I. Georg Cantor

Georg Cantor founded set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series and was the first to prove the nondenumerability of the real numbers. Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845. His family stayed in Russia for eleven years until the father's sickly health forced them to move to the more acceptable environment of Frankfurt, Germany, the place where Georg would spend the rest of his life. Georg excelled in mathematics. His father saw this gift and tried to push his son into the more profitable but less challenging field of engineering. Georg was not at all happy about this idea but he lacked the courage to stand up to his father and relented. However, after several years of training, he became so fed up with the idea that he mustered up the courage to beg his father to become a mathematician. Finally, just before entering college, his father let Georg study mathematics. In 1862, Georg Cantor entered the University of Zurich only to transfer the next year to the University of Berlin after his father's death. At Berlin he studied mathematics, philosophy and physics. There he studied under some of the greatest mathematicians of the day including Kronecker and Weierstrass. After receiving his doctorate in 1867 from Berlin, he was unable to find good employment and was forced to accept a position as an unpaid lecturer and later as an assistant professor at the University of Halle in1869. In 1874, he married and had six children. It was in that same year of 1874 that Cantor published his first paper on the theory of sets. While studying a problem in analysis, he had dug deeply into its foundations, especially sets and infinite sets. What he found baffled him. In a series of papers from 1874 to 1897, he was able to prove that the set of integers had an equal number of members as the set of even numbers, squares, cubes, and roots to equations; that the number of points in a line segment is equal to the number of points in an infinite line, a plane and all mathematical space; and that the number of transcendental numbers, values such as pi(3.14159) and e(2.71828) that can never be the solution to any algebraic equation, were much larger than the number of integers. Before in mathematics, infinity had been a sacred subject. Previously, Gauss had stated that infinity should only be used as a way of speaking and not as a mathematical value. Most mathematicians followed his advice and stayed away. However, Cantor would not leave it alone. He considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite numbers. By considering the infinite sets with a transfinite number of members, Cantor was able to come up his amazing discoveries. For his work, he was promoted to full professorship in 1879. However, his new ideas also gained him numerous enemies. Many mathematicians just would not accept his groundbreaking ideas that shattered their safe world of mathematics. One of these critics was Leopold Kronecker. Kronecker was a firm believer that the only numbers were integers and that negatives, fractions, imaginaries and especially irrational numbers had no business in mathematics. He simply could not handle actual infinity. Using his prestige as a professor at the University of Berlin, he did all he could to suppress Cantor's ideas and ruin his life. Among other things, he delayed or suppressed completely Cantor's and his followers' publications, belittled his ideas in front of his students and blocked Cantor's life ambition of gaining a position at the prestigious University of Berlin. Not all mathematicians were hostile to Cantor's ideas. Some greats such as Karl Weierstrass, and long-time friend Richard Dedekind supported...