Q&A: Gabriel’s Horn
Gabriel’s Horn
Question
I read the Wikipedia entry on Gabriel’s Horn
https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A4%D7%A8_%D7%92%D7%91%D7%A8%D7%99%D7%90%D7%9C
The solution they give to the painting paradox seems problematic to me. The fact that you can’t divide paint particles infinitely is a physical property of our world, but the paradox is at the abstract conceptual level. One can describe a world in which paint particles can be divided infinitely, and the paradox would still remain. I would be glad to hear your opinion. Thanks.
Answer
I’m not familiar with it, and I don’t have time to get into the details. From what I understood from a quick reading, the paradox begins with an object whose volume is finite but whose surface area is infinite. The solution they suggest is that the paradox exists in physics, not in mathematics. From a mathematical standpoint, there is no problem at all with an infinite area enclosing a finite volume; it’s only a question of physical painting.
But it seems to me that you’re right that this is not connected to the molecular (discrete) structure of the paint. Even if it were continuous, in my opinion there should not be a contradiction. But it requires more thought.