Splitting a Pea into Two: The Mind-Bending Banach-Tarski Paradox

Proven by Stefan Banach and Alfred Tarski in 1924, this paradox relies on the controversial axiom of choice, a fundamental principle in set theory. It states that a three-dimensional ball can be decomposed into a finite number of non-overlapping subsets, which can then be reassembled by rigid motions (translations and rotations) to form two identical copies of the original ball. Crucially, the 'pieces' involved are not solid chunks like those from a physical object, but rather highly abstract, non-measurable sets, rendering a physical demonstration impossible and highlighting the counter-intuitive nature of infinity.