Around the turn of the century, mathematicians such as Emile Borel and Henri Lebesgue were looking for a precise definition of the measure of a subset of the real n-dimensional space Rⁿ. Four considerations bounded their search. First, the measure of any 'normal' object such as a line segment in R¹, a square in R², or a cube in R³ should be equal to its length, area, or volume respectively. Second, if A and B are disjoint subsets then the measure of A union B should be equal to the measure of A plus the measure of B. Third, if A and B are congruent sets then their measures should be equal. Fourth, it should be possibly to measure 'most' sets or at least all 'nice' ones. In 1924, Stefan Banach and Alfred Tarski dramatically demonstrated that not all subsets could be measurable if...