Every Weird Math Paradox 

The trend to say 'every' is weird, as everyone know it's not everything

first one isn't even a paradox and was never thought to be, wtf

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.

In my topology course in college the Hairy Ball theorem was summarized as "Somewhere the wind isn't blowing."

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.

only Russel's paradox was an actual paradox and even that was fixed by setting new axioms 😭

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

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 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"

was waiting for a manscaped sponsorship

The Hilbert hotel would have to deny entry to Akira. An uncountably infinite blob of person would not be able to fit inside.

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.

(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

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.

Banach tarski is just "infinity/2=infinity" And you can't forget Borsuk-Ulam

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.

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.

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.

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).

Dichotomy Paradox isn't a paradox, its been solved.

i can confirm the first theorem in about 15 minutes

"Birthday Attack"- never ever thought to hear about it.

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.

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.

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.

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

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 love the banach tarski paradox! I wonder if you could make a longer vid about it

Probably the most important missing one is the Borel-Kolmogorov paradox.

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.

If number of guests in the infinite hotel has coordinality greater than countable infinity, it can't give a room to everyone

Dichotomy Paradox seems like a good argument for discrete space.

A hairy ball might not work, but a torus would.

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

Your upload schedule is pretty insane. Nice

The hotel paradox stems from the way the problem is initialized, you say there is an infinite number of rooms, but that every room is filled. If every room is filled this means the ratio of ppl to rooms (lets say max one person in a room) should be 1. From the initial definition of the problem, this implies inf/inf = 1 but this is not true.

Zeno of Elea was such a beta. He kept stating "new" philosophical discoveries, that all turned out to be the same thing. Even his peers were annoyed by it.

The second paradox is just a case of ancient Greeks nor knowing limits

With the St. Petersburg paradox, the video keeps mixing up heads and tails... :/

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!

Power tower paradox tho

Add the end he mentions that the Banach-Tarski Paradox vanishes for locales. Has anyone got a reference for that?

I remember some of these from Vsauce2 wow how has it been years

But what if I say "every live person on Earth is alive." or something else like that

is that the grand budapest hotel in the hilbert hotel drawing

At 6:22, what's the significance of "anything times infinity is infinity"?

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.

9:54 this basically is turning ∞/2 = ∞ into a paradox

zeno's isn't really a paradox he was just wrong and shortsighted. he failed to consider that the cheetah complete's each catchup in shorter and shorter time such that the cheetah is able to complete the "infinite catch-ups" in finite time

Can you make a video about physical phenomena such as the Quantum Tunneling, the zeeman effect or the casimir effect?

@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).

All throughout my childhood, rhere was always someone in my class who shared my birthday: and the shocking thing is, my twin had the same experience!

Nice compilation of paradoxes. Would be clearer without the background music, in particular at higher speed playback.

wtf is the first one that caugth me off guard

I missed the cool paradox from Bertrand, but overall great video

So - how is the hairy ball theorem at all paradoxical? Zeno's paradoxes all rely on a naive notion of 'infinity' - the example you give is solved by noting that the time decreases by a tenth each step, and that such an infinite sum is finite - so in that finite time, the cheetah catches the snail. The birthday problem is also not a paradox, just unexpected. How is Gabriel's horn paradoxical? The surface area (and length) is infinite, but the volume is finite. The elevator thing is also not a paradox. If you are in the middle, the odds even out. I'm not even sure how it's even surprising. The St. Petersburg paradox. No... It's not a paradox - although it is interesting. I would not play the game - there is a 50% chance you lose on the first flip - and while the expected value is infinite, so is the variance (or _risk_ in financial terms) of the game. I note that there are also several other reasons why this is not really a paradox - quite appealing to me is that there is no-one who would offer a game with an expected loss (to them) of infinity. Given the inherently finite nature of the game, and cutting the odds of at about 1 in 10,000 leaves an expected value about a little over $10, so the intuition people have of about $10 seems to fit this bit of human behavior. How is the Hilbert Hotel paradoxical? It is counterintuitive - but that's because infinity is not a number and we should not expect it to act like a number (i.e. something finite). Its properties (which the Hotel was created to illustrate) are logical and consistent. Finally - an actual _paradox_! Russel's paradox is a true paradox. The Banach-Tarski paradox is not really a paradox, but a consequence of measure theory and the required existence of unmeasurable sets. The pieces used are such that they do not have a measure (under the usual measure in R^3). This, I think, leads to some deep problems in just what it means to measure the area (or length, or volume, whatever) of an object. Individual points have no length, but a line is just made up of points - so where does the quality of 'length' come from? Measure theory (to me) kind of provides an answer, but we are then left with some collections of points that do not have a measure (they are not of measure zero, nor a measure of any value). This still fascinates me. It's not _set_ theory, it's _measure_ theory. I haven't heard of locales - so I will have to look into that! Do you have a reference?

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]

These are not paradoxes, just counter intuitive results

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..

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$, which is approximately 1,000,000$

1:35 this is just straight-up overthinking

6:20 at a very high buy-in cost, the odds of you losing tons of money are way greater than the odds of gaining anything??? How is the expected gain infinite?

How is the Hairy Ball Theorem a Paradox?

I don't get how the hairy Ball theorem is a paradoxe ? Nice video tho

St. Petersburg paradox actually starts with 2 dollars, not one.

everyone goes up from the ground floor. no one goes up from 2nd or 3rd floor. this is the solution

Schrodinger cat paradox coin have probability p=1/2 heads and q=1/2 tails, in B. Russel paradox extension if exist set can we choose not participate not choose any element from the set. Set of elements can be chosen 1 or all elements or choose groups of elements but not choose any is forbidden by probability definition, if you not coin toss it means probability of such event have p=1 at same times coin is in the state of heads and tails. That is non-sense, if bacteria observes radioactive decay (bacteria is conscious creature) do wave function collapses inside box with cat? Schrodinger trolling Copenhagen interpretation of quantum physics

Counterintuitive ≠ paradox.

Which horror movie was it taken from? 0:19

1:50 well that's my birthday lmao

1:46 the fact that you chose my birthday startled me for a second

Most of these aren't paradoxes they are just mathematically proven facts?

the hairy ball theorem is in no way a paradox. It's a funny named theorem, but it's so obviously true. You got the sain petersburg paradox a little wrong. It's not true that you should bet regardless of cost. The problem is that the expected value is the mean. so if you played the game an infinite number of games you would on average gain money regardless of bet size. However if you only play once it is unreasonable to bet a lot because the probabilty that you actually gain money is incredibly small even if the amount of money is significantly large with a very low probability. You also got the hilbert hotel slightly wrong. It's true it can accomodate any countably infinite number of people. But if an uncountably infinite number of people show up it cannot accomodate this. For example if there are so many people that they label every person with an infinitely long binary string, and every infinitely long binary string is used by some person, then there is no way to accomodate this (based on Cantor's Diagonalization argument. You also loosely said in the russels paradox part that the problem is that you defining some object based on itself. but the recursive theorem says this part is fine. However the problem is we constructed a object in a manner that is not possible (in this case, we assumed that such a set exists that contains everything that doesn't contain itself.) originally we assumed this was possible to do for any sets, however Russel showed there must be some sets that do not exist. To fix this, Zermalo and Franklin constructed ZFC which fixes this problem. [note this doesn't mean we no longer have this problem as we cannot construct this set using the axioms of ZFC] For the Banach Tarski Paradox it's not a few pieces, it's explicitly an infinite number of pieces. and it's the reason many people question C in ZFC. (without Choice its not true. but choice is hard to question because it's appears very obvious in other circumstances.)

George Gamow=Gamov or Gamoff

What a nice video

Great videos. The music is very distracting, though. Especially the last minute.

6:35 wow I picked 10 too

well done !

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.

*maths

remove background music please. it makes speech harder to understand.

it's okay to leave some breathing room between sentences you know

‘solutions’: 1. That’s just how hairy balls work 2. Zeno didn’t know calculus 3. Combinations grow with factorial 4. This like asking ‘how can a finite line segment contain infinite points’ because that’s how math works; you can’t sensibly compare an infinity in dimension D with a finite number in dimension D-1, although I’m happy to be shown otherwise. 5. This is common sense? Lol 6. People are poor (as video explains) 7. Infinity +- N = infinity 8. One of my favorites! Allowing self-reference enables undecidables. 9. AoC is independent of all other axioms of ZFC and makes no sense 😂

the Dichotomy paradox is wrong because the speed that the cheetah runs relative to the snail is 9 m/s so to find time you simply divide the distance by 9 and get the answer