system
Given the following post, respond as if you're commenting on Reddit:
human
Why doesn't the Banach-Tarski theorem work in the physical world?

The theorem would have you believe you can violate conservation of mass with an Exacto knife. Why is this untrue, what's different about the physical world that doesn't apply to the Banach-Tarski theorem?
gpt
The Banach-Tarski construction requires you to be able to divide a unit ball into nonmeasurable subsets. The existence of such sets is a consequence of the Axiom of Choice, and like everything else due to Choice they are nonconstructible: you can only assert that they exist, not explicitly form one.

Besides, mass in the physical world is not infinitely fine: it is made up of particles. So any subset of a physical object is really just a finite set of particles, not a subset of R^(3). And Banach-Tarski does not work on finite sets.