[¦bä‚näk ¦tär·skē ′par·ə‚däks]
(mathematics)
A theorem stating that, for any two bounded sets, with interior points in a Euclidean space of dimension at least three, one of the sets can be disassembled into a finite number of pieces and reassembled to form the other set by moving the pieces with rigid motions (translations and rotations).

(mathematics)
It is possible to cut a solid ball into finitely many pieces (actually about half a dozen), and then put the pieces together again to get two solid balls, each the same size as the original.

This paradox is a consequence of the Axiom of Choice.
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References in periodicals archive ?
Readers will painlessly learn about concepts such as first-order and second-order logic, the Banach-Tarski paradox, the Von Neumann universe, G|delAEs incompleteness theorem, and the Cantor-Bernstein theorem.
The Banach-Tarski paradox, string theory, Klein bottles, and universes where time runs in reverse are some subjects explored.
Far more mind-blowing is a mathematical result known as the Banach-Tarski paradox after two Polish mathematicians, Stefan Banach and Alfred Tarski.
Grit or Gunk: Implications of the Banach-Tarski Paradox, PETER FORREST
His argument relies on the Banach-Tarski paradox, namely that given the orthodoxy and the Axiom of Choice it can be shown that a spherical region of unit radius is the sum of rive regions each congruent to rive regions whose sure is two spheres of unit radius.
Pawiikowski (1991), The Hahn-Banach theorem implies the Banach-Tarski paradox. Fundamenta Mathematicae l38, 21-22.
However, even by the time he came to the US, Tarski was already established as a master in such matters as the Banach-Tarski Paradox (in which a sphere of any size can be cut up into a finite number of pieces and re-assembled into a sphere of any other size) and his advances in logic and set theory.

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