The German mathematician Georg Cantor (1845-1918) began a revolution in mathematics when invented the theory of sets in the 1870s. The parts of his theory dealing with infinite sets were the most controversial at that time. In an 1874 article, Cantor suggested that there were different kinds of infinite sets and different orders of infinity. This idea stirred up tremendous controversy among some editors of the journal and among mathematicians in general. Prior to this publication, all infinite sets had been considered alike. No one had ever suggested that there were different orders of infinity. By 1890 most of the furor over Cantor's ideas had settled down, and many mathematicians saw Cantor's set theory as an ideal foundation for the rest of mathematics. In particular, Gottlob Frege (1848-1925) used elements of Cantor's theory in producing a formal system of mathematics purportedly derived from logic. Unfortunately, during...