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Banach tarski paradox

Banach Tarski Paradox

Banach-Tarski Paradox is a mathematical theorem used in set theoretic geometry. This theorem was stated with an example of solid sphere. This paradox doesn't talk about the real physical sphere,but it deals with the geometrical spheres.

The theorem states that a solid sphere has been cut into a finite number of non over lapping pieces. With the obtained pieces, rearrange it to form a two identical sphere as like the original sphere, without stretching or expanding the pieces in an way. Here Mass is created out of nothing.

In simple terms, can a ball be decomposed into a finite number of point sets and re grouped into two balls identical to the original?

This theorem can be told in a informal manner. When the pea is cut into a finite number of pieces, it can be resembled in the sun. So it is called Pea and Sun paradox.

This theorem is stated as Paradox because it contradicts the basic geometric intuition. Because it is impossible to preserve the volume of the sphere when it is doubled in the size. Unlike other theorems in geometry, this can be proved only by using the Axioms of Choices, which allows for the construction of non-measurable sets.

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