In my topology course in college the Hairy Ball theorem was summarized as "Somewhere the wind isn't blowing."
@flleaf6 ай бұрын
Makes sense to me
@newwaveinfantry83626 ай бұрын
Because the Earth's surface is a sphere and wind can be considered surface level. Genius.
@blableu45197 ай бұрын
3:25 You forgot to point out the most important part of the Gabriel's Horn paradox. If you can fill the inside of the horn with a limited amount of paint, you would also manage to paint the interior of the horn, with an infinite surface area (since it's equal to the exterior surface area). Thus, you are at the same time affirming that the horn CAN and CANNOT be painted by a limited amount of paint.
@alexzhukovsky83617 ай бұрын
Same thing that the guy on numberphile forgot
@wicowan7 ай бұрын
nope, actually not true, because when we say it's impossible to paint the exterior, we are assuming we paint it with a fixed thickness, and then you need infinite paint (bc the surface is infinite). If you want to paint the interior, you can't choose a fixed thickness because there exists somewhere very far along the "tube" (or is it a trompet?) where its radius is thinner than then the fixed thickness you chose (it's a bit like epsilon delta analysis). And we have to assume that you need a thickness to even define the fact of painting something, otherwise any 3d drop of paint could paint any area. It's confusing I know. So no sadly, you can't paint the interior.
@coc2357 ай бұрын
The thing is, it CAN be painted by limited amount of paint, but it requires the layer of paint to get thinner and thennire the further away you go. That's exactlg what is happening in the inside - since the radius is decreasing, the "layer" of paint gets thinner..
@fsponj7 ай бұрын
No. If we assume that the horn's pointy side is pointing down & that there's no ground (somehow there'd be gravity though), it would take an infinite amount of time for all the paint that you put in it to fall
@wicowan7 ай бұрын
@@coc235 the thing is, with this definition you can basically paint anything with any amount of paint, which is absurd. For example, choose any surface, choose any quantity of paint, then there exists a function that decreases fast as fuck which enables you to paint the surface with the thickness according to this function.... Because then the amount of paint is pretty much the integral of the function you chose. Like for example, imagine you want to paint the whole plan (R^2), with let's say simply 1 unit of paint, then choose the function (1/(2*pi)))*e^-(x^2+y^2) as an indicator of the thickness and there you have it, (bc the integral over R^2 is 1) which is really fucking absurd. Hence why, in my opinion to define the act of paiting something, it has to be with fixed thickness, hence why you can't paint the interior of the trumpet. Now maybe you still want to define painting in the way you mentioned, but then there is no paradox because you can easily paint the exterior of the trumpet as well with a finite amount of paint, if the thickness decreases...
@shir_azazil7 ай бұрын
The trend to say 'every' is weird, as everyone know it's not everything
@ThoughtThrill3657 ай бұрын
Yeah 😂
@Ethan133717 ай бұрын
Now that’s the paradox of these kinds of videos
@masonboone43077 ай бұрын
Does he know about hyperbolies?
@Demongordon7 ай бұрын
is russel paradox 2.0, set of every video that contain the word "every" but doesn't contain everything
@mrosskne7 ай бұрын
@@ThoughtThrill365why did you claim the hairy ball theorem is a paradox?
@Redfox09287 ай бұрын
first one isn't even a paradox and was never thought to be, wtf
@undeniablySomeGuy7 ай бұрын
The definition of paradox is strange because it includes counterintuitive facts as well as unanswerable questions, like the birthday paradox
@fortidogi86207 ай бұрын
Like the birthday paradox, I guess it can be considered 'something that sounds like it should be wrong' by some people.
@newwaveinfantry83627 ай бұрын
The hairy ball theorem is not counterintuitive in the slightest. It's exactly what you'd expect, just a lot more difficult to prove mathematically.
@konuralpyldzkan14957 ай бұрын
@@undeniablySomeGuybirthday paradox shouldn't be counted as a paradox in the first place.
@mrosskne7 ай бұрын
it's engagement bait
@jimmea63177 ай бұрын
was waiting for a manscaped sponsorship
@ThoughtThrill3657 ай бұрын
😂😂
@fortidogi86207 ай бұрын
They could comb that ball!
@yocats99746 ай бұрын
"This ball is very hairy, but there is no reason why _your_ balls should be hairy as well"
@shivanshukantprasad7 ай бұрын
Regarding the Hilbert hotel, it cannot take in any number of guests, it can only take countably infinite number of guests. If you have uncountably infinite or more guests, you can't fit them in the Hilbert hotel.
@t0xic_g4s7 ай бұрын
This is an edited version of my previous reply. As many other comments have pointed out, it is true that you can either have a countable or an uncountable number of guests. Hilbert's Hotel however refers to countable number of guests. It provides intuition on how you can shift the natural numbers to create a bijection with other countable sets. In my previous response, I falsely claimed that all sets containing people (guests) are guaranteed to be countable. I thought since people are born sequentially in time, any set containing people would have to be countable. But since there can always be a magician conjuring up an uncountably large amount of people, that is not the case.
@ethos88636 ай бұрын
the thing is that you can fit as many guests as you want, you just can't check them in
@AkiraTheCatgirl06 ай бұрын
@t0xic_g4s What if the guests have every height between 5' and 6' exclusive? No one can be the shortest. This can even be the case with countably many guests. Define guest n to have height 5 feet + (1/n) inches. Then, once again, there is no smallest guest. Even if you could find a way to have any set of people have a "smallest" person, this still says nothing about the cardinality since any set--and thus this set--has a well-ordering. This is assuming you're taking your people from a set and not just a class and, of course, assumes AOC.
@josecarlosmoreno97316 ай бұрын
What's strange about the hotel is that it gets around actually having the guests in the room by making them change rooms. As in if there are an infinite number of rooms all filled, then everyone moves over 1 room to make room for a new guest, all the guests are NOT now in a room but instead there will always be 1 person in transit from their old room to the new one meaning there is always 1 person temporarily without a room and who that is is just being passed on infinitely rather than assigning that roomless state to 1 person permanently.
@jem56366 ай бұрын
Shhhh, we're not ready for uncountable infinities. (I was really confused about the difference for a while, but it made a lot more sense to me once I realized I normally view all infinities as unaccountably infinite... And I still struggle to not see countable infinities as uncountable.)
@vincentb54317 ай бұрын
Lots of people don't seem to understand that paradoxes aren't meant to suggest or prove anything, but they show that we can reach a seemingly irrational solution from rational reasoning, and that there therefore must exist a gap in our understanding. Obviously the cheetah will outrun the tortoise, but using what the ancient Greeks knew at the time, we can reach the seemingly irrational solution that the cheetah will never outrun the tortoise, which showed that we had a gap in our reasoning and knowledge. It wasn't until calculus was invented and we got a better understanding of the infinite that we could bridge that gap in our reasoning.
@MultiRobotnik7 ай бұрын
Counterintuitive ≠ paradox.
@alnea6 ай бұрын
Paradox: a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true.
@zanti41326 ай бұрын
@@alneaIf a statement believed to be absurd turns out to be true, then the problem isn't with the statement, it's with the analysis that led to the absurd conclusion. The birthday problem, for example, isn't a case of numbers acting weirdly, it's a demonstration of how poorly we understand numbers.
@alexmason55216 ай бұрын
@@zanti4132no one said the problem is with the statement Einstein.
@davidm20316 ай бұрын
@@alexmason5521Well someone hates their life
@marissonsoneur87006 ай бұрын
depends of the definition. Fortunately, jan Misali classified all types of paradox, and "counterintuitive but perfectly logical and explainable fact" is one type
@mallninja98057 ай бұрын
Zenos paradoxes never seemed particularly paradox-y. At some point one cheetah-sized step exceeds the total distance the snail was able to travel. It sounds like the sort of "profound" stuff stoners come up after a night of smoking.
@newwaveinfantry83627 ай бұрын
Well, it considers the movement as a constant, continuous function, and not a discrete set of steps. Even then it's not a contradiction, as both the distance traveled relative to time, as well the time needed to travel a certain distance, in relation between the two, can be broken down into an infinite geometric series. Since the series converges, the cheetah therefore passes the snail.
@ThePondermatic6 ай бұрын
My way of coming to terms with it is that Zeno's paradox was solved, so to speak, when we discovered calculus.
@ThomasMeeson6 ай бұрын
It proves that time and space is continuous as theres an infinite amount of points between the cheetah and snail before the cheetah catches up but its a massive logical oversight by zeno to then conclude that this means that the cheetah never catches up. I guess he hadn't discovered limits yet
@yaboiferret86816 ай бұрын
Maybe I’m just simple, but couldn’t this be resolved with just addition? Snail moves 1 m/s and starts at 9. Cheetah moves 10/s and starts at zero. At 1 second both are at 10 meters. At 2 seconds the snail is at 11, and the cheetah would be at 20. The cheetah passes at 1.01 seconds ( that last part is more of guess than actual math but you get the point )
@giddycadet6 ай бұрын
you can solve the whole thing by realizing that the logarithmic scale you've been using is creating a limiting function that has no reason to be there. just switch to a linear graph - stop zooming in on the infinitely tiny steps and see what happens when you add one whole extra second (thus completely bypassing the function's limit).
@__________g58946 ай бұрын
The elevator paradox makes way more sense when discussing floors NEAR the top or bottom, not on the actual top and bottom floors
@ralphinoful7 ай бұрын
Gabriel's horn stops being a paradox, once you consider how much surface area one drop of paint can cover. In theory, any 3-dimensional drop of paint, can cover an infinite amount of surface area.
@Aufenthalt7 ай бұрын
I would say that the solution of the paradox is the time you need to Paint the walls...
@mehdimabed41257 ай бұрын
The thing I still don't understand with this paradox appears more clearly if you make the object transparent : once filled in, you should see its surface covered with paint... A finite amount of paint...
@michielhorikx98637 ай бұрын
But that is not a problem. The key here is that the thickness of the layer of paint will decrease more and more as you go further along the horn. The only reason you would need an infinite amount of paint to paint the infinite surface area is that you assume some constant thickness of paint. If the paint layer gets thinner as you go further along the horn, there is no paradox, and that is exactly what happens when you fill the thing up with paint. This is similar to the dichotomy paradox - a sum of an infinite number of things can still be finite, if the things become small enough quickly enough.
@igorjosue89577 ай бұрын
So basically, it takes an infinite amount of 2D paint to cover it, but finite 3D paint?
@erinzaharris21627 ай бұрын
It also sort of is pedantic to say you could fill it. like sure there will be a point at which the hole becomes too small for a particle of matter to go through allowing you to fill it. That literal point is measurable though and any horn afterwards is just redundant horn to the idea. why even say it can be filled? Its like saying a wine glass with an infinitely long stem can be filled. yeah? cool?
@thomasrad52027 ай бұрын
the conclusion that was reached about the st. petersburg paradox is nonsense. a rational person should never play this game for a large sum of money. Yes the expected value over an infinite number of games is infinite, however the more you bet the more games you need to play in order to have even marginally good odds of breaking even. This is like saying you have a 1 in a trillion chance to win 2 trillion dollar lottery and the cost of playing is 1 dollar. technically if you had a trillion dollars you are guaranteed to double your money because you can buy every lotto ticket, but no one has enough money, so you are almost guaranteed to lose money. This has nothing to do with people being flawed in their perception of money, or the way they value it. No matter what the payout is, even if it is a near infinite sum, the odds dictate that you will in fact lose, every time. There is a certain threshold where an event is so unlikely that it is never expected to happen even in the entire universe's expected life span. Don't let the math fool you
@littlefishbigmountain6 ай бұрын
EXACTLY!! I was looking for a comment on this one. This is so absurd, I thought I must be misunderstanding something. Why in the world should someone bet $500 to play when they need to flip 9 times just to make $12 profit? It’s unspeakably ridiculous. And then they go on about “poor people have less money” and “a rational person should pay any amount for a ticket in this game” like LOL just shows how out of touch this bs is from basic sense, it’s unbelievable. It’s so bad, in fact, that I still think that we MUST have it wrong somehow because this cannot be the “paradox”. Surely it’s too stupid, even if at the very least because the people who made the game didn’t realize how poorly they wrote the rules and everyone who answered was thinking what we were thinking and they couldn’t comprehend that.
@matthewb23656 ай бұрын
@@littlefishbigmountain What is true is that the expected payout is unbounded (colloquially can be thought of as "infinity dollars on average"). However, a rational person would only pay this much if they had no risk aversion. Would you rather have a billion dollars, or a 1% chance of a 100 billion dollars? A risk-neutral person would see those as equally good options, but a risk averse person would greatly prefer to have a billion dollars for sure. A billion dollars would be life-changing, and another 99 billion wouldn't make that much difference in the scheme of things; certainly not enough to be willing to sacrifice the original billion in 99% of the outcomes. If your utility function is linear in wealth, sure you'd be willing to pay any finite amount to play. If your utility function is sqrt(wealth), you'd pay about $3.50...
@Storiaron6 ай бұрын
It doesnt help that the dude who made this video messed up his explanation and said heads double your payout, ad infinitum, ans then a second later that heads means the game is over
@littlefishbigmountain6 ай бұрын
@@matthewb2365 If you had $300 billion dollars, would you pay $100b for one round?
@spirou20126 ай бұрын
@@littlefishbigmountain This is why it is called a paradox. It sounds absurd, but somehow the math works out. It's not because the math is right, but because there is a gap between the theory and the reality. If I had a theoretical infinite amount of money, then I would surely apply this strategy and I would be sure (I have a pobability of 1) to win money eventually. Mathematics are coherent (I hope so), so paradoxes don't really exist if you dig deep into them to find the flaw. But I think there is some beauty in just accepting paradoxes as they are.
@martimlopes88337 ай бұрын
Another cool one is Skolem's paradox: there's a countable model of set theory. This is weird because inside this countable model, which only has as many elements as natural numbers, sets with strictly more elements than the number of natural numbers can be defined.
@newwaveinfantry83627 ай бұрын
Yes. Lowenheim-Skolem is probably my absolute favourite theore.
@rarebeeph17837 ай бұрын
Re: the hairy ball; the fact that you can't comb flat an ordinary sphere, a 4-sphere, a 6-sphere, etc., is less interesting to me than that you *can* comb flat the circle, 3-sphere, 5-sphere, etc. The Hopf fibration describes one way to do so for the 3-sphere (the surface of the 4-D ball), and I'm still getting used to the way it does so.
@newwaveinfantry83627 ай бұрын
The circle is very easy to imagine.
@jazzabighits44736 ай бұрын
@@newwaveinfantry8362 How? Wouldn't there be a tuft in the middle?
@newwaveinfantry83626 ай бұрын
@@jazzabighits4473 What? A circle doesn't have a middle. Are you talking about a disk? That can be coumbed, too. Let F(x,y)=(-2,0) be a constant function on R^2, a vector field. Then clearly, no point in the unit disk is mapped to itself. Everything is moving uniformly to the left.
@anonl58777 ай бұрын
You can also think of the elevator one from a majority-rules perspective. If you are closer to the bottom floor, there is a high probability that the last person to have called it was on a floor above you, so it has to go down to pick you up. If you are closer to the top floor, there is a high probability that the last person to have called it was below you, so it has to go up to you.
@coolnoah81836 ай бұрын
This doesnt make sense though because what if the elevator was near your floor or on your floor and someone below has called it
@codycast6 ай бұрын
1:30 that’s the dumbest thing I’ve ever heard.
@3millionyengirl6 ай бұрын
ok, cody.
@areebsheikh63606 ай бұрын
If it's dumb, why is it still a debated topic in philosophy and physics?
@TejasShastri-lh2mq6 ай бұрын
@@areebsheikh6360it's not. It's just interesting to say with your friends on a table. It ain't no "actively discussed scientific problem".
@Banana-anim8ions4 ай бұрын
Yeah I know
@tusharkaushalrajput3 ай бұрын
Only 3 million.@@3millionyengirl
@MeepChangeling2 ай бұрын
Actually the Hilbert Hotel is impossible. You see, there is a precisely 0% chance that every single person is going to be willing to move rooms in the middle of their stay just because more people arrived.
@maxkalentsov80857 ай бұрын
I understand that the defenition of paradox is unclear, but almost all of facts mentioned are just somewhat counterintuitive if you hear them for the first time in your life. And in my opinion there is a big difference between "this fact can not be explained" and "I think this fact can not be explained", so it's not justified to call any not-obvious thing "a paradox". I recently saw a video from Jan Misali on types of paradoxes and I think it is a great piece of discussion on that "what is a paradox" thing, would recommend.
@McWirst7 ай бұрын
The dichotomy paradox isnt really a paradox since it boils down to "The cheetah can never catch the snail if the cheetah cant go in front of the snail"
@bycmozeszymon7 ай бұрын
It also avoids the elephant in the room that time between each "catch-up" is getting increasingly smaller and smaller, and paradox resolves when you stop assuming time slows down somehow.
@konuralpyldzkan14957 ай бұрын
@@bycmozeszymonor if you stop assuming that time can be divided infinitely
@machalot7 ай бұрын
@@bycmozeszymon The key insight of calculus that resolves it is that an infinite number of things (time steps) can still add up to a finite sum.
@Diego-kk5uw7 ай бұрын
the thing with Gabriel Horn and paint is that what infinite area means is that you cannot paint it with an UNIFORMLY THICK coat of paint using a finite amount of paint (because this will imply a usage of area*thickness volume of paint). So there is no paradox, the thing is that if you consider some of paint inside when the filled horn as a coat of paint for the inside, this coat will have a decreasing thickness (or no thickness at all, which means using 0 liters of paint).
@anonl58777 ай бұрын
The Hilbert hotel would have to deny entry to Akira. An uncountably infinite blob of person would not be able to fit inside.
@AkiraTheCatgirl06 ай бұрын
Wow, ok, I see how it is >:|
@lkjkhfggd5 ай бұрын
I never understood the hotel paradox. Saying the hotel is fully booked doesn't make sense since it has infinite rooms. Shuffling people around is just evidence that they indeed weren't fully booked.
@prototypeinheritance5154 ай бұрын
Shuffling around is a completely natural thing to do, even in a finite hotel you could move everyone to the next room and the person in the last room moves to room 1
@singularity37246 ай бұрын
Except for Russell's paradox, none of the others are paradoxes, you just don't know the required maths. They just aren't intuitive.
@quentind19247 ай бұрын
6:37 There is another reason to not bet too much : if the other person has a finite amount of money (which will most likely be the case), the expected result will be finite and not very big. If you want an expected result of at least 20$ for example, the other person has to have at least 2²⁰$, which is approximately 1,000,000$
@qracy-kun52886 ай бұрын
Found this channel today, its visual and explanation is simple and brief which is good for me.Thanks for the video keep going.And also comment section is fascinating how people are adding their knowledge about the things in the video which is interesting for me
@theimmux30347 ай бұрын
only Russel's paradox was an actual paradox and even that was fixed by setting new axioms 😭
@LevinFroggo-fs7uu6 ай бұрын
There were other paradoxes like the gabriels Horn paradox or the birthday paradox. Paradox does not mean that there is no solution, just that it is counterintuitive
@LeNoLi.6 ай бұрын
Paradox doesn't mean unsolved
@UnCavi6 ай бұрын
Paradox means counrerintuitive, not a logical contradiction
@not_porter7 ай бұрын
1:46 the fact that you chose my birthday startled me for a second
@joelsummerfield43747 ай бұрын
Same here 😂
@kmyc897 ай бұрын
(8:35) Sorry, but Hilbert's Hotel can in at least 1 case not welcome all guests: " _How An Infinite Hotel Ran Out Of Room_ " ~Veritasium
@MrKillerMichael6 ай бұрын
You've ressurected my now undead desire to explain to people (now unfortunately you) that video. More aptly, that proof, is wrong. (Understand I'm not heated at you, rather I'm passionate it doesn't make sense) The proof proven absurd as follows: Pair each real positive integer with itself exactly, 0 inclusive; so 0 with 0, 1 with 1, 2 with 2, 3 with 3, and so on. Incriment each digit of the second, identical set of positive integers (0 inclusive), by 1, {in the same manner as the Veritasium video} (the wrap around if 9 rule exists but isn't used). The result is a number that is "dIfFeReNt FrOm EvErY nUmBeR pRiOr." Therefore, the set of all positive (zero inclusive) integers is larger than itself. edit: obsurd->absurd, and {text}
@asheep77976 ай бұрын
@@MrKillerMichael...that only works if you're talking about p-adic. we're not.
@kiwi_2_official6 ай бұрын
@@MrKillerMichael absurd*
@kiwi_2_official6 ай бұрын
there are infinite cases
@MrKillerMichael6 ай бұрын
@@asheep7797 Well, I wasn't talking about p-adic so if the reasoning is wrong I would like to know why.
@dwarky7 ай бұрын
Which horror movie was it taken from? 0:19
@ThoughtThrill3657 ай бұрын
😂 😂
@TheKivifreak7 ай бұрын
Your upload schedule is pretty insane. Nice
@ThoughtThrill3657 ай бұрын
😄
@abhigshek7 ай бұрын
@@ThoughtThrill365 pls keep it up with such intellectual stuff, educate urself as well
@Lord_Volkner6 ай бұрын
The Gabriel's Horn Paradox is based on mathematical sophistry. When the math is done correctly, the paradox disappears.
@joshuagraham38542 ай бұрын
Came for the hairy ball jokes, stayed for the fascinating educational content. You were even confident enough in your content to put the hairy ball one first!
@al_semenovАй бұрын
Russel's paradox in simple words of our teacher was: "let's say there's a city with law that every man should be bald, but the only person that is allowed to do haircuts is the barber. then who would shave the barber?"
@Who8mydamnoreos4 ай бұрын
2:06 they asked this question in class and not only did the person have the same birthday as me but the same name.
@nerdcorner26805 ай бұрын
Dichotomy Paradox is no longer a parodox thanks to the planck length. There is a smallest unit of distance that cannot be divided by 2. This means that the entire paradox no longer has any real meaning as the more intuitive answer of “the cheetah moved forard and caught the snail” is true mathmatically
@smoldragon3395 ай бұрын
You knew exactly what you were doing, starting with "The Hairy Ball Theorem" right outta the gate lol
@stefandemerov84236 ай бұрын
Dichotomy Paradox is easy to solve... if for time X the snail moves less distance than its own length, that means the back end of the sail is still within the space, that was occupied by its front during the previous period. In such case the cheetah will catch it guaranteed during the next period of X.
@wanderer3143 ай бұрын
I learned about Zeno's paradox in a philosophy class in college and thought it was so stupid that I never took another philosophy class in my life
@kikook2226 ай бұрын
1:37 That paradox only works if time doesn't exist. Speed is distance over time and this paradox is distance without time.
@Schnorzel13375 ай бұрын
What? When the snail moves, the cheetah has to take N seconds to reach the point where the snail was. The snail is moving aswell so there is a new smaller distance. The cheetah has to take N seconds to reach the point where the snail was. There is time. The "solution" is that a infinite sum can reach a finite number.
@senorpepper34055 ай бұрын
@Schnorzel1337 if the snail starts 9 meters ahead and travels 1m/s the 10 m/s cheeta will catch it. I agree with the op, there's some weird thing going on here that's over my head. Even if the snail has a fraction of a second head start, for some reason.
@matthewb23656 ай бұрын
With the St. Petersburg paradox, the video keeps mixing up heads and tails... :/
@thebradler516 ай бұрын
Is the snail’s velocity constant at 1 m/s? If so, then obviously the cheetah will catch it, how is this a paradox?
@randomxnp5 ай бұрын
The hairy ball theorem shows that somewhere in the world at any moment in time there must always be at least one spot with zero (horizontal) wind.
@AOOA9262 ай бұрын
1:41 they’re not going the same speed tho
@Kalo_agori2 күн бұрын
You have what's called a common sense. So you realised this. The person who thought of this as a paradox... Was dumb.
@franz009franz6 ай бұрын
When the KZbinr says its a math Problem but its actually just middle school math
@Freytana6 ай бұрын
Great video, I like the way you communicate these ideas! I have always had an issue with Zeno's Paradox (The Dichotomy Paradox) because of how it is framed. The discrete units at each step get smaller with each iteration so it makes complete sense that any finite action would trend towards infinity. We experience time linearly but the characters in the paradox are having their units of time reduced an order of magnitude each step. So 1 second then (approx.) 0.1 then 0.01 and so on, so Zeno's paradox is really just saying: "The number 1.11111 repeating is infinite." Or "The point at which the fast runner overtakes the slow runner is when T is larger than 1.1111 repeating." It is just doing so in a round about way that can come across as disingenuous or counter-intuitive.
@edminchau8117 ай бұрын
A hairy ball might not work, but a torus would.
@Gumballcom4 ай бұрын
"the hairy ball theorem" is a crazy way to start a math video ☠️☠️☠️
@drxyd4 ай бұрын
With Gabriels horn a drop of paint is enough to paint the entire external surface because you can spread the paint out to an infinitesimal thickness.
@theokurpierz4 ай бұрын
As a hotel worker I dissagree with David Hilbert, probably the first guest who you want to move will not be willing to
@diegomandragora43275 ай бұрын
Well this fuzzy ball paradox explains a lot...😂
@MegaMeister1236 ай бұрын
1:35 Yeah it doesn't catch the snail, because the problem entirely is one from definition. If you dont have a dynamic time and instead look at it in full seconds, its obvious that the snail will never be caught, because its moving at each point.
@JakubWaniek7 ай бұрын
6:22 Correction: the layout of the game is never infinity, the payout is always finite (2^n for some n). The *expected value* of the payout is infinite. The point of this problem is to illustrate how expectation can flawed
@Sideshownicful5 ай бұрын
Birthday problem: during our first semester in physics, we realised that 3 of us shared the same birthday (in our group of 12 friends). During our second semester, when we took probability and statistics with math majors, they were stunned at learning it took 23 people to have a 50% chance, whereas we had the reaction of 'really? that many?'
@selsickr7 ай бұрын
Hilbert’s hotel is not really a paradox. We could just say that as all the rooms in the infinite hotel are taken we cannot just move everyone into the next room. We could probably make another branch of mathematics by assuming this. Hilbert just assumed an axiom ( ie we can move everyone into the next room ). This axiom should have been clearly stated as such.
@KD-jk6yo7 ай бұрын
i dont get how everyone couldnt move over. can you explain?
@prototypeinheritance5154 ай бұрын
we can move everyone to the next room, it's called the successor function also known as n+1. It's one of the must fundamental properties of natural numbers that each number has a successor.
@giddycadet6 ай бұрын
gotta be the slowest elevator i ever heard of
@roryb.bellows86176 ай бұрын
It seems a lot of people are misunderstanding Zeno’s paradox. It’s about infinity, it’s not a literal observation. If you are in a race, you have to get to the finish. To get to the finish you have to get to the half way point. To get the half way point you have to get to the quarter mark. This can go on infinitely. The point is about motion and infinity. It’s not about a cheetah and a snail. Your school system failed you.
@vanthomias55386 ай бұрын
I dont understand your St.Petersburg-Paradox's Game explanation. How does your amout of bidding influence the winning? And How can you loose?
@AnkurDhawan-zq4ll5 күн бұрын
1:34. This paradox only works when both the Cheetah and Snail are considered as point masses. In real life, this paradox fails because both Cheetah anf snail are not points but have some length. The cheetah just has to reach a point where the diastance between it and snail is leass than or equal to the snail's body length.
@bilbobaggins8906 ай бұрын
For the cheetah and snail, assuming the speeds listed in the video.. wouldnt you just calculate the snails and cheetahs movement simultaneously and then the answer is when both distances become equal? So if the cheetah moves 10 m/s and the snail moves 1 m/s and the snail starts 10m ahead of the cheetah then the cheetah would simply catch the snail in ~1.11 seconds or at the 11.1m mark. The reason it would repeat decimals infinitely is simply because we arent stopping the calculation upon contact but rather trying to "chase" a slower target with a faster target with no end to the calculation. If the calculation ends upon contact then the answer is just 11.1 meters because any distance less than .1 meters is negligible to the scenario, but if precision mattered then it would only ever matter up to the degree necessary and then every decimal beyond that is just theoretical and no longer practical. Meaning this math problem is already practically solved and only a brain teaser for math nerds.
@placek71257 ай бұрын
4:08 what i fill it with paint and immiadetly empty this shape? Wouldnt I paint it from the inside, despie it having infinite surface area? Surface area from the inside is the same as outside.
@Schnorzel13375 ай бұрын
No you would not paint the inside of a volume completely full of paint. Interesting isnt it.
@stefanbergung55146 ай бұрын
No, the reason you shouldn't pai to much to Saint Petersburg is that the mathematically expected value depends on the very rare extremely high returns. Even if the host only quits after 200 coin flips, your expected value is still less than 5.88$.
@horatiohuffnagel79786 ай бұрын
Pretty sure a cheetah catches the gazelle and devours it.
@fluffyfang42134 ай бұрын
Dichotomy paradox annoys me because it only works if you assume all infinite series are divergent. Or perhaps more generally and intuitively, assuming that every line is infinitely long because it has an infinite number of points.
@disgracedmilo7 ай бұрын
i can confirm the first theorem in about 15 minutes
@aidbeno64097 ай бұрын
I remember some of these from Vsauce2 wow how has it been years
@PerriPaprikash5 ай бұрын
The birthday problem is not a paradox. It is simply an unintuitive result, due to false preconceptions about how probabilities work.
@菁_冬蓝6 ай бұрын
wasn't the Dichotomy Paradox (Zeno's paradoxe says same thing)already solved by calculus? The infinite small interval
@aroundandround3 ай бұрын
If a student today said what Zeno said, we would just call him math stupid.
@alwaysxl6 ай бұрын
The second paradox has to do with the 'scope' of the scenario we're talking about. The example you provided is pretty good, however with calculus we're able to figure out exactly when they catch up. The example in the video assumes the cheetah stops as it's reaching the position the snail is at (because it believes it to be stationary when actually the snail is moving).
@cabbagebutterfly8007 ай бұрын
i never get why the gabriel's horn was even a paradox, the paradox fix itself within it's own definition, it can fill a finite amount of a liquid but can't be painted with a finite amount of paint. so what if i fill the horn with paint? yeah that's right, it would pain itself frm the inside, and since it's a infinitely thin horn, the area outside is equal to the inside.
@coleozaeta63446 ай бұрын
You’d be taking a surface area function versus a volumetric function. Since we’re going all the way to infinity, the volume function experiences a 3rd “1/inf” type division, whereas the surface area function only experiences 2. This is why the surface area is infinite and the volume is pi units cubed.
@jeffreyjdesir5 ай бұрын
song: Piano Sonata No. 11 K. 331 3rd Movement, “Rondo alla Turca”
@giuseppenonna21487 ай бұрын
Hi, nice video :) But at 1:34 that sculpture is not Zeno of Elea, the one you are probably referring to, but Zeno of Citium Just a small detail though, great video!
@jaggerbushOG5 ай бұрын
Hilbert Hotel to me always gets a "thats stupid - of course- its ∞" reaction from me.
@drdca82637 ай бұрын
Is the birthday paradox, even that paradoxical seeming? It seems more, “initially a little bit counterintuitive/surprising”. 6:18 : this assumes that one cares about expected value of money, rather than expected value of money. Also, winning “infinity money” is not one of the possible outcomes.. [edit: oh nvm you do mention utility afterwards]
@firozabegum43737 ай бұрын
"Birthday Attack"- never ever thought to hear about it.
@_Heb_7 ай бұрын
At 6:22, what's the significance of "anything times infinity is infinity"?
@bigbread98865 ай бұрын
if you bet X the payout is X times infinity so the value of X doesn’t matter
@Metalhed1300p5 ай бұрын
Yeeeah, I don't think that whole cheetah catching up to the snail thing was really as clever as that guy thought it was. Bro must've been featured on Iamverysmart
@rrbk60257 ай бұрын
If it's possible to fill inside wouldn't it be able to fill with paint hence paint on inside. if thickness approaches zero wouldn't inside and outside area be same..
@allaware19715 ай бұрын
The earth's magnetic field works through the first "paradox" which means it's not a paradox but a law of physics.
@alieser77705 ай бұрын
Markov chains are all you need for St. Petersburg
@unflexian6 ай бұрын
i love how the music cues you in when the video's ending!
@lastofthewieldersoflight7 ай бұрын
Dichotomy Paradox seems like a good argument for discrete space.
@guotyr25024 ай бұрын
I don't get the second ,it works only if the cheeta has the same speed as the snail or slows down every time it arrives to it's last location
@Khanate19234 ай бұрын
What is the first musics name
@senorpepper34055 ай бұрын
My hairy ball is so smooth with no tufts anywhere😂
@FajorMuckup5 ай бұрын
The elevator one seems super obvious? If you're up top, you have to wait for it to come up, obviously? Am I missing something?
@angryyordle46406 ай бұрын
The dichotomy problem is basically just an exponential function
@AaronDennis11116 ай бұрын
Number 1! If you follow the contours, it'll be smooth. It just isn't easy to do without a microscope. You could also cut the hair to an even length that did not allow for variation. Waiting for no 2
@Rustybucketgames226 ай бұрын
Most of these would only be a true paradox if you only look at them from a purely mathematical standpoint. Without applying any other form of thinking
@oxbmaths7 ай бұрын
Nice compilation of paradoxes. Would be clearer without the background music, in particular at higher speed playback.
@ThoughtThrill3657 ай бұрын
Noted
@JonathanBartlesSWBGaming5 ай бұрын
"at least" two of them share a birthday
@Patralgan6 ай бұрын
If the trumpet is infinitely long, how can you ever fill it if there's no bottom?
@bigbread98865 ай бұрын
calculus. As you go along the X axis, the volume increases less and less and it doesn’t ever go past a certain value
@turanbirligi69697 ай бұрын
Dichotomy Paradox isn't a paradox, its been solved.
@LevinFroggo-fs7uu6 ай бұрын
Doesn't mean it's not a paradox. Paradox simply means, that the result that you get when calculating it is different than what you would expect when just thinking about it
@coledavidson56307 ай бұрын
1:35 this is just straight-up overthinking
@EmosGambler6 ай бұрын
I have a feeling like some of the paradoxes needs a time variable or some sort of consideration of periods of time (cycles?) for which the paradoxes can exist. IDK, too late.
@milokiss82765 ай бұрын
these videos always seem to put me to sleep. im counting on it this time.
@pirate322421 сағат бұрын
Gilbert can run out of rooms
@trufflefur7 ай бұрын
The birthday paradox I can everyday of work check it is true. I work as a vigilant in a parking lot building and around 20~25 sleep there and when I'm counting and reading their plates it's like if the come in "families" with the same letters and same numbers like for example I could have a "KXV 1534" "KKV 1688" "HLV 1734" and another "PPL 1022" "TPL 2102" They always have a pattern sometimes I think I'm going crazy.
@BalthazarMaignan7 ай бұрын
I don't get how the hairy Ball theorem is a paradoxe ? Nice video tho
@barutjeh7 ай бұрын
In another part of topology there's the fixed point theorem would be a neater example. Holding a map inside the region it depicts, there must always be at least one point exactly above the corresponding point in the region. Or: if you have two pieces of paper and you scrunch one up, put it on top of the flat sheet, there's at least one point of the scrunched up sheet right above the same point on the flat sheet. Or, assuming a continuous liquid, if you shake a bottle of water, there'll always be at least one point that is not displaced.
@jffrysith43657 ай бұрын
it's not, like I've never even heard anyone call it a paradox before this video. It's just a funny named theorem.
@vincentb54317 ай бұрын
Back then, the hairy ball paradox showed us that there was a gap in our understanding of topology until Poincaré solved it.
@jaytravis24877 ай бұрын
@9:36 An interesting point about the definition of words and CIRCULAR definitions. But there comes a point where you MUST be able to understand some words or else you can never participate in ANY LANGUAGE GAME (~Wittgenstein's term). This might be a bit HALF-BAKED...but try it out for yourself! You can start with a high-level concept like "lordship" and define all the words contained in it's SIMPLEST DEFINITION. Defining all those words will eventually lead you from the top-floor or high concepts to the BASEMENT OF ENGLISH where the 'SUBSTANTIVE/AXIOMATIC' WORDS and concepts exist. Ive been toying around with this idea for years but the undertaking is so arduous I'd challenge anyone to look at it and not blink. It's akin analyzing the entire dictionary. But take a word like "ownership","space"(i.e.: volume), and try to define them into simpler terms. I think you'll understand what I mean by SUBSTANTIVE-Axio words. In order to participate in English speaking one must be able to understand these concepts...there is no other method to help explain what the S/A words are other than for the teacher to point at the red apple, the stop sign, the fire, and tell the student 'red'. (Of course there are other ways but I'm not getting into pedagogy here).