :)
The dart one is only contradictory because it is assumed that you will hit the dart board, but this is not always the case. Because there are infinitely many places the dart could land (in the universe, assuming the universe is infinite), the probability tends to zero.
This guys fringe tends to infinity
Cantor's fringe...
Your accent is really strong. I like it :)
The dart board example is my favourite. Although, I'd explain it like, a person throws a (real-world so it has a finite diameter at the tip) dart at an integer number line and it ends up in between two points, what real numbers have they hit?
I have a problem with the casino game bc wouldn't you reach a barrier if the probability added each time was shrunk in half meaning it would add but at couch a shrinking rate, it couldnt improve
ehhhh i dont think the hilbert hotel paradox really works since there is either, at any given point, 2 guests in 1 room, or a 1 new guest in the hallway... the problem just gets diverted for the duration of infinity... really trippy lol
unless everybody comes out their rooms at the same time and moves along into the next room. If not then there's no rate of change to the solution of the problem
that's similar to his paint in a trumpet example. Mathematically all people could move in such a coordinated way that it works out - in our sense it would already be complicated with 100 people in the hotel.
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.
The hope and wishful thinking you have for tomorrow, that is infinity, it is one direction.
The dartboard problem has a solution. Measure the area of just the tip of the dart, and the surface area of the frontside of the dartboard. Divide the two and get a percentage. That is your chance of hitting the exact spot on the dartboard. Granted it will be VERY small, but it is a finite answer.
as he already stated, that's an area which has a finite solution. but an area isn't satisfying. the infinite problem is trying to find an exact point.
He said so himself in the video. Once you think about an area (as you would in a real world problem) the issue goes away. If you, however, talk about the exact point there is 0 chance. That's actually an important rule of propability: Any point in a continuous distribution has a propability of 0, and only intervals (or in the dart-board problem: areas) can have a propability greater than 0.
If you never run out of rooms is the hotel really full or empty?
First of all, it is impossible to fill the infinite amount of rooms. One can keep adding peoples in it will never reach the infinite amount. That is why it was assumed it was filled with an infinite number of peoples. :)
Also, it wont be possible to move customers from room to room as in the video, I think, because infinite is already the largest limit (unlimited) and is not extendable. However, the idea of an extendable infinite is something very interesting.
Fabrice Duche You are 100% wrong. First of all, if you have an infinite amount of people, and they all become guests, then you can pair exactly one guest to one room. So you CAN fill it up. And then, if EVERY guest moves up one room, then every room becomes available for the guest moving up. Saying otherwise is like saying infinity is not infinite.
That said, there are different infinities of different sizes. Georg Cantor proved this over a century ago.
There's nothing strange about the second problem, a drop of paint can easily cover an infinite amount of surface if the thickness of the layer is infinite.
Mixa That's exactly the paradox though. Infinite is endless, and even with an infinite amount of paint, it will never paint the whole surface of the inside of the trumpet. An infinite amount cannot meet another infinite amount because, well, it's infinite.
+Nathan Chung No, although the surface area of the trumpet is infinite, the volume of the trumpet is not. So, if you were to fill the trumpet by pouring mathematical paint into it, you could paint the inside surface, however if you were to paint the surface area only, then you could not. This contradiction (you can both paint the trumpet and not successfully paint the trumpet) is where the paradox arises.
@@jackhooper2839 yes you could paint the outside with 0 mathemathical paint: take a cube of side length equals to the maximum diameter of the trumpet made out of mathemathical paint, take a slice off the top and wrap it around the exterior of the trumpet. You didn't reducethe height of the cube, since it's mathemathical paint and the slice is infinitely thin. You can do that infinitely many times, the volume of paint on the outside of the trumpet is exactly 0 but the entire surface is painted, and the volume of your cube of paint is unchanged.
yeah, what's paradoxical is the paint
the paradox has nothing to do with paint actually. It is just a way to illustrate things. The paradox is that the volume of the horn is pi while its surface area is infinite. And it seems paradoxical that an object can have an infinite surface area and a finite volume. Yet the maths behind it is legit and it has to do with the convergence of the series of term 1/n^2 (for the volume) and the divergence of the series of term 1/n (for the surface area).
Mathematics is about relationships between objects and at the most discrete level these objects are relationships themselves. Paradoxes are two different truths about the same object being communicated to a particular perspective the result is a phenomena that contains both truths. This gives rise to manifestations to the perspective. In the case of the Hotel the manifestation is seen as a wave of people moving from room to room off into infinity when a new person is added. Paradox resolved.
In hilbert hotel when you add a person you could say it causes an infinity interference, when one has to move to a different room that other person is displaced to another room making an infinite rate of change of 1. Add another person it becomes a infinite rate of change of 2. If the people come in at infinite regular intervals you can see the rate of "rate of change". The opposite is the same, take out a person, a person moves to a lower room, this happens infinitely creating an infinite rate of change of -1. This looks exactly how electrons work in a circuit.
By the way every finite thing is actually infinite. example 0 infinitely added to 1 will always equal 1. this is an infinitely unchanging 1 until another infinitely unchanging number comes along.
The trouble is comprehending what infinity actually is.
NO ITS NOT
I disagree, I understand what infinity is, sometimes what you can do with it is counterintuitive
isn't planck length the answer to the dart paradox?
planck length doesn't mean anything in mathematics
Nah, just remember the fundamental truth. Points DO NOT exist in the real world, only in the abstract world of mathematics.
No that is not a point! If you put a magnifier on it you will see an irregular ink blotch that has an AREA! Points don't have an area. Put that in your pipe and smoke it! :D
Yeah see thats the point, there isn't anything smaller than a quark when discussing matter. You can't even do a quark pen! You have now left the real universe and are now in the strange land of mathematics where things like infinite hotels and points exist. Thats why they say there is an infinite amount of points on a dart board. Because points do not occupy space but can hold a definite position you can say something outlandish like that.... Or get away with crazy professor hair..
Hah! I said the same thing when I was an undergrad. Undergraduates think they know everything, graduates realize they only know a few things and doctoral students realize they actually don't know anything at all! :D
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...
These blow my mind
For the third paradox, the dart one, what if the chance of hitting that one point was 1/x, where x -> infinite? then, would it still be a paradox?
+Adam Mehdi divide 1 by infinity. 0? but then the probably of hitting anywhere on the dartboard is now 0+0+0+.... = 0.
Maltager if it's 1/x where x is infinite, then 1/x * x = 1, or 100%. Even if we use your method, 0 * infinite is undetermined, not 0.
For example, 2/x is zero. Multiply by x, you get two. therefore x * 0 is 2.
Or do 3/x * x, which is 3. or 1/5x * x, which is 1/5.
+Adam Mehdi I think you're right there. But my head aches
+Adam Mehdi
I think you've forgot to put in the limit, but you are indeed correct. The assumption that 0 + 0 + 0...+0 = 0 is wrong. Infinite times 0 is undetermined, depending on the rate of growth.
Nick374a, no. 0+0+...+0=0 (a finite sum) is correct, point blank. Also, 0+0+0+...=0 (an infinite sum) is correct. There are no rate of growths, or anything else in these equations other than 0's.
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.
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.
I think Infinity of any kind is a sort of reflection or shadow of a higher physical dimension.