Leonard M. Wapner

Delightful insight into the Banach/Tarski so called duplication of a set of points inside a sphere. One of the key pieces in the proof is the axiom of choice, which is discussed briefly. Another large part of the proof is transfinite mathematics, which always leads to astoundingly counterintuitive conclusions. Since the domain of the problem is the R (the real numbers), the set included in the sphere contains elements that are both rational and irrational numbers. The rational numbers make a countable set of numbers while the irrationals are not. Georg Cantor proved this in the late 19th century. The properties of both sets are wonderfully perverse--great fun. Might I suggest Rudy Rucker's White Light?
SPOILER ALERT====>

So, the proof rests on the bizarre properties of transfinite sets.
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