1. studious and all-round development
Cantor was born in St. Petersburg, Russia, 1845. 1 1 studied at Wiesbaden liberal arts middle school at the age of. He especially likes mathematics, but he is not partial to it. His literature, music, painting and other subjects are excellent, thanks to a good family education. /kloc-When he was 0/5 years old, his father wrote him a letter, encouraging him to "master all kinds of basic scientific and practical knowledge …" and hoping that he would "become a shining star on the scientific horizon". He kept this letter with him for a long time as a spur to himself.
1863, Cantor entered the University of Berlin and studied under three world-class mathematicians: Wilstras, Cuomo and Kroneck. Four years later, he received his doctorate. He was deeply influenced by Wilstrass and was regarded as a member of the Wilstrass school. Cantor showed good social activities and organizational skills when studying in university. He joined the "Mathematical Association", a college student organization in Berlin, and served as the chairman of this association in his sophomore year, which laid the foundation for his later establishment of the German Mathematicians' Union and his participation in organizing the International Mathematicians' Congress.
2. Be good at asking questions and be determined to create.
1In the spring of 869, Cantor became a lecturer at Harley University. As soon as he arrived at Harley University, Cantor accepted Edward? 6? 1 Professor Heine suggested learning trigonometric series. This work led him to study real number sets and infinite point sets ... which laid the foundation for the establishment of set theory in the future. Cantor is good at asking profound and valuable questions, as he said in his doctoral thesis: "In the field of mathematics, the art of asking questions is more important than the art of solving problems".
1873165438+1October 29th, Cantor asked in a letter to Dai Dejin: Can there be a one-to-one correspondence between positive integer sets and real number sets? This is a big problem that really leads to set theory. A few days later, Cantor proved the negative result of this problem by reduction to absurdity: "Real number set is uncountable set", and wrote a paper entitled "A property of all real algebraic numbers", which was published in the German Journal of Mathematics 1874. This is "the first revolutionary paper on infinite set theory". In his series of papers, he defined the operations of set, infinite set, derivative set, ordinal number and set for the first time and established the set theory of the system.
Another famous question of Cantor is: How many points are there on a straight line? He extended the concept of number of elements in a set to the concept of cardinal number or potential. For example, through the correspondence: n→ 2n, the set of positive integers n * can be mapped to the even subset m of n * one by one, so the number of elements of n * and m is equal, that is, the set {1, 2, 3, 4, 5, 6...} and its parts {2, 4, 6, 8}. Being able to establish a one-to-one relationship with one's proper subset is a feature of infinite set. These views were too unexpected at the time. Even Kroneck and Poincare, the great authorities of mathematics, came out against it. They are opposed to studying infinite sets as a whole. However, Cantor's flashing thought swept away the fallacy. He bravely broke into the forbidden area and invited the real infinity back to the world. He showed people that infinity can be studied.
From 1878 to 1884. Cantor published a series of articles on the study of infinite cardinality, and founded the theory of excess number. He boldly concluded that since the real number set R can't establish a one-to-one correspondence with the positive integer set N *, then the cardinality C of the real number set R and the cardinality C of natural number set. Not equal, and n * is a part of R, so we get the famous cardinal inequality: C.< In this regard, Cantor thinks: Is there any other cardinal number between C and C in C? He found a cardinal number c 1 and proved that c1≤ c.
Cantor further asked: C 1 = C? This is the famous continuum hypothesis, which is one of the central problems of set theory. He tried his best to solve it, but failed. Decades later, this problem was solved in another way by combining the work of mathematician G6del (1938) and mathematician Cohen (1963).
Try to spread the truth regardless of illness.
Cantor's work solved many unsolved problems and reversed many predecessors' ideas, which was naturally difficult to be accepted by people at that time. Of course, Cantor has an extreme weakness in character. He always takes criticism of his works too seriously and often reacts with strong feelings. Coupled with the long-term inability to solve the problems caused by the continuum hypothesis, he finally had a mental breakdown at 1884-suffering from depression. Later, his illness was good and bad, which made him a frequent visitor to a mental hospital. On the one hand, he struggled to defend his theory, on the other hand, he struggled with cruel diseases.
With his excellent organizational ability, he undertook the historical responsibility of establishing the German Mathematicians' Federation. He had extensive contacts with mathematicians, and finally 1890 formally established the German Federation of Mathematicians. At the meeting, Cantor was elected president of the Federation. The Federation got rid of the prejudice and control of Berlin's mathematical authority, and Cantor's thoughts spread freely here. At that time, Germany was the center of mathematics in the world, and Cantor began to work and actively promoted it, so that the International Congress of Mathematicians was successfully held in Zurich, Switzerland on 1897. Set theory was publicly recognized and praised at the meeting. Hilbert, a generation of mathematics leader, praised Cantor and said, "No one can expel us from the paradise created by Cantor."
1918 65438+16 October, Cantor died of a heart attack in a mental hospital. He left a brilliant legacy to mankind-set theory and remainder theory, and he also created many exquisite research methods and concepts. For example, he created the Cantor Diagonal Method and constructed the magical Cantor Set, which has wonderful properties and has been applied to the fractal theory of modern emerging disciplines, resulting in important concepts such as Cantor Dust and Cantor Function. Cantor deserves to be a great mathematician. He smiled and died for his extraordinary contribution to human civilization.