Every Weird Math Paradox

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

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

was waiting for a manscaped sponsorship

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

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.

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.

Except for Russell's paradox, none of the others are paradoxes, you just don't know the required maths. They just aren't intuitive.

Counterintuitive ≠ paradox.

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.

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.

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.

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

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).

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.

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

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"