Georg Cantor James L. Marden Georg Cantor's Life Georg Cantor was born in 1845 in St. Petersburg to Georg Waldemar Cantor and Maria Anna Bhm. Cantors mother was very musical and encouraged him to play the violin, which he excelled at incredibly. Georgs father was a successful merchant from Denmark with an appreciation for the arts. Early in his life Georg was educated by a private tutor until attending a primary school in St. Petersburg. The Cantor family moved to Germany when Georg was eleven. After his father fell into poor health and the family had moved several times, Georg became a boarder at the Realschule in Darmstadt. In 1862, he began university studies at the Polytechnic of Zurich with dreams of becoming an engineer. His dreams ended quickly as he became very devoted to pure and theoretical mathematics. Cantor joined the Schellbach Seminar in 1868, a group for teachers of math- ematics. He continued his work at the University of Halle where he became Extraordinary Professor in 1872. His position was upgraded to full Professor in 1879, achieving this rank at 34. Cantor aspired to hold chair at a more prestigious institution, yet was restrained by the mathematical communitys resistance to his revolutionary theory. No universities in Germany wanted to take on such a radical and award him a high-ranking position. Especially at the University of Berlin, the mathematics department, headed by Leopold Kronecker, was especially conservative. Cantor was described as a corrupter of the youth and was outcast by many more traditional mathematicians. Cantor married Vally Guttmann in 1874. The two had six children together over the coming Cantor continued to work at Halle through the 1880s, strug- gling with bouts of depression after his close friend and mathematical partner Richard Dedekind. He lost his self-confidence and drive to focus on rigor- ous mathematics and transitioned into teaching philosophy. He lectured on mostly Shakespearean and Victorian literature. He eventually published two pamphlets on the subject late in the 1890s. After this he returned to true mathematics. His return was glorious, he created his most famous proof, the diagonal ar- gument. This gave him the fame and reputation to become a trustworthy and reputable mathematician. He was elected as the first president of the Deutsche Mathematiker-Vereinigung, a society of German mathematicians. He was elected over Kronecker, a major victory for the advancement of math- ematics. Cantor's struggles with depression were worsened by his challenge with the continuum hypothesis. His inability to answer the question of whether or not there is a set whose cardinality is strictly between that of the integers and the real numbers drove him to depression. This problem would eventually be shown to be unanswerable, causing it to become an axiom of set theory rather than something to be shown within gthe field. 2 Cantors youngest son Rudolph died suddenly in 1903, leading to another hospitalization for depression. In 1904, Cantor was publicly humiliated with a paper claiming to disprove set theory. The paper was shown to be flawed a day after it was presented, but still haunted Cantor for the rest of his life. He lived in and out of hospitals and struggled with depression for the rest of his life, but continued to lecture on mathematics. Most of his lectures concerned paradoxes. Cantor attended the International Congress of Mathematicians for the last time in 1904. He faded out of the mathematical community until officially retiring in 1913. He lived the rest of his life in poverty. He lived through the war mostly in sanatoriums, where he eventually died in 1918. 3 Georg Cantor's mathematical works Cantors most influential and groundbreaking work was in Set Theory. Be- fore the 18th century he only references to set theory came from Aristotle. However these were incredibly elementary ideas. They were limited to finite sets with no real application or possibility for furthering mathematically. In Bernard Bolzano (1781-1848) established Naive Set Theory. He described the concept of sets as: an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference. However, in his work, there could not be infinite sets. He effectively only worked with subsets and tried to show one to one correspondence. His work would eventually become a smaller part of Cantors larger Set Theory. Can- tor put Set Theory into a strong mathematical basis. Cantors first paper directly concerning Set Theory was an article in Crelles journal. This was an important mathematical journal at the time in Europe. He also pub- lished follow-up works in this journal. However, the editorial staff, including 4 Leopold Kronecker, did not appreciate Cantors new and revolutionary ideas. Cantor used many proofs by geometry in the beginning of his work on cor- respondence. He showed for instance that two lines of different finite lengths and endpoints had an equal number of points. This can then be used later to prove that for instance: there are an equal number of numbers in two intervals of different sizes. As counter-intuitive as this sounds, it was one of Cantor's conclusions. Think of it this way: For the numbers one to ten, there are matching multiples of ten from ten to one hundred (1-10, 2-20, etc.) Now we can split these numbers in half and still maintain correspondence. (3.5-35, 4.5-45, etc.) One can continue the splitting infinitely. 5 Cantor created his first proof of uncountability was published in 1874. This was unlike any of his other work.His article was under five pages, but pre- sented two new theorems as well as ideas on dense sequences and "Cantor Construction." His first theory addressed the roots of polynomials. He went to show that Cantor's most well known proof is his proof by diagonalization. Similar to the one outlined before, this argument show correspondence. He went on to use this form of proof in many ways. He used it to show the uncountability of real numbers, 6 Georg Cantor's Collaboration with Other Scholars Cantors most significant partnership was with Richard Dedekind. He and Dedekind met while Cantor was on vacation in Switzerland. The two recon- nected while Cantor was on his honeymoon. Cantor and Dedekind continued to send letters to one another, some discussing mathematics, others just dis- cussing life. One such letter from Cantor to Dedekind in January of 1874 reads: Can a surface (say a square that includes the boundary) be uniquely referred to a line (say a straight line segment that includes the end points) so that for every point on the surface there is a corresponding point of the line and, conversely, for every point of the line there is a corresponding point of the surface? I think that answering this question would be no easy job, despite the fact that the answer seems so clearly to be "no" that proof appears almost unnecessary. Reminiscent of Euclids fifth axiom, this idea became the basis of most of Cantors lifes work. Just three years later, Cantor proved the one to one cor- respondence of the numbers [0,1] to points in p-dimensional space. This was so remarkable to him, as he had thought the answer would be a resounding no, that he wrote to Dedekind I see it, but I dont believe it! 7 He went on to show that there was not however correspondence between the numbers [0,1] and the set of unit squares. The two often referenced one anothers work. They clearly built off one another. Cantors work in trigonometry aided Dedekinds theorem on rational and irrational numbers by Dedekind Cuts. Years later, a seat opened at the University of Halle, Cantor suggested Dedekind for the position, yet Dedekind declined. This ended their mathematical correspondence within a year. Filling that void was Gsta Mittag-Leffler. Cantor began publishing in his journal Acta Mathematica within a year. Their mathematical influence was not as deep and ended with mostly editing papers. Cantor also presented Hilbert with a paradox on sets. One of Cantors mentors, Eduard Heine was one of the pioneers of subset theory, the logical predecessor to set theory. Felix Klein, another German mathematician, specializing in group theory and geometry influenced some of Cantors publications. He aided Cantor in the publication of Mathematische Annalen, and also helped Cantor get a solid mathematical basis to set theory. Klein became the primary editor of Cantors final works at the end of the 19th century. He helped to prove Cantors work on Well-Ordered and Ordinal numbers. 8 Filling that void was Gsta Mittag-Leffler. Cantor began publishing in his journal Acta Mathematica within a year. Their mathematical influence was not as deep and ended with mostly editing papers. Cantor also presented Hilbert with a paradox on sets. One of Cantors mentors, Eduard Heine was one of the pioneers of subset theory, the logical predecessor to set theory. Historical Events that Marked Georg Cantor's Life. The world was fairly stagnant during the time of Cantor's actual work. Shortly after Cantor's birth, the French Revolution occurred. There were also uprisings across Europe, including Berlin and Vienna. Russian history is moderately turbulent at the beginning of Cantors life, when he lived in Saint Petersburg. The French and Turkish declared war on Russia in 1854. This is likely why his family left the country and moved to the far safer Germany from Saint Petersburg just two years later. Within Germany, the Reich of Otto Von Bismark was mostly tranquil. The Socialist party was on the rise, but did not ever take any action until after Cantor's career ended.