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Abstract: The Banach-Tarski paradox is an example of the hypocrisy of intuition. It says, if we assume the axiom of choice, then given a ball in R^n for n>2 of finite nonzero radius r we can cut this ball into finitely many sets, move the sets around, glue them back together and have a ball of radius 2r. Even better, we only move the pieces by rotations (this is the Algebra Dance part). Why does this mean our intuition is hypocritical you may ask? It's because of the axiom of choice. The axiom of choice says if C is any collection of nonempty sets, then we can choose a member from each set in that collection to form a new set B so that the intersection of B with any set in C is nonempty. The axiom of choice is equivalent to the following statements: If A and B are any two sets then either |A| <= |B| or |B| <= |A|. Any vector space over a field F has a basis. Any product of compact topological spaces is compact. |