## The paradox of Gabriel's horn.

• 6.1k
I am no mathematician.

I see this at Numberphile on Youtube:

Near the end a good question is asked, which causes this OP.

The idea is that an infinitely long horn, were you to pour paint into it, could be filled with a finite amount of paint, per the video π units of paint. But the surface area of the horn, itself being infinite, cannot be painted with a finite amount of paint. Which is pretty interesting. But here is the question:

If you fill the horn, have you not essentially painted the surface with a finite amount of paint? So it would seem that the surface cannot be painted, yet in fact is paintable.

Can one of our esteemed math members make tolerably clear to a num-num like me how this can be? What the trick is? .
• 1k
If it were truly infinite, in the context of decreasing in size, it would reach such a small size beyond what is possible due to the makeup of this world, ie. smaller than an atom. Or perhaps not as we're constantly talking with the implied context of "reality" ie. things as they are at the moment of observation. Before then even, once it becomes smaller than what allows a single molecule of paint, it would of course "fill up". Though it is interesting because assuming once we reach the point of smaller than a molecule, you could still paint it's exterior because even though it's girth or width may not be able to hold a single molecule of paint, perhaps it's length would... actually no. Molecules, as we're told are symmetrical. There is probably a single point after a molecule of paint would no longer fit, where due to the structure itself counting as at least one molecule or atom, there is a point where the exterior can be painted (one molecule of paint can reside on the outside past the point where one molecule of paint can no longer pass on the inside). I couldn't imagine in a million years where this "paradox" would ever come up or be relevant in.. literally anything would ever do or ever come across but, isn't free society fun. Lots of time for non-productive speculation.
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Completely missing the point is seldom so obvious. You are correct in that there are no infinite horns for sale at Walmart. But there are in the mathematics store, which is where you are - or did you miss that?
• 1k

Quite. I must have. Paint however is not mathematics, it consists of molecules of a finite and measurable size, or else it is no longer paint. Though perhaps "paint" is implied to be simply the smallest unit of measure available, ie. an atom or the smallest known subatomic particle being a quark (which is still measurable). Or does it transcend even this? Basically, how can something be infinitely long if it's slowly decreasing from a measurable state or point? Writing off the world we live in for numbers that could in theory go on forever.. it's still kinda confusing lol. Almost seems kinda like arguing over a glass being half full or half empty. Which of course is actually simple. If it was an empty glass filled up, it is half full. If it was a full glass, with half removed, it's half empty. Unfortunately, this does seem to boggle the mind. Hopefully not by design.

I never liked math. Perhaps someone who does can shine some light on this for the both of us? Basically, what is the largest number that can be reached, and is it any closer to infinity than 1? What is the smallest unit of measurement to reach infinity? What is the smallest fraction that can be reached without there being nothing left, etc. The universe, at least the world we live in, seems to disprove the existence of 'infinity' beyond "oh look a solid figure".
• 4.6k
The apparent paradox comes from equating volume as defined by an integral with volume as defined by a concrete, physical enclosed region. Minimally, the infinity of the shape breaks the correspondence between the two ideas.

There are other examples, like the Cantor set, which is dispersed throughout the interval (0,1) - like being completely dissolved in a fluid - but nevertheless has no volume whatsoever.

Physical volume and mathematical volume don't have to correspond!
• 1.8k
But the surface area of the horn, itself being infinite, cannot be painted with a finite amount of paint.

No that's not true. Divide the infinite surface area into sections 1 unit long, as for example the positive real number line is partitioned by the integers 1, 2, 3, 4, ...

Between 1 and 2 you use 1/2 gallon of paint. Between 2 and 3 you use 1/4 gallon. Between 3 and 4 you use 1/8 gallon, and so forth. You will then cover the entire infinite surface of the cone with only one gallon of paint. Of course this is a-physical, since the thickness of the paint would soon be far less than the width of an atom. But mathematically you can indeed cover an infinite surface area with a finite amount of paint.

I didn't see the video, but it was referenced on Reddit this morning with the same misconception, so I wonder if Numberphile perhaps confused people on this point.

ps -- It's perhaps easier to see this in two dimensions. Imagine the graph of y = 1/x from 1 to infinity. We know from calculus that the area under the curve is infinite. But for any positive integer n, the area under the curve on the interval [n, n+1] is finite. So you just cover each such interval with a finite amount of paint according to some convergent infinite series such as 1/2 + 1/4 + 1/8 + ... = 1.
• 568
If you fill the horn, have you not essentially painted the surface with a finite amount of paint? So it would seem that the surface cannot be painted, yet in fact is paintable.
You painted the internal surface of the horn, but not the external one, which is bigger, even if just infinitesimally. How is this accounted for?
• 2.1k
If you fill the horn, have you not essentially painted the surface with a finite amount of paint? So it would seem that the surface cannot be painted, yet in fact is paintable.
The solution lies in the meaning that we silently assume for the word 'paint', which is to cover an area with a layer of liquid paint with a constant, nonzero thickness. We cannot 'paint' the horn in that sense because the volume required would be the area (infinite) multiplied by the thickness (nonzero), which means an infinite volume.

In the filled horn, the internal surface is indeed covered by paint. The thickness of the covering layer at a particular place equals the radius of the horn at that place. Since the radius decreases towards zero, the thickness of the layer just keeps getting thinner. There is no nonzero thickness of layer with which we can say that filling the horn has 'painted' it in the above-defined sense.

This solution does not rely on any physical limits such as Planck lengths or diameters of paint molecules. It simply lies in the definition of 'paint' requiring a constant, nonzero thickness of layer. If we remove that requirement then we can 'paint' any infinite two-dimensional surface (including painting the horn externally, by the way*) with any nonzero volume of paint, however small. We might have to make some technical restrictions such as only allowing surfaces that can be smoothly embedded in 3D Euclidean space. But those restrictions would only interest mathematicians as they would not exclude any surfaces non-mathematicians might imagine.

The mathematical principle behind this is that the function f(x) = 1/x has no convergent integral from 0 to infinity whereas the function g(x) = 1/x^2 does.

Or, avoiding calculus, the sum 1/1 + 1/2 + 1/3 + ... diverges to infinity whereas the sum 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + ....converges to a value around 1.645.

* As fishfry points out, the horn has no distinction between its 'internal' and 'external' surface. But we can meaningfully define 'painting the horn internally (externally)' as follows. First define the inside I of the horn as the set of points x such that the shortest line segment connecting x to the horn's axis does not intersect the horn's surface S. The outside O of the horn is all of 3-space excluding the inside and the surface.
Then an internal (external) painting of S is a subset A of Euclidean 3-space (which we think of as being 'full of paint'), such that, for every point P on S, there exists an open neighbourhood N(P) whose intersection with I (O) lies entirely within A.
Under this definition, which we note specifies no minimum thickness of paint, we can make both an internal and an external 'painting' of S using a finite volume of paint.
• 8.6k
Ziploc bags. An object with a given surface area can be collapsed and that can reduce the volume without affecting the surface are...if you do it just right, the volume can become zero but not the surface area.
• 114

Reminds me of this: https://youtu.be/5LWfXhggC70
It's just semantic trickery of the mind in the end that creates these apparent paradoxes.
They are not very interesting (in my opinion).
• 6.1k
Is it fair to suppose that the inner and outer surfaces of the horn, in terms of area, do not materially differ? And is the video correct in claiming the horn is completely fillable with a finite amount of paint, in the particular case π units?

The inner is filled, therefore the inner surface is "painted." That is, every place or location on the inner surface is painted and nowhere not painted. It would seem, then, that if we want to paint the interior surface, we need only pour in π amount of paint - and pour out again the excess if we care to. On the other hand, if the outer surface is not paintable with a finite amount of paint, then it seems reasonable to suppose the inner is not either. So if we set out (with however much difficulty) to just paint the inner surface with π paint, that would be impossible.

It seems to say that I can fill my pint-sized glass with a pint of beer, but I cannot moisten the inner surface with 10 billion gallons, or any other amount.

I buy the idea that if described one way, one outcome, and another, another. But with the thought experiment it seems it must be one way or the other, one right, one wrong.

It occurs to me to not think in terms of paint and surfaces, but of sets of real numbers - but the cardinality of the sets would be the same, yes? In which case the outer is paintable, if the inside is fillable. So it would appear that something is wrong somewhere.
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It would seem, then, that if we want to paint the interior surface, we need only pour in π amount of paint

Correct, but the paint can't be uniform since it must be spread over an infinite surface area. The thickness of the paint has to decrease as some convergent infinite series: 1/2 gallon for the area between x = 1 and x = 2, 1/2 gallon for the next chunk, and so forth. So nothing you said is in conflict with the story, which is simply that 1/x has an infinite area but its solid of revolution has finite volume.
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but the paint can't be uniform since it must be spread over an infinite surface area. The thickness of the paint has to decrease as some convergent infinite series:
Why? Why cannot it be a consistent thickness, mathematically thin? If I wish to paint my living room, it does not occur to me that I must fill the living room with paint or that the paint must vary in thickness.

I'll attempt a more rigorous description of the paint and painting. And for that purpose I'll borrow from the argument that the cardinality of the points on the number line between zero and one is the same as the cardinality of the points in a cube measuring one mile on a side.

Let's consider the cardinality of the points that make up the inner and outer surfaces of the horn respectively. It seems to me it must be the same for both. Now, if we may, the cardinality of the points that make up the paint itself. By painting is meant an assignment of one point of paint to each point of surface. It seems to me the cardinality of the paint must be greater than or equal to that of one of the surfaces. And greater because for any cross section of the horn the inner surfaces never meet, and consequently there is always more paint in the cross section when the horn is filled than is needed to just paint the surface.

Thus on the assumption that the horn is filled with a finite amount of paint, understood to be proved, then the inner surface has been painted. Because the inner surface thus paintable, and the outer surface the same area, the outer surface must be paintable.

Is there an error? Where is it? If it depends on an if, then the if is that the horn can be filled in the first place and the rest "flows" from that, so it seems.
• 1.8k
Why? Why cannot it be a consistent thickness, mathematically thin?

Because if the thickness is constant its volume must be infinite. Think of the infinitely long real line and a rectangle above it with height 1/10. Width x height is infinite. Say the height is 1/million. Same thing. 1/zillion. Same thing. As long as the height is nonzero and uniform, the total area across an infinite surface must be infinite.

If I wish to paint my living room, it does not occur to me that I must fill the living room with paint or that the paint must vary in thickness.

Nothing to do with the previous sentence. And the whole idea of painting your living room is a red herring, since this is a mathematical and not a physical situation. It's like people complaining about Hilbert's hotel that there can't be any such hotel, or how would the maid make up all those beds, and so forth. The attempt to put the puzzle in "practical" terms ends up confusing the issue. But the point still stands that if the surface to be painted is infinite and the paint layer is uniform and of positive thickness, no matter how small, the volume of paint must be infinite.

I'll attempt a more rigorous description of the paint and painting. And for that purpose I'll borrow from the argument that the cardinality of the points on the number line between zero and one is the same as the cardinality of the points in a cube measuring one mile on a side.

Since that's true, that's a hint that a cardinality argument will be of no use here.

Let's consider the cardinality of the points that make up the inner and outer surfaces of the horn respectively.

There's a big problem, which is that there is no such thing as the "inner or outer surface" of a line. For example, what is the "upper surface" of the real line? There clearly isn't one because if the surface is any positive height above or below the line, it's not the "surface," and if it's directly on the line, it's the line.

Same argument with a circle. There is no inner and outer surface of a circle. Do you follow this point? It's crucial, since the "inner and outer surface" of a two-dimensional surface in 3-space is being bandied about in this thread and there is no such thing.

What are the inner and outer surfaces of a sphere? Can you name a point on the inner surface? No. Because if the point is zero distance from the sphere, it's on the sphere. And if it's a positive distance inside the sphere, it's not the "inner surface" because there are many points strictly between that point and the sphere.

In fact the "surface of a sphere" can only be the sphere itself. Like FDR said! The only surface of the sphere is sphere itself. Ok nevermind.

It seems to me it must be the same for both.

Bearing in mind that there is no such thing as the inner or outer surfaces of the horn, which is a tw- dimensional object living in 3-space, I'll play along. But remember the premise is false already.

Now, if we may, the cardinality of the points that make up the paint itself. By painting is meant an assignment of one point of paint to each point of surface.

Ok. But think of a circle in the plane. Can you paint it's "inner" or "outer" surfaces? No, because there's no such thing. If you paint the interior of a circle you either include the points on the circle or you don't. If you do, you've painted the circle itself. But if you don't, then there's always a space between your paint and the circle into which you could have put more paint; that is, gotten a little closer to the circile.

I hope you see this, if not please say so, because this point is essential. There are no inner or outer surfaces to a 1-D object living in 2-D, or a 2-D object living in 3-D.

It seems to me the cardinality of the paint must be greater than or equal to that of one of the surfaces.

Why? The cardinality of any circle concentric to a given circle is the same. Ancient proof, just draw rays from the center, there's your bijection.

And greater because for any cross section of the horn the inner surfaces never meet, and consequently there is always more paint in the cross section when the horn is filled than is needed to just paint the surface.

This doesn't make sense (because there are no inner and outer surfaces) and doesn't follow logically even if I granted you that there are such things. It's your own cardinality argument. Any continguous chunk of 3-space has the same cardinality as the unit interval on the line.

Thus on the assumption that the horn is filled with a finite amount of paint, understood to be proved, then the inner surface has been painted. Because the inner surface thus paintable, and the outer surface the same area, the outer surface must be paintable.

There are no inner or outer surfaces. But you haven't got an argument here even if there were. After all the interval []0,1] has the same cardinality as [0,2] which is twice as long. You already made that point earlier. I don't see that you've made an argument.

Is there an error?

At least half a dozen so far. But more to the point you haven't made an argument, only claims that you yourself refuted at the beginning by noting that every contiguous subset of 3-space has the same cardinality as the real line or as a finite segment of the real line.

Where is it?

Which of the many errors I pointed out do you disagree with? Starting from the false concept of inner and outer surfaces of 2-D shapes living in 3-space? Just think a sphere in 3-space or (easier) a circle in the plane. Name a point on the "inner" or "outer" surface? You can't. If you pick a point on the circle that's on the circle. But if the point is any positive distance whatsoever off the circle, it's not the inner or outer surface because there are points strictly between your point and the circle.

If it depends on an if, then the if is that the horn can be filled in the first place and the rest "flows" from that, so it seems.

This I didn't understand. But imagine filling the inside of a circle in the plane. If you tell me you're including the circle in your painted region, then the points on the circle are painted. But if not, then there's some positive distance between painted region and the circle, so that's not the "inner or outer" surface after all.

In the end perhaps casting this problem in terms of paint causes more problems than it solves. But just saying, "1/x is square-integrable but not integrable" (a point made earlier by @andrewk) doesn't have quite the same ring to it.
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That last question is key, it seems to me. The video, and others, tells me the volume in the horn is finite, is in fact π in appropriate units. That it is fillable, with π "mathematical paint" - whatever that means. I take it to mean analogously the same as when a paint can is filled. If a paint can is filled with paint, then it seems fair to say that the inner surface of the can is painted. Analogously if the horn can be filled, then whatever it has that passes for a surface is "mathematically" painted. Any problem with this so far?

The proposition of the paradox, as I get it from the video, is that the amount of "paint" is not enough to cover the outside of the horn because the area to be covered is infinite. But the question retains its edge: if the inside is covered by the "paint" inside, then why cannot the same volume of paint cover the outside? Is the area of the "outside" somehow different from the area "inside"?
• 1.8k
The video, and others, tells me the volume in the horn is finite, is in fact π in appropriate units.

Yes. It's a finite volume with infinite surface area. It's a veridical paradox: "A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless." In other words it's a true fact (provable in freshman calculus) that's so counterintuitive it seems impossible. But there is no actual paradox. There's no statement that's both true and false. Math is filled with these kinds of things.

That it is fillable, with π "mathematical paint" - whatever that means.

Whatever that means. It's a shape with finite volume and infinite surface area. That's all you can say. Yes it's "fillable with paint" if you want to view it that way. But if you push too hard on that visualization it gets confusing.

I take it to mean analogously the same as when a paint can is filled. If a paint can is filled with paint, then it seems fair to say that the inner surface of the can is painted.

I agree with that, about actual paint cans. But I don't agree with that about Gabriel's horn, because we're not filling anything with paint. And there is no inner surface. I went at this in detail in my previous post. You understand that for example a circle in the plane has no "inner surface." The horn is a two-dimensional surface in 3-space. It's not like a paint can. That's another source of bad intuition. A paint can has thickness, with an inner and outer surface. But the horn has no thickness and no inner or outer surfaces.

Analogously if the horn can be filled, then whatever it has that passes for a surface is "mathematically" painted. Any problem with this so far?

Well, to the extent that I accept that, and I sort of do, it's a veridical paradox. We have a finite volume that has infinite surface area, and it's a seeming common-sense paradox without being an actual logical contradiction.

The proposition of the paradox, as I get it from the video, is that the amount of "paint" is not enough to cover the outside of the horn because the area to be covered is infinite.

I didn't actually watch this particular video, I've seen this example in the past. But I disagree with that statement because we could paint the outside of the horn by using pi/2 gallons of paint on the segment between 1 and 2; and pi/4 gallons between 2 and 3; and pi/8 gallons between 3 and 4; and so forth. We'd use pi gallons of paint to cover the entire infinite surface area. Of course we can't paint it uniformly, unless we imagine paint of zero thickness, but then we can't sensibly have any meaningful volume of paint at all.

But the question retains its edge:

I agree with that. I saw this example years ago and it hasn't lost its force. It's a real puzzler. The best you can say is that it's a veridical paradox. There is no actual logical contradiction; only a violation of common sense. But the math is clear. The horn has finite volume and infinite surface area.

if the inside is covered by the "paint" inside, then why cannot the same volume of paint cover the outside? Is the area of the "outside" somehow different from the area "inside"?

I don't know the answer. It's a veridical paradox. How can an infinite surface area enclose a finite volume, or a finite volume have an infinite surface area? I don't know. It just does. I don't suppose that's satisfactory. That's why Numberphile got everyone talking about it. For what it's worth, Wiki has my solution and nothing better:

Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface. The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate (much like the series 1/2^N gets smaller fast enough that its sum is finite). In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.

https://en.wikipedia.org/wiki/Gabriel%27s_Horn

If you didn't find my explanation compelling you won't be satisfied by Wiki's identical explanation (except by virtue of Wiki's authority). And as I say, nobody on Wikipedia could come up with a better explanation. It's just one of those things in math that, as von Neumann said, you don't understand; you just get used to it.

ps -- Here's a tl;dr:

* It's just one of those things. It's counterintuitive but not a true paradox.

* The horn is two-dimensional whereas a paint can has thickness. That's throwing off our intuition. Paint cans have an inside and outside surface. The horn doesn't.
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There's another way to approach this puzzle. Whatever a Gabriel's horn is, it can be approximated as a cylinder.

Volume V of a cylinder = pi * r^2 * h

Surface area A of a cylinder = 2 * pi * r * h

for a cylinder with radius r and height h

Ratio of A to V = (2 * pi * r * h)/(pi * r^2 * h) = 2/r

As r approaches 0, V too approaches 0 but, oddly, A doesn't.
• 6.1k
The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate

That's good enough for me. "Mathematical" paint for a mathematical problem.
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If you would have listened to me in those other threads, where I explained the deficiencies in mathematical axioms, especially those which make what is mathematically indefinite into some thing definite, like the closure at the end of the infinitely long horn, you would have no problem with this issue.

It's quite obvious that the contradiction is in "infinitely long horn". If you pour paint in the top, and it is infinitely long, the paint never reaches the bottom. But if you assume the horn has a closure, a bottom, just like you might assume that .999... is 1, then the horn gets filled.

The film clip just demonstrates the inconsistencies in how "infinite" is dealt with by mathematical axioms. In one group of axioms, "infinite" is allowed to be a real mathematical object, so the infinite sutface requires an infinite amount of paint. But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure.
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But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure.

Bullpucky. There is no "closure at the bottom of the horn." You just make stuff up and claim mathematicians said it when they didn't. That's called a strawman argument.

like the closure at the end of the infinitely long horn

Bullpucky bullpucky bullpucky. YOU said that, not any mathematician, ever. As always you take your own mathematical ignorance and project it onto mathematics itself. You wield your ignorance like a weapon.

ps -- Let me not be so ill mannered. Perhaps you could explain to me your version of calculus in which the area of 1/x from 1 to infinity isn't infinite, and the area under 1/x^2 from 1 to infinity isn't finite. If you could elucidate your version of calculus then I'd be enlightened. I seek your wisdom. Or perhaps you reject calculus entirely. I'd like to hear your perspective on this.

Do you happen to understand that there is no "closure at the bottom" of the cone? That this is NOT anything that any mathematician says or thinks? That is is entirely something you made up and then claim to mock? A strawman, as they say. Do you apprehend this point?
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It's quite obvious
that there are none so blind as will not see. Present case, Which fishfry has made clear with this:
The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate....
. Which captures in a phrase what I could not convey in paragraphs. Mathematical paint. For a mathematical horn.

Math tell us this mathematical horn has a volume of π. Also that the area of the surface of the horn is infinite. Which π paint will cover, if it's mathematical paint. It seems, however, that you've been looking for this horn at the music store, and the paint at the paint store. Like a man looking to buy a real tool at a toy store. And you're free to not like it. But if you're going to take it on, then stop with the toys. Try math, because that is what it is all about, and only what it is all about.
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Do you happen to understand that there is no "closure at the bottom" of the cone?

One divided by infinity is not zero, it is indefinite. If you assume that one divided by infinity equals zero, you assume that the value for y reaches zero, therefore closure. It's very clearly stated in the YouTube video, he says we're taking the value of y to zero. However, this clearly contradicts the premise that the horn continues infinitely. The real issue is that integrals are approximations, and infinity has no place in an approximation. So that method of integration is simply not applicable to an infinitely long cylinder.
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One divided by infinity is not zero, it is indefinite.

Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity. I'm afraid this looks like yet another case of your misunderstanding the math, making up a story about what the math is saying, then arguing with your own story. A classic strawman argument.

But you know, you did say something mathematically correct, that one divided by infinity is not defined. I give you credit for that.

If you assume that one divided by infinity equals zero,

Which it doesn't. So if we assume a falsehood we can prove anything by Ex falso quodlibet". If 2 + 2 = 5 then I am the Pope.

you assume that the value for y reaches zero,

and I am the Pope, since your antecedent is false. Go in peace and sin no more. You can cos as much as you like.

therefore closure.

Likewise.

The cone or horn is NEVER closed. Pick any point on the real line and the cone is still open at the bottom. There is no "point at infinity" in this problem nor is there one in the real numbers. Again you misunderstand the math, make up a story, then argue against your own story. Your strawman has by now had all the straw beaten right out of it.

It's very clearly stated in the YouTube video, he says we're taking the value of y to zero.

Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.

The tl;dr here is that as x increases without bound, but remains at all times finite; y gets as close as you like -- "arbitrarily close" as they say -- to zero. x does NOT "become infinite" nor does y ever become zero. That's the math. It's well-known. They teach it to college freshman. These days they even teach it to high school students. Not that it's taught well at that level or that anyone understands it. I'll stipulate that nobody comes out of calculus class with a clear understanding of the logical fine points. But it's all on Wiki.

However, this clearly contradicts the premise that the horn continues infinitely.

Since your beliefs are false, you haven't got a contradiction; only confusion caused by your lack of mathematical knowledge. But now that I've explained it to you, you can no longer claim to be ignorant.

The real issue is that integrals are approximations,

Integrals are exact. The Riemann sums that define them are approximations, but the integrals are the limits of the Riemann sums and they are perfectly exact. The volume of the horn is exactly pi and the surface area is "infinite," which has a technical meaning: It's greater than any real number you could name. It's not the metaphysical infinite. I think that's a point of confusion in these conversations.

Integrals are perfectly exact, that's the takeaway.

and infinity has no place in an approximation.

Well they're not approximations. And secondly, "infinity" is not a magic thing that you should allow to cloud your mind. Infinity in this context is nothing more than a shorthand for "increases without bound." We say "x goes to infinity" meaning that x increases without bound, but is at every point finite. Mathematical infinity is not the metaphysical infinity. Perhaps that needs to be said more often. It's more of a technical gadget or shorthand for a quantity that increases without bound, or is larger than any finite number.

So that method of integration is simply not applicable to an infinitely long cylinder.

Of course it is. It's freshman calculus. It's been working well since the days of Archimedes, formalized into a systematic scientific tool in the late 17th century, and put on a logically rigorous footing since the late 19th century.
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Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.

It's not a matter of what is "meant by 'going to zero'", it's a question of what value is given in the calculation. Look at 6:25 in the video where he's doing the calculation, plugging the values into the formula. He says "well one over infinity that's zero, so you get nothing from that".

Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity.

Clearly you are using a different calculation than the one in the video then. If you know of a method to figure out the volume of that horn, which avoids rounding off the infinitely small dimeter to zero, then maybe you should present it for us.
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As r approaches 0, V too approaches 0 but, oddly, A doesn't

Nonsense :roll:

I am no mathematician.

I never liked math.

Probably best, then, to avoid topics like this one.
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Clearly you are using a different calculation than the one in the video then.

Try Wiki.

https://en.wikipedia.org/wiki/Gabriel%27s_Horn#Mathematical_definition

I can't speak for Numberphile, which I generally don't watch because the guy annoys me. Perhaps they were trying to simplify the fact that $\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$ and you were confused by their simplification. In which case it's on them and not on you. But now that I've explained it to you, it's up to you. Truly if they said that on Numberphile they shouldn't have exactly for this reason. It confuses people.

When we see the expression 1/infinity it's a shorthand for the limit and people generally know that. But if the video is intended (as it apparently is) for people who don't know that, they shouldn't have done it.

ps -- Ok I watched that section of the original video. That's a shorthand they do in integration problems, formally you'd replace it with a limit. I guess now I'd be in a position of trying to defend calculus pedagogy, that's hopeless. In general 1/infinity isn't defined but when it comes up in problems like this you can take it to be zero. I can't argue with you that there's a bit of flimflam to the whole enterprise. I can only say that the procedure can be made rigorous, but at the level of calculus problems, it rarely is. I don't expect that to be a satisfying answer, it's not to me either.

pps -- For integration problems we can consider ourselves working in the extended real numbers. These are the standard reals with special symbols $\pm \infty$.

One of the rules for these symbols is that if $a$ is a positive real number, the symbol $\frac{a}{\infty} = 0$.

That's actually the formally correct answer to your question and I probably should have thought of it earlier. The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.
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The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.

I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. Is the horn closed (limited), or is it infinite (unlimited). Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise. And that's why the appearance of a paradox arises.
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I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory.

No it's not. We have a logically rigorous theory of the calculus used in the example

Is the horn closed (limited), or is it infinite (unlimited).

It's infinite. But let me ask you this. Are you familiar with the graph of the function $y = \frac{1}{x}$? Isn't it infinite on the right? You can go out as far as you like, right? Did you need me to copy a picture from the web? No matter how far you go, like x = a zillion, there's a point on the graph at y = 1/zillion. Right? It's the cross-section of Gabriel's horn. We can do the integration to see that the area is infinite. And if you don't like calculus, there's a simple visual demonstration that I could provide.

Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to?

Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise.

You're equivocating the word limit, as in "limited," versus the mathematical theory of limits to infinity. A cheap rhetorical trick. How can you accept confusing yourself like this? Surely you know better. Or you could just ask.

And that's why the appearance of a paradox arises.

The apparent paradox arises because 1/x has an infinite integral and 1/x^2 has a finite one.
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Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to?

I'm objecting to the method employed by the person in the YouTube clip, which replaces the stated infinite limit, (approaches zero) with zero, as I referenced above. By that method, the equation for the volume of the horn resolves to pi, as stated in the op. If you have a method to figure out the volume of the horn without that substitution, then you might present it. If not, then we probably don't disagree, and we're just wasting our time talking past each other.

I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, and there is no paradox. This means that the amount of paint required to fill the horn cannot be determined. Therefore no act of pouring a determined amount of paint into the horn will fill it
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the volume is necessarily indefinite, rather than finite
Eh? The specification of a number between zero and one is indefinite, but certainly finite, yes? The proof in question says it's π. If you have found a problem with that, please make clear what it is.
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Check my reference above, 6:25 in the video, where one over infinity is taken to be zero. Otherwise you do not get pi.
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