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The Banach-Tarski paradox Stan Wagon
Publisher: Cambridge University Press

By Stan Wagon Publisher: Cambridge University Press. If you put many dots in a sphere, you may find infinite pattern. One of the interesting features of this branch of mathematics is the 'Axiom of choice' which to the beginner can seem to be the intuitive way to go, until they come to the 'Banach-Tarski paradox'. The common axioms of set theory lead to the troubles such as the Banach-Tarski Paradox (http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). @charm Simmons: The Banach-Tarski paradox explains how to disassemble a solid sphere into a finite number of congruent parts, then reassemble those same parts into two solid spheres of the same size as the original. The Banach-Tarski paradox : PDF eBook Download. Q.3), The Banach Tarski Paradox: Can a sphere be cut into a finite number of pieces, and then reassembled into two spheres that are identical to the original? February 26th, 2013 reviewer Leave a comment Go to comments. There are two spheres, X=(1,2,3,4,・・・・・∞) and Y=(0,1,2,3,4,・・・・・∞). In 1924, Stefan Banach and Alfred Tarski Published an article in FundamentaMathematicae, entitles, “Sur la decomposition des ensembles de points en partiesrespectivement congruents” (Feferman and Solomon 43). Your dots would be smaller and smaller in it. Much of it is not useful at all, like the Banach-Tarski paradox mentioned by Murphy or Fermat´s last theorem.