In this abstract setting, it is possible to have subspaces without point but still nonempty. Īs proved independently by Leroy and Simpson, the Banach–Tarski paradox does not violate volumes if one works with locales rather than topological spaces. It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices. Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The theorem is called a paradox because it contradicts basic geometric intuition. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the " pea and the Sun paradox". Īn alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. The reconstruction can work with as few as five pieces. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. "Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?"
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