Tue, 11 Nov 2008
Presumably people think it's paradoxical that the thing should have a finite volume but an infinite surface area. But since the horn is infinite in extent, the infinite surface area should be no surprise.
The surprise, if there is one, should be that an infinite object might contain a merely finite volume. But we swallowed that gnat a long time ago, when we noticed that the infinitely wide series of bars below covers only a finite area when they are stacked up as on the right.
The pedigree for that paradox goes at least back to Zeno, so perhaps Gabriel's Horn merely shows that there is still some life in it, even after 2,400 years.
[ Addendum 2014-07-03: I have just learned that this same analogy was
also described in this
math.stackexchange post of 2010. ]