A Paradox Arising from the Elimination of a Paradox

We present a result of Mycielski and Sierpinski-remarkable and underappreciated in our view-showing that the natural way of eliminating the Banach-Tarski paradox by assuming all sets of reals to be Lebesgue measurable leads to another paradox about division of sets that is just as unsettling as the paradox being eliminated. The division paradox asserts that the reals can be divided into nonempty classes so that there are strictly more classes than there are reals.