Basic information
1845 was born in a Jewish businessman's family in Petersburg, Russia. 1856 On March 3rd, his family moved to Frankfurt, Germany. Cantor studied at the University of Zurich, the University of G? ttingen, the University of Frankfurt and the University of Berlin, mainly studying philosophy, mathematics and physics. At the University of Berlin, influenced by the famous analyst Wilstrass, he became interested in pure mathematics. +He took the integer solution of the indefinite equation A * X 2+B * Y 2+C * Z 2 = 0 (where A, B and C are arbitrary integers) as his doctoral thesis and obtained his doctorate. 1869 came to Haller university and has been a teacher, associate professor and professor since childhood. Cantor has been interested in mathematics since he was 23 years old.
Major achievements
1874, Cantor's concept of infinity shocked the intelligentsia. With the help of the infinite thought in ancient and medieval philosophical works, Cantor derived a new thinking mode about the nature of numbers, established the basic skills of dealing with infinity in mathematics, and greatly promoted the development of analysis and logic. He studied number theory and used trigonometric series to express functions uniquely, and found amazing results: he proved that rational numbers are countable, but all real numbers are uncountable.
When Cantor was 29 years old (1874), he published his first set theory paper in a mathematical magazine, and put forward the mathematical concept of "infinite set", which attracted great attention in the mathematical field. He introduced some concepts of infinite point set, such as cardinal number, potential and ordinal number. And try to distinguish different infinite discrete point sets from infinite continuous point sets in some way. He also built. Cantor sequence. In 1874, the countability of algebraic number set and rational number set and the uncountability of real number set are proved, and the axiom of real number continuity is established, which is called cantor axiom. In 1877, it is proved that the points on the line segment can correspond to the points on the square one by one, thus proving the set of all points in the straight line, plane, three-dimensional space and even high-dimensional space. They all have the same potential. 1879- 1884 focused on the theory of infinite number and transcendental number. The most important work is the theoretical basis of transcendental number (1895- 1897).
Academic debate
Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Even the mathematician Poincare, who is famous for his "profundity and creativity", compared set theory to an interesting "morbid situation", and even his teacher Kroneck retorted that Cantor was "neurotic" and "walked into a hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said: "My theory is as firm as a rock, and anyone who opposes it will shoot himself in the foot." He also pointed out: "The essence of mathematics lies in its freedom, and it is not bound by traditional ideas." This argument lasted for ten years. Cantor suffered from schizophrenia on 1884 and finally died in a mental hospital.
The World Identity of Set Theory
However, after all, history fairly evaluated his creation. At the beginning of the 20th century, set theory gradually penetrated into all branches of mathematics and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. Hilbert, the greatest mathematician in the world in the early 20th century, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Russell praised Cantor's work as "the most amazing product of mathematical thought".
Because the study of infinity often leads to some logical but absurd results (called "paradox"), many great mathematicians are afraid of getting stuck and take an evasive attitude. During 1874- 1876, Cantor, who was less than 30 years old, declared war on the mysterious infinity. With hard sweat, he successfully proved that points on a straight line can correspond to points on a plane one by one. It can also correspond to points in space one by one. In this way, it seems that there are as many points on the line segment 1 cm long as there are points in the Pacific Ocean and the whole earth. In the following years, Cantor published a series of articles about this kind of "infinite set" and drew many amazing conclusions through strict proof.
Cantor's creative work has formed a sharp conflict with the traditional mathematical concept, which has been opposed, attacked and even reviled by some people. Some people say that Cantor's set theory is a kind of "disease", Cantor's concept is "fog in fog", and even Cantor is a "madman".
Great mental pressure from the authority of mathematics finally destroyed Cantor, making him exhausted, suffering from schizophrenia and being sent to a mental hospital. Many of his outstanding achievements in set theory were obtained during the period of mental illness.
However, truth is invincible, and many outstanding mathematicians are deeply moved by Cantor's set theory. 1897 At the first international congress of mathematicians held in Zurich, two mathematicians Hurwitz and Adama stood up and pointed out the important application of Cantor's set theory in analysis. Hilbert is also one of the mathematicians who most support Cantor's theory. He shouted, "No one can drive us away from the paradise created by Cantor." He wrote an article praising Cantor's transfinite arithmetic as "the most amazing product of mathematical thought and the best expression of human activities in the purely rational category." The famous philosopher Russell described Cantor's works as "probably the greatest works that can be boasted in this era."
On set theory from a modern perspective
The development of modern mathematics tells us that Cantor's set theory is the first time in the history of human cognition to establish an abstract formal symbol system and definite operation for infinity since the ancient Greek era, which reveals the characteristics of infinity in essence, revolutionizes the concept of infinity, permeates all branches of mathematics, fundamentally transforms the structure of mathematics, promotes the establishment and development of many new branches of mathematics, and becomes the theory of real variable function, algebraic topology and group.
True gold is not afraid of fire, and Cantor's thoughts finally shine. At the first international congress of mathematicians held in 0897, his achievements were recognized. Russell, a great philosopher and mathematician, praised Cantor's works as "probably the greatest works that can be boasted in this era." However, Cantor is still in a trance, unable to get comfort and joy from people's reverence. 38660.6886868866 1