George Cantor

George Cantor George Cantor, the founder of transfinite set theory revolutionized mathematical thinking in this area. For centuries, the concept of infinity had been a highly controversial one both mathematically and philosophically and his ideas could not get the full impact in his life time. George Ferdinand Ludwing Philipp Cantor was born in St. Petersburg, Russia. His father George Woldemar Cantor was a stockbroker while his mother Maria Anna came from a musical family. Cantor was raised in an intensely religious atmosphere. In 1856, when young Cantor was 11, the family moved from St. Petersburg to Germany where he attended the Gymnasium school. Cantor showed all round ability there but he had special affinity with mathematics and science and his father decided that he should train as an engineer. He was therefore enrolled in Polytechnikum at Zurich but seeing his special interest in mathematics his father later decided that he should study mathematics instead of engineering and he was admitted to University of Berlin, where he proved to be a good but not excellent student. Cantor moved to the University of Berlin where he became friends with Hermann Schwarz who was a fellow student. Cantor attended lectures by Weierstrass, Kummerand Kronecker. He spent the summer term of 1866 at the University of Göttingen, returning to Berlin to complete his dissertation on number theory De aequationibus secundi gradus indeterminatis in 1867. After obtaining his doctorate in 1867, he became a school teacher for a short period and later joined University of Halle where he remained for the rest of his entire carrier. He married Vally Guttmam in 1874 and they had 6 children. While he was in Berlin, he became interested in Number Theory and came to the notice of Kronecker. His first achievement was in traditional field of trigonometric series, especially the question of uniqueness for the Fourier representation of a given function. He gradually showed that the 19th Century idea in the connection between the dimension of a set and the number of its elements were dangerously fallacious. He showed that the concepts of Cardinal and Ordinal Number could be defined mathematically in such a way that it made a good sense to talk of infinite or transfinite numbers. In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts to decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874. It is in this paper that the idea of a one-one correspondence appears for the first time, but it is only implicit in this work. Cantor went on to develop his revolutionary ideas in a series of paper published between 1879 to 1884. Cantor got the first part of his major work Beitrage zur Begrundung der transfiniten Mengenlehre (Contributions to the foundation of Transfinite set theroy) published in 1895. This book led to the circulation of Cantor’s idea to the world. Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:- I see it, but I don't believe it! In 1910 Cantor received an invitation from the University of St Andrews in Scotland to attend the 500t anniversary of the founding of the University as a distinguished foreign scholar .Cantor had hoped to meet with Russell who had just published the Principia Mathematica. However ill health and the news that his son had taken ill made Cantor return to Germany without seeing Russell. The following year Cantor was awarded the honorary degree of Doctor of Laws by the University of St Andrews but he was too ill to receive the degree in person. The second half of his life, his ideas was attacked by many mathematicians. Leopold Kronecker launched a particular vitriolic attack on Cantor that led to the decline in Cantor’s mental age and he died in a mental hospital on 6th January 1918. SEND YOUR VALUABLE COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com