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.
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.
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..
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
@@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...
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.
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.
@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.
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.
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.)
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.
@@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.
depends of the definition. Fortunately, jan Misali classified all types of paradox, and "counterintuitive but perfectly logical and explainable fact" is one type
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.
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.
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
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 )
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).
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.
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...
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.
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?
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.
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.
@@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.
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
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.
@@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...
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
@@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.
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.
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.
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).
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
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!
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"
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.
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
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.
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
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
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.
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$, which is approximately 1,000,000$
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.
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}
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.
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?'
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.
@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.
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$.
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.
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.
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.
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.
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.
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
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
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
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).
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.
I'd say Bernoulli was being very smart, but in the end was overengineering his explanation of the paradox. The problem, when not listened to or thought through properly, presents itself as a gamble, and most normal people are not willing to stake much on a game.
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
No, it's weirder than that. Of course there is a bijection between a single ball and a pair of disjoint balls, since they are sets of the same cardinality. That isn't surprizing. Banach-Tarski says tha you can actually split the ball into a finite set of disjoint congruent subsets, whose union simultaneously gives you the ball as well as two identical copies, without chainging the elements of those sets.
@@newwaveinfantry8362 That you can separate the ball in to a finite (I think as few as five?) pieces, and then reform them only using rigid motions in R^3, yielding two distinct balls. I remember seeing a very nice you tube video detailing the process - but that was a couple of years ago and I cannot remember the channel...
At this point these mathematical paradoxes are simply semantic trickery and limitations of the human brains. Imagine x, now imagine not x. What if I defined them as the same. Oh no, paradox.
I don't get any of the comments saying that some of these aren't real paradoxes. I think one thing that very much justifies calling these paradoxes, which ThoughtThrill touches on for each paradox, is that these ideas were important to our understanding of mathematics. Some of them are "just" counterintuitive, sure. And some of them make perfect sense to us now that we're comfortably in the 21st century. But all of these spurred on significant advancements, refinements, or deeper explorations in our understanding of math, precisely because they were unexpected or genuinely confusing when mathematicians first encountered them.
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.
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!
Gabriel's Horn can't exist, though. Even assuming it sprang into being, there is a minimum diameter the horn could have, given it's made of baryons and they have a measurable size. Therefore the concept of something that keeps shrinking forever as you go along is incoherent.
Some of these math paradoxes are kinda stupid because they make some assumptions that they don't tell you about. For example, the first one assumes that time slows down as the cheetah gets closer to the snail, and time would stand still if the cheetah caught up to the snail, which is why you would say the cheetah never approaches the sail. It's the exact same reason why an object with mass can't reach the speed of light. It's not a paradox.
On the St. Petersburg paradox: The way mathematicians calculate probabilities for investments is wrong. You should not count the absolute gain, but the relative gain. This makes a big difference. The justification for this is that to make up for a 50% loss, you need to earn +100%. So, geometric means are better for calculating investment odds rather than arithmetic means. I often calculate probabilities in speculative investments, and my default is always to guess what the highest possible gain is vs the highest possible loss, and calculate a geometric mean. So for instance, if I believe an asset can do a 0.5x, or a 10x, and is equally likely to do anything in between, then I figure my expected gain is sqrt(0.5*10) = sqrt(5) = 2.23, NOT (0.5+10)/2 = 5.5. If there is a possibility that an asset can go to 0, then no plausible gain can justify going all-in. When dealing with assets that can go to 0, you have to consider them as a part of your portfolio in order to make the calculation (like maybe some % gold, which you assume can't go to 0, and some % of some alt coin which could go to 0 but could go to infinity). In the case of the St. Petersburg paradox, if people are paying $10 and are only allowed to play once, they have an 87.5% of losing money. This means it is realistic and practical that people aren't willing to spend a lot of money on it. The theoretical arithmetic mean is infinity (1/2*2+1/4*4+1/8*8...= 1+1+1...), but if someone spent his whole net worth on the game, there's an almost guaranteed chance that he'd end up broke. It is thus practical that people are not willing to spend a lot of money on it. Real people only have one life, so it makes sense that they play to win the median outcome rather than the average outcome that would happen if they had infinite lives to play this game. If you want to generalize geometric means without even probability distribution (like 75% chance that something happens), then the result is that effect1^(chance of effect1)*effect2^(chance of effect2) and so on, with however many possible effects there are. Edit: After some googling and messing around, I was able to solve for 2^(1/2)*4^(1/4)*8^(1/8)... I believe the answer is equal to 4. So, I believe this game is worth $4. Edit 2: After messing around with a random number generator and a large number of trials, it appears to me that the game might actually be worth something like $7.5. I wonder if I made a mistake in the above calculation. Maybe $4 is like the median amount you'll earn, and $7.5 is the mean. I will investigate this more later because it is an interesting puzzle.
The under a geometric averaging (is there some kind of measure/function for probability distributions that gives such a thing? I suppose there must) is lim_(n->inf) (prod(1 to n) 2^(i-1))^1/n. Taking the product inside the exponent, with triangukar number formula we get the inside of the limit is equal to 2^((n-1)/2), which still goes to infinity. In fact, it goes to infinity faster, which makes me think I've made a mistake somewhere
St Petersburg seems stupid. If you get just one head in the first four tosses, the most you can make is $16, and your chances of getting $100 or more are 1 in 128, why should you be paying more than $100 to enter? Its kind of like a weird scratch off ticket bet.
In my topology course in college the Hairy Ball theorem was summarized as "Somewhere the wind isn't blowing."
Makes sense to me
Because the Earth's surface is a sphere and wind can be considered surface level. Genius.
first one isn't even a paradox and was never thought to be, wtf
The definition of paradox is strange because it includes counterintuitive facts as well as unanswerable questions, like the birthday paradox
Like the birthday paradox, I guess it can be considered 'something that sounds like it should be wrong' by some people.
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.
@@undeniablySomeGuybirthday paradox shouldn't be counted as a paradox in the first place.
it's engagement bait
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.
Same thing that the guy on numberphile forgot
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.
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..
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
@@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...
The trend to say 'every' is weird, as everyone know it's not everything
Yeah 😂
Now that’s the paradox of these kinds of videos
Does he know about hyperbolies?
is russel paradox 2.0, set of every video that contain the word "every" but doesn't contain everything
@@ThoughtThrill365why did you claim the hairy ball theorem is a paradox?
was waiting for a manscaped sponsorship
😂😂
They could comb that ball!
"This ball is very hairy, but there is no reason why _your_ balls should be hairy as well"
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.
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.
the thing is that you can fit as many guests as you want, you just can't check them in
@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.
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.
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.)
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.
The elevator paradox makes way more sense when discussing floors NEAR the top or bottom, not on the actual top and bottom floors
Counterintuitive ≠ paradox.
Paradox: a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true.
@@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.
@@zanti4132no one said the problem is with the statement Einstein.
@@alexmason5521Well someone hates their life
depends of the definition. Fortunately, jan Misali classified all types of paradox, and "counterintuitive but perfectly logical and explainable fact" is one type
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.
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.
My way of coming to terms with it is that Zeno's paradox was solved, so to speak, when we discovered calculus.
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
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 )
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).
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.
I would say that the solution of the paradox is the time you need to Paint the walls...
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...
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.
So basically, it takes an infinite amount of 2D paint to cover it, but finite 3D paint?
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?
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.
Yes. Lowenheim-Skolem is probably my absolute favourite theore.
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.
The circle is very easy to imagine.
@@newwaveinfantry8362 How? Wouldn't there be a tuft in the middle?
@@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.
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
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.
@@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...
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
@@matthewb2365
If you had $300 billion dollars, would you pay $100b for one round?
@@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.
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.
This doesnt make sense though because what if the elevator was near your floor or on your floor and someone below has called it
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.
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).
The Hilbert hotel would have to deny entry to Akira. An uncountably infinite blob of person would not be able to fit inside.
Wow, ok, I see how it is >:|
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
1:30 that’s the dumbest thing I’ve ever heard.
ok, cody.
If it's dumb, why is it still a debated topic in philosophy and physics?
@@areebsheikh6360it's not. It's just interesting to say with your friends on a table. It ain't no "actively discussed scientific problem".
Yeah I know
Only 3 million.@@3millionyengirl
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!
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"
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.
@@bycmozeszymonor if you stop assuming that time can be divided infinitely
@@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.
only Russel's paradox was an actual paradox and even that was fixed by setting new axioms 😭
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
Paradox doesn't mean unsolved
Paradox means counrerintuitive, not a logical contradiction
Your upload schedule is pretty insane. Nice
😄
@@ThoughtThrill365 pls keep it up with such intellectual stuff, educate urself as well
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.
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
2:06 they asked this question in class and not only did the person have the same birthday as me but the same name.
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
Except for Russell's paradox, none of the others are paradoxes, you just don't know the required maths. They just aren't intuitive.
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.
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$, which is approximately 1,000,000$
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.
"the hairy ball theorem" is a crazy way to start a math video ☠️☠️☠️
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.
You knew exactly what you were doing, starting with "The Hairy Ball Theorem" right outta the gate lol
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.
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
Well this fuzzy ball paradox explains a lot...😂
(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
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}
@@MrKillerMichael...that only works if you're talking about p-adic. we're not.
@@MrKillerMichael absurd*
there are infinite cases
@@asheep7797 Well, I wasn't talking about p-adic so if the reasoning is wrong I would like to know why.
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.
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?'
The Gabriel's Horn Paradox is based on mathematical sophistry. When the math is done correctly, the paradox disappears.
1:37 That paradox only works if time doesn't exist. Speed is distance over time and this paradox is distance without time.
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.
@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.
I remember some of these from Vsauce2 wow
how has it been years
Markov chains are all you need for St. Petersburg
As a hotel worker I dissagree with David Hilbert, probably the first guest who you want to move will not be willing to
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$.
gotta be the slowest elevator i ever heard of
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.
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.
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.
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.
i dont get how everyone couldnt move over. can you explain?
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.
i can confirm the first theorem in about 15 minutes
1:41 they’re not going the same speed tho
i love how the music cues you in when the video's ending!
Is the snail’s velocity constant at 1 m/s? If so, then obviously the cheetah will catch it, how is this a paradox?
Hilbert Hotel to me always gets a "thats stupid - of course- its ∞" reaction from me.
The earth's magnetic field works through the first "paradox" which means it's not a paradox but a law of physics.
A hairy ball might not work, but a torus would.
The dichotomy problem is basically just an exponential function
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
Dichotomy Paradox seems like a good argument for discrete space.
1:46 the fact that you chose my birthday startled me for a second
Same here 😂
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
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
If a student today said what Zeno said, we would just call him math stupid.
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).
When the UA-camr says its a math Problem but its actually just middle school math
song: Piano Sonata No. 11 K. 331 3rd Movement, “Rondo alla Turca”
these videos always seem to put me to sleep. im counting on it this time.
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.
With the St. Petersburg paradox, the video keeps mixing up heads and tails... :/
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
Pretty sure a cheetah catches the gazelle and devours it.
I'd say Bernoulli was being very smart, but in the end was overengineering his explanation of the paradox. The problem, when not listened to or thought through properly, presents itself as a gamble, and most normal people are not willing to stake much on a game.
I find the dichotomy paradox lame : if you add up the distances travelled by the cheetah, it adds up to the distance travelled by the snail.
The birthday problem is not a paradox. It is simply an unintuitive result, due to false preconceptions about how probabilities work.
Which horror movie was it taken from? 0:19
😂 😂
The birthday ones crazy. The church group I’m in has like 18 people in it and 4 people share a birthday.
Dichotomy Paradox isn't a paradox, its been solved.
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
"Birthday Attack"- never ever thought to hear about it.
ooh what an interesting video *click*. THE HAIRY BALL THEOREM
"at least" two of them share a birthday
Banach tarski is just "infinity/2=infinity"
And you can't forget Borsuk-Ulam
No, it's weirder than that. Of course there is a bijection between a single ball and a pair of disjoint balls, since they are sets of the same cardinality. That isn't surprizing. Banach-Tarski says tha you can actually split the ball into a finite set of disjoint congruent subsets, whose union simultaneously gives you the ball as well as two identical copies, without chainging the elements of those sets.
@@newwaveinfantry8362 That you can separate the ball in to a finite (I think as few as five?) pieces, and then reform them only using rigid motions in R^3, yielding two distinct balls. I remember seeing a very nice you tube video detailing the process - but that was a couple of years ago and I cannot remember the channel...
Xeno’s paradox isn’t really a paradox since the cheetah never moves backwards.
My hairy ball is so smooth with no tufts anywhere😂
At this point these mathematical paradoxes are simply semantic trickery and limitations of the human brains.
Imagine x, now imagine not x. What if I defined them as the same. Oh no, paradox.
Zeno must of been high AF to come up with that Paradox. either that or he clearly doesn't understand basic physics.
I don't get any of the comments saying that some of these aren't real paradoxes. I think one thing that very much justifies calling these paradoxes, which ThoughtThrill touches on for each paradox, is that these ideas were important to our understanding of mathematics. Some of them are "just" counterintuitive, sure. And some of them make perfect sense to us now that we're comfortably in the 21st century. But all of these spurred on significant advancements, refinements, or deeper explorations in our understanding of math, precisely because they were unexpected or genuinely confusing when mathematicians first encountered them.
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.
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!
Probably the most important missing one is the Borel-Kolmogorov paradox.
I dont understand your St.Petersburg-Paradox's Game explanation. How does your amout of bidding influence the winning? And How can you loose?
Gabriel's Horn can't exist, though. Even assuming it sprang into being, there is a minimum diameter the horn could have, given it's made of baryons and they have a measurable size. Therefore the concept of something that keeps shrinking forever as you go along is incoherent.
Some of these math paradoxes are kinda stupid because they make some assumptions that they don't tell you about. For example, the first one assumes that time slows down as the cheetah gets closer to the snail, and time would stand still if the cheetah caught up to the snail, which is why you would say the cheetah never approaches the sail. It's the exact same reason why an object with mass can't reach the speed of light. It's not a paradox.
On the St. Petersburg paradox: The way mathematicians calculate probabilities for investments is wrong. You should not count the absolute gain, but the relative gain. This makes a big difference. The justification for this is that to make up for a 50% loss, you need to earn +100%. So, geometric means are better for calculating investment odds rather than arithmetic means. I often calculate probabilities in speculative investments, and my default is always to guess what the highest possible gain is vs the highest possible loss, and calculate a geometric mean. So for instance, if I believe an asset can do a 0.5x, or a 10x, and is equally likely to do anything in between, then I figure my expected gain is sqrt(0.5*10) = sqrt(5) = 2.23, NOT (0.5+10)/2 = 5.5. If there is a possibility that an asset can go to 0, then no plausible gain can justify going all-in. When dealing with assets that can go to 0, you have to consider them as a part of your portfolio in order to make the calculation (like maybe some % gold, which you assume can't go to 0, and some % of some alt coin which could go to 0 but could go to infinity).
In the case of the St. Petersburg paradox, if people are paying $10 and are only allowed to play once, they have an 87.5% of losing money. This means it is realistic and practical that people aren't willing to spend a lot of money on it. The theoretical arithmetic mean is infinity (1/2*2+1/4*4+1/8*8...= 1+1+1...), but if someone spent his whole net worth on the game, there's an almost guaranteed chance that he'd end up broke. It is thus practical that people are not willing to spend a lot of money on it. Real people only have one life, so it makes sense that they play to win the median outcome rather than the average outcome that would happen if they had infinite lives to play this game.
If you want to generalize geometric means without even probability distribution (like 75% chance that something happens), then the result is that effect1^(chance of effect1)*effect2^(chance of effect2) and so on, with however many possible effects there are.
Edit: After some googling and messing around, I was able to solve for 2^(1/2)*4^(1/4)*8^(1/8)... I believe the answer is equal to 4. So, I believe this game is worth $4.
Edit 2: After messing around with a random number generator and a large number of trials, it appears to me that the game might actually be worth something like $7.5. I wonder if I made a mistake in the above calculation. Maybe $4 is like the median amount you'll earn, and $7.5 is the mean. I will investigate this more later because it is an interesting puzzle.
The under a geometric averaging (is there some kind of measure/function for probability distributions that gives such a thing? I suppose there must) is lim_(n->inf) (prod(1 to n) 2^(i-1))^1/n.
Taking the product inside the exponent, with triangukar number formula we get the inside of the limit is equal to 2^((n-1)/2), which still goes to infinity. In fact, it goes to infinity faster, which makes me think I've made a mistake somewhere
It's bigger on the outside! (Gabriel's Horn)
The dichotomy one literally isn’t a paradox, if I run up to my walking friend I know for a fact I’ll catch up with them. His math is just wrong
1:35 this is just straight-up overthinking
8:17 I pity the clients of chamber 7 384 104 who have to do all the way to their new chamber
St Petersburg seems stupid. If you get just one head in the first four tosses, the most you can make is $16, and your chances of getting $100 or more are 1 in 128, why should you be paying more than $100 to enter? Its kind of like a weird scratch off ticket bet.