The Painter's paradox or Gabriel's horn paradox

[@MrGentleWombat]

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?

[@i-ska]

Exactly, it's a funnel, well spotted you are far from stupid, your intelligence, sense of logic is superior to many of the worlds top mathematicians believe it or not. Pi is not finite it is transcendental. Yet the mathematician Penrose did get a Nobel prize for sci-fi math in 2022 for using math to describe an imaginary singularity in a stationary black hole which the known laws of physics show cannot exist in reality. The Nobel prize is worth $1m but it has little value in science any more. It is generally doing more harm than good much like a lot of the sci-fi theoretical math which has become more popular than real science yet which is promoted as fact. Look outside the mainstream if you want the real physics.

[@nancyhey1012]

Well, I’ve heard that the reason there is no Nobel prize for mathematics is because Mr. Nobel’s lover jilted him in favor of a mathematician.

[@nancyhey1012]

Poor Gabriel, he will never be able to blow his own horn!

[@parkershaw8529]

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??

[@jotasietesiete4397]

Mathematicaly perfect paint would require zero paint to cover any surface, so it definitely can paint this shape too

[@facedasire]

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???

[@leonsunu1209]

I think the main issue here is that the Planck units are only of relevance in like physics and stuff. To plain mathematics they don't mean anything and we're at a simular point where he started talking about molecules.

[@Mernom]

That depends on the size of the base unit.

[@alec006a15]

I mean, limiting the size of the end of the bell to y=+-planck length/2 would put constraints in the function and so yes, the area would be calculable. Thats practically true. its important to note that this shape *theoretically* has unlimited surface area. As soon as real world limitations are imposed in such a dilemma, such as the size of a paint molecule, the hypothetical falls apart

[@RhysOlwyn]

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

[@mahmoudattalla2972]

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.

[@LearningCurveScience]

Thank you for taking the time to write such a detailed comment. Like virtually all paradoxes, Gabriel's horn is only really a paradox due to a quirk of either measurement or language. I just find them fascinating because they cause me to stop and think and ponder. I can stop and think for more time than I really should about the consequences of paradoxes, their intricate nature and how the universe works. Thank you again.

[@Mernom]

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.

[@milapravda3916]

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

[@morbideddie]

It's infinitely precise, but is finite in value.

[@bosshogg8621]

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. : )

[@RhysOlwyn]

But does mathematical paint have surface tension? One for the philosophers, there

[@Wesley_H]

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?

[@jamestagge3429]

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.

[@LearningCurveScience]

Yes infinity is a very strange concept. Maybe I'll make a video about it someday.

[@RF-fi2pt]

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.

[@aaronwolbach9880]

This makes an excellent homework problem for calculus students.

[@dineshvyas]

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

[@lizard_man4379]

Very well explained Pretty neat

[@LearningCurveScience]

Thank you. I do my best at explaining stuff.

[@roberthuismans3533]

is this not also the basis for Zenos paradox?

[@Fine_Mouche]

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

[@metehan9185]

we form an antiderivative

[@petevenuti7355]

Is this like a black hole?

[@jamestagge3429]

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.

[@LearningCurveScience]

Thank you for watching the video. Just like most paradoxes they don't really work in actuality. Gabriel's horn is simply a mathematical model, and thus the size of the universe is immaterial. As for the volume being finite, imagine this. Add together 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ... to 1/infinity. Your answer will tend towards but never reach 1, that is how the volume can be finite. In reality there would come a point at which the width of the horn would become smaller than the size of any paint molecules, but since these are merely paint molecules of the imagination, we don't need to worry about their size. Like most mathematical paradoxes, Gabriel's horn is just an interesting quirk of mathematics. Hilbert's hotel just shows the impossibility of finite beings like ourselves trying to imagine something infinite. We can't even conceptualise infinity because it has no end.

[@jamestagge3429]

@@LearningCurveScience Thank you for your response to my comment. I really appreciate it. I have so much fun with this stuff. Anyway, my point in my response is that these are not paradoxes at all but conceptual contradictions which fail even theoretically. I don’t see the utility in making mathematical points from contexts so glaringly fraudulent. For example, by definition, conceptually, Gabriel’s horn cannot be infinite in surface area and finite in volume, literally or theoretically. By claiming so the very means of the definition of the infinite surface area is contradicted so what is the point at all? This may be redundant but…if the horn were removed from the volume it encompassed (especially if the horn is theoretical) the surface area of the volume left behind would be exactly the same as the surface area of the horn so how is it that it would be finite and the horn, not? My objection has nothing to do with this not being possible materially, but not theoretically either. This supposed paradox makes absolutely no sense. So too with Hilbert’s hotel, which I don’t believe shows the impossibility of our imagining something infinite. I believe we can imagine it quite well and in the process of that act, we can see that Hilbert’s hotel is sophomoric and quiet one dimensional. Again, I might be redundant, but he sank his own boat by stating in the formulation of his paradox that “all the rooms were full”. That there were infinite rooms is not a factor that can be manipulated in terms of the infinite guests, each of whom would have his own room. If in claiming that he could shift the guests in their rooms to make accommodations for even a single new guest, let alone infinite new guests means that he would appeal to some sort of elasticity of the infinite number of rooms, thus, he would have to make the corresponding manipulation of the infinite guests because “all the room were full” and that condition, being part of that which defined the context of the formulation would have to remain intact. No matter what he did, “all the rooms were full”. That would not change so there is no shift of guests that could be made. Now I didn’t write the damned thing, Hilbert did and did a piss poor job of it to boot. All I am saying is that these supposed paradoxes are not paradoxes at all and none but one I have yet seen is worthy of applause. They are piffle and nothing more. What do you think??

[@DoesRocksFloat]

You can’t divide by zero. It should have been obvious from where he showed the formula instantly.

[@tedytarrify]

How does this only have sub 100 likes?!

[@ryanbenge7824]

"That's like a paradox squared" bahahaha

[@rolandkarlsson7072]

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.

[@LearningCurveScience]

As with most paradoxes, they aren't really paradoxes, just mathematical/linguistic/logical ideas of interest. This paradox is nothing more than a mathematical curio. It couldn't exist as the length would be infinite (impossible), and there would come a point at which the diameter of the tube would be smaller than paint molecules so it would be impossible to paint anyway. It's a nice curio nonetheless.

[@rolandkarlsson7072]

@@LearningCurveScience - correct. It is a bit annoying though with a "paradox" that is so simple to dismiss and that still people seem to have problems getting their heads around. I would understand if the ancient Greeks had a problem with this. But not we, 2022. Its like a finite area that has a border that is a fractal of infinite length. Do such a thing exist? Not in real life, of course, but yes, as a mathematical object. And there is nothing strange with that. But then it will take an infinite time to walk around it? Nah, you cannot walk on a mathematical object. It will also take an infinite time to construct the fractal if you want to walk on it. And you need to be able to draw a line that has zero width.

[@RF-fi2pt]

@@rolandkarlsson7072 See this object, finite value at 2 dimensions contains an infinite at 1 dimension: circle have finite area to one given R (although the precision is given by the π decimals). One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. 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 Contains a 2D infinite .

[@i-ska]

Since when is pi finite? It is transcendental not finite. Therein lies the lie in the math. The horn can never be filled even to cover the inside surface area because the hole in the math allows the paint to slip imfinitely further down into the bottomless horn. A good demonstration however of why maths tricks misused and misrepresented have led to the crisis in physics and cosmology. Such a horn cannot exist of course no more than a singularity with infinite density in a black hole can. It's conman mathematics, even the Nobel foundation have either lost the plot to the conmen or are deliberately lying to the public too.

[@lto1964]

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

[@ferretappreciator]

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 💙

[@4dirt2racer0]

lol paradox² :p

[@DoesRocksFloat]

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.