Georg Cantor Biography

Georg Cantor, a German mathematician, developed a number of ideas that profoundly influenced 20th-century mathematics. Among other accomplishments, he introduced the idea of a completed infinity, an innovation that earned him recognition as the founder and creator of set theory. His revolutionary insights, however, were accepted only gradually and not without opposition during his lifetime. The praise for his work was best epitomized by the famous mathematician David Hilbert, who said that "Cantor has created a paradise from which no one shall expel us." Besides being the founder of set theory, Cantor also made significant contributions to classical analysis. In addition, he did innovative work on real numbers and was the first to define irrational numbers by sequences of rational numbers.

Georg Ferdinand Ludwig Philipp Cantor was born on March 3, 1845, in St. Petersburg, Russia, the first child of Georg Woldemar Cantor and Maria Böhm. The family moved to Frankfurt, Germany, in 1856 when the father became ill. His father, born in Copenhagen, had moved to St. Petersburg at a young age and had become a successful stockbroker there. His mother came from an artistic family. Cantor's brother Constantin was an accomplished piano player and his sister Sophie had drawing talents. Cantor himself sometimes expressed regret that he had not become a violinist. Of Jewish descent on both sides, Cantor was nevertheless raised in an intensely Christian atmosphere. The breadth and depth of his knowledge of the old masters, theologians, and philosophers was brought about by his religious upbringing and became evident in his more philosophical writings.

At a young age, while still in St. Petersburg, Cantor showed clear signs of mathematical talent. Though he wanted to become a mathematician, his father had charted out an engineering career for him. He attended several schools along the lines of his father's wishes, including the Gymnasium in Wiesbaden and, from 1860, a Technical College in Darmstadt. Cantor finally received parental approval to study mathematics in 1862. He started his studies in the fall of that year in Zurich, but moved to Berlin after one semester. Cantor was a solid student. He spent a summer semester in Göttingen in 1866 and successfully defended a Ph.D. thesis in number theory on December 14, 1867, in Berlin. Cantor then moved to the University of Halle as a Privatdozent, becoming an associate there in 1872 and a professor in 1879. He remained at Halle for his entire career.

A friend of his sister's, Vally Guttmann, became his wife in 1874. During their honeymoon in Switzerland, the couple met Richard Dedekind, from then on a friend and mathematical confidant of Cantor's. Georg Cantor and Vally Guttmann had six children.

Discovers Set Theory

When Cantor arrived at Halle, the leading mathematician there was Heinrich Heine, under whose influence Cantor began to study Fourier series. His analysis of the convergence of these trigonometric series eventually led to far-reaching innovations. What started as a slight improvement of a theorem on the uniform convergence of Fourier series contained the first seeds of set theory. Cantor's first paper on set theory proper was published in 1874 under the title Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen and dealt with algebraic numbers. An algebraic number is any real number that is a solution to an equation with integer coefficients. Cantor's paper contained the proof that the set of all algebraic numbers can be put in a one-to-one correspondence with the set of all positive integers. Moreover, Cantor proved that the set of all real numbers cannot be put into a one-to-one correspondence with the positive integers. As he later explained it, the set of positive integers has the same power (called Mächtigkeit in German) as the set of algebraic numbers, while the power of the set of real numbers is different from either. The 1874 paper was accepted for publication only after Dedekind's intercession.

The set of all algebraic numbers, containing, for example, the square root of 2, is properly larger than the set of all rational numbers (that is, quotients of integers). In turn, the set of rationals contains infinitely many more elements than the set of positive integers. In spite of that, Cantor showed that the three sets--the rationals, the algebraic numbers, and the positive integers--have the same power. Sets like the algebraic numbers or the rationals are said to be countable, and Cantor furnished proof that this is so. However, he discovered that the set of all real numbers is not countable. Encouraged by these successes, Cantor introduced the notion of equipollency of sets in his next paper, written in 1878. Two sets are equipollent if a one-to-one correspondence exists between them. Where he had previously shown that the set of algebraic numbers is equipollent with the set of positive integers, Cantor then proved that the set of points on any surface such as a plane is equipollent to the set of all real numbers. He finished the 1878 paper with the conjecture that every infinite subset of the set of real numbers is either countable or equipollent to the set of all real numbers. That conjecture became known as the continuum hypothesis. The possibility of a one-to-one correspondence between an infinite set and one of its proper subsets had been observed earlier by scientists such as Galileo and Gottfried Leibniz. The novelty and courage of Cantor's contributions are in his refusal to consider this a contradiction and in using it to define infinite sets of equal power.

Consolidates Set Theory

Cantor's next paper was published in six installments between 1879 and 1884. Where he had previously come to grips with the countable and had realized the gap between the countable and the continuum, his ideas about infinite sets in general had ripened. The paper broaches the idea of a proof of the continuum hypothesis. In 1882, Cantor defined another main concept of set theory, that of well-ordering. In 1883 he wrote that we may assume as a law of thought that every set can be well-ordered. Earlier, in 1878, Cantor had stated, without proof, "If two point-sets M and N are not equipollent then either M will be equipollent to a proper subset of N or N will be equipollent to a subset of N." This principle later became known as the trichotomy of cardinals. However, Cantor was not able to give a solution to the continuum hypothesis or a proof of the trichotomy of cardinals.

In 1884, Cantor had a nervous breakdown; several of such mental crises would follow. He had applied for a professorship in Berlin but was turned down, strongly opposed by Leopold Kronecker, a former teacher of his. In spite of his illness, Cantor remained active. He worked to institute the German Mathematical Society, founded in 1889, and was instrumental in establishing the first International Congress of Mathematicians in 1897 in Zurich. Between 1895 and 1897 Cantor published his last paper, Beiträge zur Begründung der transfiniten Mengenlehre ("Contributions to the Foundation of Transfinite Set Theory") in two parts. In these he defined the transfinite numbers that measure the magnitude of infinite sets.

While there was little enthusiasm for his discoveries within his own country of Germany at this time, Cantor's ideas were gaining support in the world mathematical community. The eventual recognition of sets as a notion underlying all of mathematics led to new fields like topology, measure theory, and set theory itself. Developments at the turn of the century reflected the importance of Cantor's work. At the Second International Congress of Mathematicians in Paris in 1900, the continuum hypothesis was first among 23 problems that Hilbert proposed as central to the development of twentieth-century mathematics. Not much later, in 1904, the mathematician Ernst Zermelo established that every set can be well-ordered, using the so-called axiom of choice. From it followed the trichotomy of cardinals. The earlier controversy between Kronecker and Cantor intensified into a new rage about what was and what was not permitted in mathematics. Much of the debate was later settled by the work of Kurt Friedrich Gödel and Paul Cohen. The first book on set theory was published in 1906 by William H. Youngand Grace C. Young. These years also showed the beginnings of the study of topology. In 1911, L. E. J. Brouwer proved the topological invariance of dimension, at which Cantor himself had tried his hand earlier. In 1914, Hausdorff published the first book on topology, entitled Grundzüge der Mengenlehre ("Principles of Set Theory").

Toward the end of his career Cantor's achievements were recognized with various honors. He became honorary member of the London Mathematical Society (1901) and the Mathematical Society of Kharkov and obtained honorary degrees at several universities abroad. A bust of Cantor was placed at the University of Halle in 1928. Perhaps more fittingly, one special subset of the real numbers that he introduced is now known under his name, the Cantor set. Cantor died on January 6, 1918, at the psychiatric hospital in Halle.