Dirk: The paradox sits squarely on the axioms we give to set theory—particularly the axiom of choice. If you accept that axiom, you can construct non-measurable sets that defy our usual intuition about volume. From a purely logical standpoint, the construction is airtight: you partition the sphere into finitely many pieces, then use rotations that are not commensurable with any rational angles to reassemble them. The steps are all valid within the framework of ZFC. The intuition fails because we normally think of "volume" as a continuous, additive measure that must be preserved under rigid motions. The key is that the pieces you’re moving around are not measurable in the Lebesgue sense, so the usual theorems about invariance don’t apply. In short, the paradox doesn’t break any mathematical rule; it simply exploits a loophole in our definition of measurable sets. If you’re looking for an explanation that satisfies both intuition and strict analysis, you have to give up either the axiom of choice or your expectation that “reasonable” sets behave well under decomposition. Either way, the paradox remains a fascinating illustration of how expanding the set-theoretic universe can upend classical geometry.

Dirk: I’d call it a very tidy proof, even if the proof itself is tidy but the object it constructs is not. If you want a mirror that doesn’t shatter, you have to stop using the axiom of choice. The paradox is a logical consequence of a choice principle that lets you build pathological sets. So either you accept those sets and the paradox, or you reject the principle and your universe of sets gets simpler. No clever trick will reconcile the two; the choice is the real issue.

Dirk: I’ll say it straight: the prism is exactly what choice does— it cuts the universe into pieces that don’t recombine nicely under volume. If you want the glass to catch light, you let the choice shine. If you want a single, smooth sphere, you don’t. Both sides are logically consistent; it’s just a matter of which axioms you like to keep in your toolbox.

Dirk: So whether you slice or not, the crystal’s outline is still defined by the same set of rules. The only difference is whether you see the pieces or the whole. The choice of rule determines the view, not the underlying physics.