> makes you go looking for some kind of loophole. But there isn't one.
Doesn't it require fractal cuts? Seems like a loophole to me. It only seems paradoxical because you assume the resulting pieces are smooth at some scale, like real cuts are.
That’s opening a can of worms. Fundamentally, the real number system is larger and weirder than “we” mostly think it is (and hence its extensions to higher dimensions). If you start trying to remove objects it easily becomes a game of whack-a-mole; some of your nice, intuitive definitions elsewhere become muddy. Colin alludes to this in the article (e.g. should all sets be measureable, etc.). If there were easy and satisfying fixes for this, it would have been sorted out long ago!
This isn’t just aesthetic. Although probability is one of the oldest areas of mathematical thought, dating back millennia, it took measure theory to put it on a really solid basis, several decades ago. These approaches are powerful and useful, but some of the corners are certainly counterintuitive.
Fractal is understating it. It requires non-constructive cuts -- if you limit yourself to cuts that can be constructed through any finitely-expressed process then the theorem does not hold.
Doesn't it require fractal cuts? Seems like a loophole to me. It only seems paradoxical because you assume the resulting pieces are smooth at some scale, like real cuts are.