Every Infinity Paradox Explained

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Timestamps:

0:00 Hilbert’s Hotel
1:07 Cantor’s Diagonal Argument
3:22 Thomson’s Lamp
5:16 Gabriel’s Horn
7:23 Ross-Littlewood Paradox
8:51 Dartboard Paradox
10:47 Sponsor
11:50 St. Petersburg Paradox
13:18 Riemann Series Theorem

Thanks for watching :)

This video was sponsored by Brilliant

— DISCLAIMER —

This video is intended for entertainment and educational purposes only. It should not be your sole source of information. Some details may be oversimplified or inaccurate. My goal is to spark your curiosity and encourage you to conduct your own research on these topics.

[hardworkingcriminal]

Note to self never stay at Halbert's hotel. I'm not tryna keep switching rooms everytime a new schmuck shows up

[Pocketnemo]

The lamp is gonna be off by the end of the 2 minutes, because if it's not, Dad's gonna yell at me for wasting electricity.

[ultimazilla]

when you've been walking down the hall in Hilbert's Hotel for the past 10^6875 years to reach your room and an announcement plays saying "please move to the room number that doubles yours"

[erikm]

I hate staying at Hilbert's Hotel. I had this great room right by the pool, and right across the hall from the where they serve the complementary breakfast, then this uncountably infinite group of guys shows up and they make _me_ move rooms. Now I have to wake up crazy early if I want to get to the breakfast before the heat death of the universe.

[PrzemyslawSliwinski]

The paradox of the Gabriel's horn is that you need an infinite amount of dye to paint it inside with a brush, but only a finite amount of dye to fill it (and paint it anyway).

[Lawh]

I feel that we created infinity and then began complaining about infinity lore.

[zanakil]

I LOVE that you just hit the road running and started the vid w/o intro. Thanks for respecting our time.

[billpines]

How does this affect the local trout population

[Sairuken]

The biggest reason why these paradoxes exist is because they assume and treat infinity like it's a finite number. It's the same kind of logic as when people say 0 divided by 0 is 1, because of the observed rule that every other number divided by itself results in 1, which while true, doesn't mean that it also necessarily applies to 0. It's an interesting thought experiment though.

[Halo]

6:38 change yo smoke detector batteries

[andrewmichel]

For Gabriel's horn you were supposed to calculate the surface area of the area of revolution, not the area beneath the curve of 1/x. The surface area integral is the integral from 1 to infinity of ((1/x)sqrt(1+(1/x^4)))dx which you can bound below by the integral you did, which diverges.

[dawica]

My thinking on the Ross-Littlewood paradox is that if the balls are removed in order, whenever any ball of number ‘n’ is removed, the vase will still contain ball n+1. So even though for all values of n, ‘ball n’ will be removed, it is impossible for the vase to be empty.

[klikkolee]

My long-time statistics hot take is that the "expected value" operation is given undue significance. Its name is very suggestive of it being the inherently correct way to judge risk and reward, but the fact of the matter is that it is fairly uncommon for it to actually yield a value which you can, in any meaningful way, expect. The St Petersburg paradox shows this well. There is less than a 7% chance of getting $32 or more from the game. A person simply cannot, in the ordinary sense of the word, expect more than $32, regardless of the value of the operation which has been named "expected value".

Pascal's mugging is another "paradox" that just comes down to the expectation operator being treated as gospel. A mugger says if you give them your money, they will give you x amount back later. No matter how improbable you believe it to be that they will uphold the deal, they can give an x for which the "expected value" of taking the deal is positive, giving muggers a surefire strategy against people who treat the "expected value" operation as the ultimate arbiter.

"Expected value" is only genuinely accurate to its name when the risk is taken infinitely many times. Humans aren't immortal. The risks also often have an ante of some kind, and if the risk failed enough times, you can't afford the ante anymore and can't try again. Even with large corporations, which can absorb many more and much larger failed risks than people can, risk assessment goes beyond expected value -- they qualify or quantify the level of risk itself and compare that to a risk tolerance

[pedramhashemi]

Your production quality is getting better and better! keep up good work :)

[chaotickreg]

6:35 *BEEP* change your smoke detector battery dude

[herrhartmann]

The spot you choose on the dartboard may be infinitely small.

But the dart itself is not!

Therefore, the dart does not actually hit any one spot. Instead, it hits a small area.
And you can totally calculate the probability of your chosen spot lying inside that area.

[giddycadet]

the probability of hitting any individual point on a dartboard changes quite a lot when you remember that the point of the dart itself ALSO has an area to it, and thus its own infinite number of points

[emperormegaman]

The furniture and laundry budget to keep Hilbert's hotel running must be a nightmare.

[Koroistro]

I find the St. Petersburg paradox very interesting, it highlights (as many other paradoxes here do) how "right" results are wrong when we ignore other variables applicable to that context.
For example looking at the variance.
While the EV is theoretically infinite we can plot every price of the game to the odds of turning a profit, and how much total capital we would need to ensure a profit in the long-run.

[kennymartin]

The real question about Gabriel's horn is "does it go doot?"