Banach-Tarski paradox

Banach-Tarski paradox

[¦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).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

Banach-Tarski paradox

(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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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.
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.