Gabriel's Horn was cool till someone pointed out to me that you can have a line of infinite length within a square (trivially).
When comparing something of a certain dimension with something of a higher dimension, it's not at all surprising that the lower one can be infinite and the higher one finite.
Usually it's phrased as "a finite amount of paint can paint an infinite area." But why do I need the Horn to realize this? It works in the Horn only if there is no lower limit to the thickness of paint. If you accept that, then I can take a drop and paint an infinite plane with it. Why do I need the Horn to demonstrate this?
The way I'd heard the paint comment was along the lines that "Gabriel's Horn can hold only a finite quantity of paint, but requires an infinite quantity of paint to cover the surface".
So if you think of it as a bucket that can't hold enough paint to cover itself, that is at least a little surprising.
But that's exactly my paint. If you allow for infinitely thin paint, then a finite volume of paint can always cover an infinite surface - you don't need Gabriel's Horn to show that.
If you don't allow for infinitely thin paint, then no - Gabriel's Horn surface cannot be painted even with an infinite amount of paint.
It's not supposed to be especially tricky. Yes, as soon as you realise that the horn is a bucket with a very deep section that is narrower than the assumed thickness of a coat of paint, the surprise dissipates.
I believe the point of the exercise is an introduction to curves with infinite length but finite area under them, in order to expand one's intuition about such objects, which is then transferable to other examples like space-filling curves.
When comparing something of a certain dimension with something of a higher dimension, it's not at all surprising that the lower one can be infinite and the higher one finite.
Usually it's phrased as "a finite amount of paint can paint an infinite area." But why do I need the Horn to realize this? It works in the Horn only if there is no lower limit to the thickness of paint. If you accept that, then I can take a drop and paint an infinite plane with it. Why do I need the Horn to demonstrate this?