German mathematician, founder of set theory,/kloc-one of the great achievements of mathematics in the 9th century. 1845 was born in Petersburg, Russia on March 3, and his father was a wealthy businessman. 1856 The whole 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. 1867 He obtained his doctorate in philosophy by seeking the integer solution of the indefinite equation ax2+by2+cz2= 0 (where A, B and C are arbitrary integers). 1869 Up to now, teacher, associate professor and professor of Haller University. Cantor was interested in mathematics since he was a child. He received his doctorate at the age of 23 and has been engaged in mathematics teaching and research ever since. The set theory he founded has been recognized as the basis of all mathematics.

The concept of infinity put forward by Cantor in 1874 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 the problem of number theory and the unique representation of trigonometric functions to functions, and found amazing results: he proved that rational numbers are countable, but all real numbers are uncountable.

In the 29th year of Cantor (1874), he published his first set theory paper in a mathematical magazine, and put forward the mathematical concept of "infinite set", which aroused great concern 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 the real one. 1874 proves the countability of algebraic number set and rational number set and the uncountability of real number set, and establishes a continuity axiom called "Cantor Axiom". 1877 proves that the points on the line segment can establish a one-to-one correspondence with the points on the square, thus proving the set of all points in a straight line, a plane, a three-dimensional space or even a 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).

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 Pi Aucar, who is famous for his "profound and creative", compared set theory to an interesting "morbid situation", and even his teacher Kroneck hit back at Cantor as a "mental derangement" 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 at 1884 because of frequent depression, and finally died in a mental hospital.

However, after all, history has fairly evaluated his creation. Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. At the beginning of the 20th century, Hilbert, the greatest mathematician in the world, 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 works as "the greatest works that can be boasted in this era".

Because the study of infinity often leads to some logical but absurd results (called "paradox"), many great mathematicians are afraid of falling into it and adopt 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, and can also correspond to points in space one by one. In this way, it seems that there are "as many" points on the 1 cm long line segment 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 a sharp conflict with the traditional mathematical concept, and some people oppose, attack and even abuse it. 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 the role of set theory initiated by Cantor. 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 that Cantor has created for us." He also 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 category of pure reason." The famous philosopher Russell described Cantor's works as "probably the greatest works that can be boasted of this era."

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 the infinity of more than 2,000 years since the ancient Greek era. In essence, it reveals the characteristics of infinity, 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 basis of real variable function theory, algebraic topology, group theory and functional analysis, and also has a far-reaching impact on logic and philosophy.

True gold is not afraid of fire, and Cantor's thought finally shines. At the first international congress of mathematicians held in 1897, his achievements were recognized. Russell, a great philosopher and mathematician, praised Cantor's work as "probably the greatest work that can be boasted in this era." But Cantor is still in a trance, unable to get comfort and joy from people's reverence. 1918 65438+1October 6th, Cantor died in a mental hospital.