[crimsonalucard]

The paradox is not the convergence of infinite long curves..

Read the paradox description. To fill the horn with paint you need a finite amount. To paint the outside of the horn you need an infinite amount.

That is the paradox. The surface area of the inside is equal to the outside, yet one side requires a finite amount of paint while the other side requires an infinite amount.

At least that's how I understand the paradox.

[om2]

That's not really the paradox. The amount of paint required depends on the thickness of the coat, and the inside is forced to get a thinner and thinner coat (since the space available gets thinner) while the outside is assumed to be painted with an even thickness of paint. If you painted the outside with a layer of paint whose thickness is proportional to the thickness of the curve, you could do it with a finite amount. (This is all assuming idea "paint" that is continuous and arbitrarily subdivisible, real paint is made of molecules.)

The intended "paradox" is that the surface area is infinite but the volume is finite.

[xerophyte12932]

From the article:

>Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its INNER surface

[gohrt]

It's the same thing. Imagine that you need one blob of paint for every tick-mark on the real axis. Infinite blobs of paint. (The resolution, in both scenarios, is that the size of a blob is not constant, and goes to infinitesimal)