Gabriel's horn is not a paradox.

 Gabriel's horn is not a paradox.

The "Gabriel's horn paradox" (sometimes called "the painter's paradox") is called a paradox because the result feels contradictory. I think I can explain it in a way that is intuitive; the feeling of contradiction will disappear.

This post assumes you already have an understanding of the paradox and the proof that Gabriel's horn has a finite volume and an infinite surface area. If not, then see https://tomrocksmaths.com/wp-content/uploads/2022/05/gabriels-horn.pdf for a written description, or https://www.youtube.com/watch?v=yZOi9HH5ueU for a video treatment.

The real problem is that the paradox description contains a mistake.

"Make a horn shape by the rotation around the X axis of the curve y=1/x from x=1 to x=infinity. The volume of this horn is finite, and so can be filled with a finite amount of paint. But the surface area of the horn is infinite, and therefore requires an infinite amount of paint. How can painting it require more paint than filling it?"

The mistake is in the claim that painting it requires infinite paint.

Let's say I have a bucket of paint. I can paint a wall with it. How thick is the layer of paint? Depends on how heavily I painted it on. Maybe 1mm?

So now I'm asked to paint a wall that's twice as long but use the same amount of paint. No problem, I just lay down a thinner layer - 0.5 mm.

Double the length again, halve the thickness, and the same bucket paints that even larger wall.

Now there is no such thing as infinitely big things, but we like to talk about them in math. We generally talk about infinities in terms of a limit. I.e. what does an equation do as you "approach" infinity. Take the sequence:

sum1 = 1/2 + 1/4 + 1/8 + 1/16 ...

For a finite number of terms, the value of sum1 approaches 1. We say that the *infinite* series converges to exactly 1. In contrast, consider the sequence:

sum2 = 1/2 + 1/3 + 1/4 + 1/5 ...

For a finite number of terms, the value of sum2 does not approach any particular value. As you add more terms, the value will grow without an upper limit. We say that the infinite series diverges to infinity.

These concepts of converging and diverging infinite series are why Gabriel's horn can have both finite volume and infinite surface area. (See the above links for formal proofs.)

My point is that you can intuitively see that a finite amount of paint can cover a surface that approaches infinity, simply by reducing the thickness of the paint layer to approach zero. A bucket of paint can both fill the horn and paint it.

The reason the original description feels like a contradiction is that in real life, you can't have an arbitrarily thin layer of paint. There's a minimum practical layer thickness, and therefore a maximum practical area of coverage. But this is a math puzzle, where practical matters don't intrude. We can have values that approach infinity and approach zero, no problem.

Switching from Painting to Filling

Here's an interesting thought experiment. Imagine having the horn in front of you, extending downward towards infinity. We have to imagine that gravity is constantly downward for the entire length of the horn, which does not correspond to anything our universe could do, but work with me here.

Now imagine I have my bucket of paint and I let one drop fall into the horn. Let's imagine it doesn't actually stick to the surface of the horn and simply slides down the side of the horn. At some point, the diameter of the horn will equal the diameter of the drop, so the drop will continue falling, but will reshape itself into a thin thread of paint, falling down this ever-narrowing horn. As it continues to fall, the diameter will continue to decrease, meaning the length will continue to increase as it stretches out. So the front of the paint thread will be moving down faster than the back of the thread. But both will continue moving down.

Forever.

Here's the interesting thing: as time tends toward infinity, the front of the paint thread will keep traveling down the horn toward infinity. But the rear will travel more and more slowly, approaching a "convergence" point that it won't ever cross. This is because the original paint drop has a finite volume, and that volume will exactly fill a section of the horn starting at that convergence point and continuing to infinity. 

If you speed up time, you would see the front of the paint thread rocketing down the horn and the back of the paint thread slowing down and appearing to stop.

I find that thought experiment fascinating.