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.Check my reference above, 6:25 in the video, where one over infinity is taken to be zero. Otherwise you do not get pi. — Metaphysician Undercover
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. — Metaphysician Undercover
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. — Metaphysician Undercover
I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, — Metaphysician Undercover
and there is no paradox. — Metaphysician Undercover
This means that the amount of paint required to fill the horn cannot be determined. — Metaphysician Undercover
Therefore no act of pouring a determined amount of paint into the horn will fill it — Metaphysician Undercover
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. — fishfry
Go back and read your own post. You just slipped up there. Not a mistake, just an error, a goof. — tim wood
Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself. — TheMadFool
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. — Metaphysician Undercover
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. — fishfry
Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile: — TheMadFool
We now return you to your normal programming. — fishfry
By the way, what's "normal programming"? Do you have one yourself? — TheMadFool
And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue. — TheMadFool
1∞=01∞=0 in the extended real number system, which is always the implicit domain of integration problems. — fishfry
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. — fishfry
Then what were you talking about? — TheMadFool
I feel like a drunk driver being asked to conduct a sobriety test on himself. — TheMadFool
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". — Metaphysician Undercover
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. — Metaphysician Undercover
Then what were you talking about? — TheMadFool
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.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. — Metaphysician Undercover
No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits. — fishfry
No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area... — InPitzotl
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. — InPitzotl
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.The shape is only said to have finite volume because of the method employed to determine the volume. — Metaphysician Undercover
The method makes no such assumption.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. — Metaphysician Undercover
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.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. — Metaphysician Undercover
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.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. — Metaphysician Undercover
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. — TheMadFool
hat's tricking you to use your intuitions of paint to compare a volume ("filling") to an area ("painting"). — InPitzotl
It depends on what you mean by paint.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? — tim wood
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.Because filling the inside means the inside is painted, — tim wood
So it seems to me no paradox at all, but just confusion on the meaning of terms, the paradox coming out of the confusion.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. — InPitzotl
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. — InPitzotl
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?I don't see how this is relevant. Are you forgetting that it's infinitely long? — Metaphysician Undercover
...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?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, — Metaphysician Undercover
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. — tim wood
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. — InPitzotl
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