## The paradox of Gabriel's horn.

• 6.1k
Check my reference above, 6:25 in the video, where one over infinity is taken to be zero. Otherwise you do not get pi.
Sorry, this not allowed. Whether or not the fellow's remark was infelicitous - and it may have been - is of zero relevance. If you disagree with the mathematically derived answer, you have to provide a better answer on a better argument or you become just annoying noise.
• 1.8k
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 I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity. I could be wrong, I only looked at a couple of seconds of the vid. Either way though, I already explained that he's implicitly using the extended real number system as a shorthand for the cumbersome use of limits.

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.

That was already clear two years ago. But I'm surprised to see you still complaining about something I already explained to you -- that we're working in the extended reals -- and that you're rejecting freshman calculus. I must admit you're making progress ... from being confused about 2 + 2 = 4 to being confused about integrating 1/x.

I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite,

Nonsense.

This means that the amount of paint required to fill the horn cannot be determined.

So you don't agree with the determination of the volumes of solids of revolution. Ok. Whatever. An intellectual nihilist. Throw out all the engineering, all the physics, all the physical science simply because you aren't willing to take the time to understand it.

Therefore no act of pouring a determined amount of paint into the horn will fill it

Yeah yeah.

ps -- You know, if you said, "Modern calculus does get the right answers and it's useful for physics and engineering; but the mystery of the ultimate nature of infinitesimal quantities, whether mathematical or physical, is not satisfactorily addressed by the formalism," that would be an intelligent criticism.

But to deny the mathematical result of computing the volume of the solid of revolution ... that's just ignorance for its own sake.
• 8.6k
Nonsense

:lol: Why?
• 6.1k
Go back and read your own post. You just slipped up there. Not a mistake, just an error, a goof.
• 8.3k
If I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity.

The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.
• 8.6k
Go back and read your own post. You just slipped up there. Not a mistake, just an error, a goof.

Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself.
• 1.8k
Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself.

As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.
• 1.8k
The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.

$\frac{1}{\infty} = 0$ in the extended real number system, which is always the implicit domain of integration problems. The Numberphile guy didn't mention it since it's taken for granted: either glossed over, in the case of freshman calculus; or explicitly formalized, in the case of a more rigorous class in real analysis. Either way it's perfectly rigorous. We could live without it by substituting the phrase, "increases without bound" instead of infinity, and plugging in limits as needed. But that's far more messy and confusing than simply defining the extended real numbers as a notational shorthand.

https://en.wikipedia.org/wiki/Extended_real_number_line

And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi.
• 8.6k
As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.

Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile:
• 1.8k
Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile:

We now return you to your normal programming.
• 8.6k
We now return you to your normal programming.

:ok: :up: By the way, what's "normal programming"? Do you have one yourself? And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue.
• 1.8k
By the way, what's "normal programming"? Do you have one yourself?

Whatever my vatkeepers have scheduled for me today.

And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue.

I scrolled back and did not find the argument you're referring to, can you please repeat it?
• 8.3k
1∞=01∞=0 in the extended real number system, which is always the implicit domain of integration problems.

So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".

And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi.

The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.
• 8.6k
Then what were you talking about?
• 1.8k
Then what were you talking about?

Do you have a cat? Because someone in control of your account typed:

I feel like a drunk driver being asked to conduct a sobriety test on himself.

Not so? If it wasn't you, perhaps your cat used your keyboard while you were otherwise occupied.
• 8.6k
Thank you. Have a g'day.
• 1.8k
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".

No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits.

The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.

That's now how limits are defined. What's literally true here is that as $a$ gets arbitrarily large, the volume of the solid between 1 and $a$ gets arbitrarily close to $\pi$. Since that is the case, we say that the limit is $\pi$. That's the definition of a limit.

https://en.wikipedia.org/wiki/Limit_(mathematics)
• 1.8k
Then what were you talking about?

You mentioned sobriety checkpoints, and I went off-topic for a moment to express my ambivalence about them in my dual societal role as both a civil libertarian and a driver. Like the scene in Full Metal Jacket where Joker is confronted by a Colonel for wearing a peace sign and having "Born to Kill" written on his helmet, and he responds, "The duality of man. The Jungian thing Sir!" That was another little off-topic excursion.

https://rolandscivilwar.wordpress.com/2017/06/24/full-metal-jacket-pvt-jokers-born-to-killpeace-and-the-jungian-duality-of-man/

And now that you mention it ... perhaps that's what Gabriel's horn is ultimately about. A shape that reflects aspects of both the finite and the infinite. I think Jung would approve.
• 411
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox.
No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area; that those are two completely different kinds of things; and that our intuitions about paint "connect" the two. We just "know" that if we buy a gallon of paint and start painting the walls, we'll run out at some point and cover a certain area of the walls. But the reason that is, as already mentioned by @andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint.

You make it sound like the problem is that Gabriel's horn stretches out for an infinite while, but that's actually a red herring. You get the same exact problem with any shape of finite inner volume/infinite area, such as extruding a Koch snowflake and giving it a bottom. The only reason Gabriel's horn is noteworthy is that it's a curiosity with a built in toy homework exercise for people taking a calculus class.
• 8.6k
:up: :ok:
• 8.3k
No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits.

It's quite simple. The radius is y, which is represented at the one limit as y=1/x=0. From someone who adamantly argues that = means "the same as", I don't see that you have anything to argue. The diameter is taken to be 0 at that point in the solution. This means that the horn is closed at that point, represented within that method of figuring out the volume. There's nothing to discuss, it's clear and obvious.

The nature of "a limit" is irrelevant and just a ruse. What matters is the values that are plugged into the equation, the values which bring about the apparent paradoxical conclusion. If y=0 then the radius is zero at that point, which is what is employed in that method. If the premise is that the radius never reaches zero, then the solution does not truthfully adhere to the premise.

No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area...

The shape is only said to have finite volume because of the method employed to determine the volume. As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.

If we were to properly consider that infinite extension of the horn, then despite the fact that it is an infinitely small circumference, being an infinitely long extension of that circumference gives us an infinitely large volume. That volume could not ever be filled.
But the reason that is, as already mentioned by andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint.

Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.

If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed. Therefore if we are assuming that the paint can go around the tiny surface on the outside, it can also go around the tiny surface on the inside, because we are not discussing the physical properties which would prevent the thought experiment. If that were the case, we'd just reject the whole idea of an infinitely long horn as ridiculous in the first place.
• 411
The shape is only said to have finite volume because of the method employed to determine the volume.
The questioning of the shape's volume is only said to be problematic because the questioner thinks that the questioner knows what he is talking about.
As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.
The method makes no such assumption.
$\lim_\limits{x\rightarrow \infty}{f(x)}=L \Rightarrow \forall \epsilon \gt0, \exists M:x \gt M \Rightarrow L-\epsilon \leq f(x)\leq L+\epsilon$
In other words:
$\lim_\limits{x\rightarrow \infty}{f(x)}=0 \Rightarrow \forall \epsilon \gt0, \exists M:x \gt M \Rightarrow -\epsilon \leq f(x)\leq \epsilon$
Limits don't need points at infinity to work.
Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.
So? Areas have no volume. The paint analogy tricks you into thinking they at least relate when, in fact, they don't. A cubic foot of paint is roughly 7.5 gallons. A gallon of paint can paint about 400 square feet of wall; thus our cubic foot can paint about 3000 square feet of wall. Such paint has a specific layer width of 1/3000 feet. The intuitive equating of a particular area of paint to a particular volume requires such a specific nonzero layer; in the case of actual paint, 1/3000 feet thick layer.

But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000.
If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed.
The physical properties are implied by the intuitions, so they're imported by a back door; the paradox is always phrased about paint filling versus painting the horn... that's tricking you to use your intuitions of paint to compare a volume ("filling") to an area ("painting"). You don't need a shape extending into the infinite to make this paradox "work"; you just need a finite volume/infinite area, and to be tricked into thinking areas relate to volume. The actual thickness of the surface of Gabriel's horn is 0 units, so that infinite surface area actually doesn't consume any meaningful volume at all. Comparing the infinite surface area to the finite volume is simply a false comparison.
• 6.1k
Volume V of a cylinder = pi * r^2 * h
Surface area A of a cylinder = 2 * pi * r * h
....
As r approaches 0, V too approaches 0 but, oddly, A doesn't.

So you have V = r times whatever. As r goes to zero, V goes to zero.
And you have A = r times whatever. As r goes to zero, A goes to zero.
• 6.1k
hat's tricking you to use your intuitions of paint to compare a volume ("filling") to an area ("painting").

In terms of the so-called paradox, do you not agree that if the paint could fill the horn, then necessarily that same volume of paint paints the surface (boundary) of the horn? Because filling the inside means the inside is painted, and the outside is equivalent to the inside, and in the sense of being a mathematical boundary, identical.

But perhaps the question ought to be, even if you have "mathematical paint," does being able to fill the horn mean that the inner surface is painted? If the thing is fractally crenellated, for example, then the surface becomes unpaintable with anything less than infinity paint? Even though the volume is finite and the container closed?

In the real world, volume implies fillability as a possibility. Maybe with this mathematical horn, concepts of fillability have to be well-defined.
• 411
In terms of the so-called paradox, do you not agree that if the paint could fill the horn, then necessarily that same volume of paint paints the surface (boundary) of the horn?
It depends on what you mean by paint.
Because filling the inside means the inside is painted,
If filling the inside means the inside is painted, then there's no positive minimal thickness of paint required to paint things; i.e., paint can be 0 thick (quick proof; assume there is such a thickness t... then if you go out 1/t on the horn, and toss the finite part, you're left with a horn whose insides are too thin to paint). At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn.

If you just think of this as whether or not you can spread a drop of mathematical paint indefinitely thin, then I don't quite see any intuitive conflict to build a paradox from.
• 6.1k
At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn.

If you just think of this as whether or not you can spread a drop of mathematical paint indefinitely thin, then I don't quite see any intuitive conflict to build a paradox from.
So it seems to me no paradox at all, but just confusion on the meaning of terms, the paradox coming out of the confusion.

If it comes to mapping points to points, which is what "mathematical paint" seems to do, there is still the question of whether such a paint can fill any (mathematically defined) container that has a finite volume. As noted above, a container whose inside boundary is a fractal appears to have an unbounded surface area. Does the paint fill it and paint the inside? I see no clear single intuitive answer.
• 8.3k
But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000.

I don't see how this is relevant. Are you forgetting that it's infinitely long? It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity, and therefore enough paint to make an infinite number of those "less than layers worth" layers on the inside of that horn..
• 411
I don't see how this is relevant. Are you forgetting that it's infinitely long?
You've got this backwards. I have 400 square feet of wall coated with 1/3000th of a foot of paint. How much paint is that? Well, given that the entire area is covered with a 1/3000th foot layer, we can just multiply 400 by 1/3000 and we get about 1/7.5. Now what's this you were saying about the horn?
It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity,
...ah, yes. Tiny bit of a problem you have there, though. It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies. The rest of the infinite horn (the infinite portion) is too small to have an inner layer that thick. You can't just multiply your paint's thickness by infinity if it isn't covering the infinite area. So where are you getting this crazy notion that you can multiply your layer's thickness by infinity?
• 8.6k
So you have V = r times whatever. As r goes to zero, V goes to zero.
And you have A = r times whatever. As r goes to zero, A goes to zero.

What about the ratio between A (surface area of the cylinder) and V (the volume of the cylinder)?

A = 2 * pi * r * h

V = pi * r^2 * h

A / V = 2 / r ??!!

One way of making sense of this is as below:

2 * pi * r = C = Circumference of a circle with radius r [1 dimensional object]

pi * r^2 = = E = Area of a circle with radius r [2 dimensional object]

A = C * h

V = E * h

As r -> 0, the 2 dimensional object (E) collapses to a point and subsequently, all objects based on E, like the 3 dimensional object V, vanishes.

But as r -> 0, there's still h to deal with; the cylinder becomes a 1 dimensional object, a line with length h to be specific.
• 8.3k
It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies.

The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place.
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal