But the surface area of the horn, itself being infinite, cannot be painted with a finite amount of paint. — tim wood
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?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. — tim wood
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.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. — tim wood
It would seem, then, that if we want to paint the interior surface, we need only pour in π amount of paint — tim wood
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.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: — fishfry
Why? Why cannot it be a consistent thickness, mathematically thin? — tim wood
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. — tim wood
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. — tim wood
Let's consider the cardinality of the points that make up the inner and outer surfaces of the horn respectively. — tim wood
It seems to me it must be the same for both. — tim wood
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. — tim wood
It seems to me the cardinality of the paint must be greater than or equal to that of one of the surfaces. — tim wood
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. — tim wood
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. — tim wood
Is there an error? — tim wood
Where is it? — tim wood
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. — tim wood
The video, and others, tells me the volume in the horn is finite, is in fact π in appropriate units. — tim wood
That it is fillable, with π "mathematical paint" - whatever that means. — tim wood
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. — tim wood
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? — tim wood
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. — tim wood
But the question retains its edge: — tim wood
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"? — tim wood
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.
But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure. — Metaphysician Undercover
like the closure at the end of the infinitely long horn — Metaphysician Undercover
that there are none so blind as will not see. Present case, Which fishfry has made clear with this:It's quite obvious — Metaphysician Undercover
. Which captures in a phrase what I could not convey in paragraphs. Mathematical paint. For a mathematical horn.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....
Do you happen to understand that there is no "closure at the bottom" of the cone? — fishfry
One divided by infinity is not zero, it is indefinite. — Metaphysician Undercover
If you assume that one divided by infinity equals zero, — Metaphysician Undercover
you assume that the value for y reaches zero, — Metaphysician Undercover
therefore closure. — Metaphysician Undercover
It's very clearly stated in the YouTube video, he says we're taking the value of y to zero. — Metaphysician Undercover
However, this clearly contradicts the premise that the horn continues infinitely. — Metaphysician Undercover
The real issue is that integrals are approximations, — Metaphysician Undercover
and infinity has no place in an approximation. — Metaphysician Undercover
So that method of integration is simply not applicable to an infinitely long cylinder. — Metaphysician Undercover
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. — fishfry
Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity. — fishfry
Clearly you are using a different calculation than the one in the video then. — Metaphysician Undercover
The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity. — fishfry
I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. — Metaphysician Undercover
Is the horn closed (limited), or is it infinite (unlimited). — Metaphysician Undercover
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. — Metaphysician Undercover
And that's why the appearance of a paradox arises. — Metaphysician Undercover
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? — fishfry
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.the volume is necessarily indefinite, rather than finite — Metaphysician Undercover
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