The Painter's paradox or Gabriel's horn paradox 

Since a shape with finite volume can have infinite surface area, then who is to say a finite amount of paint can NOT cover an infinite surface area??

Also surely in the real world the planck length means the horn would eventually close up by any metric making it finite

I found your channel through the videos on planck units, which led me to this video. That got me thinking that if we consider the horn to be a "real" object, the end of the horn can only have a radius of (planck length)/2. If we consider that value as the limit for y, we should only integrate x from 1 to 2/(planck length). I have no idea what happens at that point since there would no longer be nice cancellations in the integration due to the use of infinity while calculating the answer. This begs the question, what would be the answer???

Evaluating Gabriel’s Horn Paradox . Area of horn= infinite square units. . Volume of horn= pi cubic units. . Paint required to cover the area= infinite × thickness approaches zero = zero cubic units. . The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi. . Note: no surprising for an object has area bigger than volume, for instance, a cube with one unit length, has one volume unit and six area units. .The misunderstanding of Gabriel's Horn has existed after We had compared volume unit with area unit. We should not compare different units.

This is why mathematicians get no Nobel prize. I am relatively stupid, but even I can see that this one is a funnel... So do funnels have any volume? Can this shape be considered a container?

This reminds me of something I was once told about black holes - that, because of the way the singularity warps spacetime, a black hole's event horizon has finite diameter but infinite radius. Sounds neat, and I can see the logic, but my grasp of maths & physics isn't strong enough to debunk or to verify it

Very well explained Pretty neat

The thing is that PI is not quite finite either. There are more and more decimal places popping up without an end.

The flaw with assuming mathematical paint with zero thickness is that you can use zero paint to paint an infinite area. Since no matter how much paint you use, it would have had zero volume when collected. Since it has zero thickness.

This makes an excellent homework problem for calculus students.

you could never be able to blow into the horn because there is no end with which to do it. The horn is infinite in length so there is no end.

So if we’re calculating the volume based on a quantity of cylinders within the infinite length of the horn, shouldn’t that quantity be infinite, thus making the volume infinite?

see this other object, finite value at 2 dimensions paradox an infinite at 1 dimension: circle have finite area to one given radius R . One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. Is the polar coordinates . The Integral from 0 to R of 2πr, gives exactly πR^2, but trying to see the integration process as that increasing swirl see the 1D line Length going to infinite. At Gabriel Horn is the same. 3D finite paradox a 2D infinite, one dimension below.

3:48 : how we go from line 2 to line 3 ? (i don't understand from where come from the minus (-) )

is this not also the basis for Zenos paradox?

How does this only have sub 100 likes?!

I am working on this Gabriel’s horn business because I don’t think it makes sense. Note…if the surface area of the horn is infinite, remove it from the volume it bounds and the surface area of that volume is likewise infinite, not finite. IF the surface area of the volume is in fact infinite as it must be if the surface area of the horn which touches it is, then the volume cannot be finite for the volume, as with the horn would extend out into infinity and though the volume would grow smaller in some proposition to its extension, it would never end. This is NOT the same as a 1 x 1 square (often used as a validation of the horn paradox) in which the square is subdivided infinitely along with the 1 x 1, finite volume it bounds (in this the boundary is NOT infinite but just as the volume it contains, is subdivided infinitely). The horn is an infinity in extension. So, unless the volume of the horn contained is reduced by the same proportions as the volume in the 1 x 1 square (reduced by half with each subdivision) which it is not for it cannot be (note the formula for the definition of the surface area of the horn), there is no way that it could even be considered finite. This is simply a conceptual reality and it matters little what the math claims. In this context, math formulae which demand the volume be finite simply must be in error. As for infinity, there is no such thing in materiality. The very means of existence requires finitude, i.e., that all that exists from sub atomic particles to the entities a composite of them, must be delineable and quantifiable. For example, consider the claim in recent past of many cosmologists and physicists that the universe is likely infinite in scope. It cannot be. Consider…if the universe expanded, as they all also claim, from the size of 10 - 15 meters or some such, a quantifiable measure, to about the size of a grapefruit via inflation (after which inflation ceased but the expansion continued), also a quantifiable measure, what was that last measure of expansion after which the very next one was infinity? As you can see, infinity is merely an abstraction. Additionally, many of these same physicists and cosmologists believe that the universe has existed in infinity and has “bounced”, expanding then contracting to expand again, ad infinitum. This is even more ridiculous and cannot be, in part for the reasons above which show it cannot be infinite in scope. If we have infinite rocks and take away a trillion, trillion, trillion, trillion times a trillion, trillion, trillion, what would we have left? We would still have infinite rocks. So how does an infinite universe contract to any quantifiable measure? What is it to contract? The reverse of expansion. The infinite, cyclical universe is a fraud for other reasons as well but that is another post. I have not finished with it yet, but I think (think at this point) that Gabriels horn is as big a fraud as Hilberts hotel or the Russell paradox. As for your video, it was interesting but you did NOT clarify how or why the volume of the horn is finite. Showing your average viewer a page of complex math is meaningless, unless you don’t care to impart the truth of this paradox to them and only to those who understand the math. If you would, please deconstruct that aspect and explain it in a metaphor which demonstrates the claim via “physical” or “mechanical” means. Thanks.

Is this like a black hole?

Maybe this has been observed already in the comments, idk. But, my solution lies in the fact that it make no sense to say "how much mathematical paint" it takes to paint any surface. Infinitley thin paint has no volume. Another approach: I have a volume of 1mm^3 of paint. The area it covers at 1mm thickness is 1mm^2. vol=area*thickness therfore area=vol/thickness. as thickness approaches zero, so does volume. infinite thinness IS infinite surface area. Saying that the surface area is infinite is not the same as saying it takes an infinite "volume" of paint to paint it. Math paint has no volume. Great video and paradox none the less. : )

"That's like a paradox squared" bahahaha

At the end I thought you were just gonna say that if you blew the horn it would sound like shit. I like my ending more tbh but the rest of the video more than makes up for it 💙

At some time it will reack plank length and that will be end of it, not infinite.

lol paradox² :p

This is no paradox and have never been a paradox. Yes, the volume is finite and the surface infinite. But, as the surface has 0 thickness, the surface has no volume.

You can already see the problem. You can’t divide by zero so x can only get close to zero but never be zero, infinitely.

Your explanation(s) of the paradox does not make any sence. Math Physics