The '''Banach-Tarski paradox''' is that statement that a sphere can be cut into 5 [[Non-measurable set|non-measurable sets of points]], which can then be reassembled into two spheres of equal volume; however, this cannot be done with an actual object. It is called a paradox because it violates our physical intuition that this is impossible. It is very similar to the [[Vitali set]], which was developed to illustrate possible flaws in using the [[Axiom of Choice]] (an element of [[Zermelo-Fraenkel]] set theory) in order to prove mathematical claims. Since the Vitali set is less easily discussed with a layman, the Banach-Tarski paradox is often used to illustrate the "weirdness" of the Axiom of Choice. | The '''Banach-Tarski paradox''' is that statement that a sphere can be cut into 5 [[Non-measurable set|non-measurable sets of points]], which can then be reassembled into two spheres of equal volume; however, this cannot be done with an actual object. It is called a paradox because it violates our physical intuition that this is impossible. It is very similar to the [[Vitali set]], which was developed to illustrate possible flaws in using the [[Axiom of Choice]] (an element of [[Zermelo-Fraenkel]] set theory) in order to prove mathematical claims. Since the Vitali set is less easily discussed with a layman, the Banach-Tarski paradox is often used to illustrate the "weirdness" of the Axiom of Choice. |