But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error. — InPitzotl

Most of the area is on the "needle" portion but most of the volume is enclosed by the "horn" portion. I find it odd that you're focusing on the needle portion. The typical area and volume calculations for GH are based on limits as a approaches infinity (not the other way around) - in other words, as the horn gets longer and longer. But sure...

We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube. — InPitzotl

You lost me here. Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle?

Already had that thread with MU. — InPitzotl

Fair enough.

1/x cannot be less than 0, whereas it can be less than any other positive real number. — InPitzotl

Consider the Stern-Brocot tree. If L=1/2, LL=1/3, LLL=1/4, and so on, then L_repeated is a "real number" which it cannot be less than. I suspect you'll argue that L_repeated = 0, which brings us to the classical debate of whether 0.9_repeated=1. I would argue that 0.9_repeated describes a potentially infinite process and is not a number in the same sense that 1 is (dare I say that 0.9_repeated is not a rational number), but that's a debate for another time. In any case, consider this: if a number at the 'bottom' of the Stern-Brocot tree is equivalent to both of its neighbors, are any of the real numbers actually distinct from each other?

What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. — Metaphysician Undercover

Given that y continually approaches 0 as x increases, the limit is 0. We don't need a point at (∞,0) for the limit to be equal to 0. I think the problem is that your definition of limit doesn't match the standard definition. With that said, I sympathize with your view and I think your argument would be stronger if you focused on the limit used to calculate the volume of GH. Such a limit is a whole different beast since it converges to pi - an irrational 'number'. Just because the interval corresponding to V can get arbitrarily small as a approaches infinity, it doesn't necessarily mean that V has a definite value. Just as it is impossible to explicitly compute all decimal digits of pi, the best anyone can do here is either (1) provide a small interval for V or better yet (2) leave V in algorithmic form (i.e. pick your favorite formula for pi and don't bother to compute it).

But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?" — InPitzotl

You are essentially saying that it takes <3.15 ft3 to paint the finite horn AND it takes >3.15 ft3 to paint the finite horn. If film thickness doesn't explain the apparent contradiction, then I would conclude that your problem definition is invalid. Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall. I think focusing on physical paint is a distraction.

I don't see the original paradox; a square foot of area has no meaningful volume. — InPitzotl

I agree that an area has no volume, so a single drop of mathematical paint could paint any surface of arbitrarily large size. However, we are not justified to claim that it can paint a surface of actually infinite area. That requires a leap of though which we are not in a position to make.

we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 gallons of paint, but takes over 1000 gallons to paint the outside. — InPitzotl

I don't see the paradox - as long as we're allowed to play with the thickness we can come up any combination of numbers for the finite horns. What's important is that the numbers in your example are all finite.

The best anyone can do to offer a measure of the volume — Ryan O'Connor

Let me clarify this...there's something better than giving a numerical measure....the best we can do is to stick with an algorithm which defines a potentially infinite process. And that's how we define pi.

Because it cannot be measured. That's what infinite means. — Metaphysician Undercover

The surface area of the horn has no limit - as we consider larger and larger horns that area continues to increase, so we can loosely say that it has 'infinite' area. But as MU notes, I think it's more appropriate to say that we cannot measure the area (or I would argue that the horn simply doesn't exist). What's paradoxical is that the volume does have a limit. But that's not to say that the horn has a definite volume. We cannot complete the potentially infinite process associated with the limit any more than we can explicitly list all decimal digits of pi, so we cannot claim that it has a definite volume. The best anyone can do to offer a measure of the volume is to prematurely terminate the potentially infinite process and output a rational approximation of the volume.

As everyone on this thread knows, the reason why calculus was reformulated based on limits was to avoid talking about infinitesimals (and actual infinity). Instead of actually computing the volume of an infinitely long horn, we break the calculation down into a potentially infinite process, where we compute the volumes of an endless sequence of horns of increasing length (all of which have a finite volume - and so there is nothing paradoxical about any of these horns). The problem is that we're not interested in any of these horns, we're interested in the horn of actually infinite length. And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite. I'm inclined to believe that Gabriel's Horn doesn't exist any more than the "number" 1/∞.