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 – an apparent paradox. In fact, in a theoretical mathematical sense, a finite amount of paint can coat an infinite area, provided the thickness of the coat becomes vanishingly small "quickly enough" to compensate for the ever-expanding area, which in this case is forced to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant thickness of paint, no matter how thin, would require an infinite amount of paint.[2]