1. Cantor's life
1845 On March 3rd, George Cantor was born into a Danish Jewish family in Russia. 1856 Cantor moved to Frankfurt with his parents. Like many excellent mathematicians, he showed special sensitivity to mathematics in middle school and came to surprising conclusions from time to time. His father urged him to study engineering, so Cantor entered Berlin University with this goal in 1863. At this time, the University of Berlin is forming a mathematics teaching and research center. Cantor has always longed for this one of the world's mathematical centers occupied by Storis. So at the University of Berlin, influenced by Wilstrass, Cantor turned to pure mathematics. 1869 obtained the teaching qualification of Haller university, and soon after 1879 was promoted to associate professor and full professor. 1874, Cantor published the first revolutionary article about infinite set theory in Koehler Journal of Mathematics. In the history of mathematics, it is generally believed that the publication of this article marks the birth of set theory. The originality of this article attracted people's attention. In later research, set theory and remainder became the mainstream of Cantor's research, and his papers in this field were published until 1897. Excessive mental fatigue and strong external stimuli made Cantor suffer from schizophrenia. For more than 30 years, this stubborn disease has been affecting his life intermittently. 1918 65438+16 October, Cantor died in the mental hospital of Haller University.
2. The background of set theory
In order to clearly understand Cantor's work on set theory, firstly, the background of set theory is introduced.
/kloc-the basic reason for the birth of set theory in the 0/9th century comes from the critical movement of the basis of mathematical analysis. The development of mathematical analysis inevitably involves infinite concepts such as infinite process, infinitesimal and infinity. 18th century, due to the lack of precise definition of the concept of infinity, calculus theory not only encountered serious logical difficulties, but also discredited the concept of real infinity in mathematics. /kloc-In the first half of the 9th century, Cauchy gave an accurate description of the concept of limit. On this basis, the theories of continuity, derivative, differential, integral and infinite series are established. It is the limit theory developed in19th century that perfectly solves the logical difficulties encountered in calculus theory. However, Cauchy did not completely complete the rigor of calculus. Cauchy's thought has certain fuzziness, and even produces logical contradictions. /kloc-mathematicians in the late 0/9th century found that the reason for Cauchy's logical contradiction lies in the concept of limit, which laid the foundation of calculus. Strictly speaking, Cauchy's concept of limit does not really get rid of geometric intuition, but is really based on pure and rigorous arithmetic. As a result, many mathematicians affected by analyzing the basic crisis are committed to the rigor of analysis. In this process, it involves the description of continuous function, the basic research object of calculus. In the definition of number and continuity, there is a theory about infinity. Therefore, the existence of infinite sets in mathematics has been put forward again. This naturally leads to the search for the theoretical basis of infinite set. In a word, seeking the thorough and strict arithmetical tendency of calculus is an important reason for the emergence of set theory.
3. The establishment of set theory
Cantor's tutors at the University of Berlin are Stores, Man Ku and Kroneck. Professor Man Ku is an expert in number theory. He is famous for introducing ideal numbers and greatly promoting the study of Fermat's Last Theorem. Kroneck was a great mathematician, and many people were proud of his praise at that time. Wal-Mart Stores is an excellent teacher and a great mathematician. His speech laid an accurate and stable foundation for mathematical analysis. For example, he first introduced the famous concepts in calculus. It is because of the influence of these people that Cantor became interested in number theory earlier and concentrated on the problems left by Gauss. His graduation thesis is about prime numbers with ++=0. This is an unsolved problem put forward by Gauss in arithmetic research. This paper is well written, which proves that the author has profound insight and the ability to inherit excellent ideas. However, the establishment of his super-difference set theory did not benefit from the early study of number theory. On the contrary, he quickly accepted the advice of mathematician Heine and turned to other fields. Heine encouraged Cantor to study a very interesting and difficult question: Is the expression of trigonometric series of arbitrary function unique? For Cantor, this problem is the most direct reason for his establishment of set theory. Function can be expressed by trigonometric series, which was first proposed by 1822 Fourier. Since then, the study of discontinuity has attracted more and more attention in the field of analysis. Since 1930s, many outstanding mathematicians have been engaged in the study of discontinuous functions, and they are all related to the concept of set to some extent. This created conditions for Cantor to finally establish set theory. In 1870, Heine proved that the trigonometric series representing a function is unique if it uniformly converges in the rest of the interval [-π, π] after removing any small neighborhood of the discontinuous point of the function. As for the function of discontinuity, Heine did not solve it. Cantor began to solve the problem of uniqueness expressed in such a concise way. In this way, he took the first step of set theory.
Cantor suddenly showed stronger research ability than Heine. He decided to lift the restrictions as much as possible, which will certainly make the problem more difficult. In order to give the most general solution, Cantor introduced some new concepts. In the following three years, Cantor published five articles on this topic. In 1872, when Cantor weakens Heine's condition of uniform convergence to the case that the function has infinite discontinuous points, he extends the uniqueness result to the case that the allowable outliers are infinite sets. Cantor's 1872 paper is an extremely important link from discontinuous point problem to point set theory, which integrates infinite points into a clear research object.
The center and difficulty of set theory is the concept of infinite set itself. Since Greek times, infinite sets have naturally attracted the attention of mathematicians and philosophers. However, it is difficult to grasp the nature of this set and its seemingly contradictory nature as a finite set. So the understanding of this set has not progressed. As early as the Middle Ages, people have noticed the fact that if you draw a ray from two concentric circles, then this ray establishes a one-to-one correspondence between the points of the two circles, but the perimeters of the two circles are different. In the16th century, Galileo also said that, for example, one-to-one correspondence can be established between two line segments ab and cd with different lengths, so as to imagine that they have the same point.
He also noticed that positive integers can form a one-to-one correspondence with their squares, as long as each positive integer corresponds to their squares:
1 2 3 4 … … n … …
2 3 4 … … n … …
But this leads to the infinite "order of magnitude" difference, which Galileo thought was impossible, because all infinity is the same size.
Not only Galileo, but also mathematicians before Cantor mostly disapproved of one-to-one comparison between infinite sets, because it would lead to the contradiction that part equals all. Gauss made it clear: "I am against taking an infinite quantity as an entity, which is absolutely not allowed in mathematics." Infinity is just a saying ... "Cauchy also denied the existence of infinite sets. He can't allow parts to correspond to the whole. Of course, latent infinity is very useful under certain conditions, but it is one-sided to regard it as infinity. The development of mathematics shows that it is impossible to admit only the potential infinity and deny the real infinity. Cantor used his time to think deeply about the research object. He wants to explain the problem with facts and convince everyone. Cantor thinks that an infinite set can correspond to its parts one by one, which is not a bad thing, but only reflects an essential feature of an infinite set. For Cantor, if a set can form a one-to-one correspondence with its parts, then it is infinite. It defines concepts such as cardinal number and countable set. It is also proved that the real number set is uncountable and the algebraic number is countable. Cantor's original proof was published in an article entitled "On the Characteristics of All Real Algebraic Numbers" in 1874, which marked the birth of set theory.
With the establishment of uncountability of real numbers, Cantor raised a new and bolder question. In 1874, he considered whether there could be a one-to-one relationship between points on a plane and points on a straight line. Intuitively, there are obviously more points on the plane than on the line. Cantor himself knew from the beginning. But three years later, Cantor announced that one-to-one correspondence can be established not only between plane and straight line, but also in general N-dimensional continuous space! This result is unexpected. Even Cantor himself felt "incredible". However, this is an obvious fact, which shows that intuition is unreliable, and only by reason can we find truth and avoid fallacies.
Since N-dimensional continuous space and one-dimensional continuous system have the same cardinality, Cantor concentrated on linear continuous systems from 1879 to 1884, and published six series of articles in succession, which were collected into On Infinite Linear Point Sets. The first four chapters directly establish some important results of set theory, including the application of set theory in function theory. The fifth article was published in 1883, with the longest length and the richest content. It not only goes beyond the research scope of linear point sets, but also gives a completely universal theory of super-finite numbers, in which the whole pedigree of super-finite ordinal numbers is introduced by means of the ordered form of well-ordered sets. At the same time, it also discusses the philosophical problems arising from set theory, including answering the critics' criticism of Cantor's infinite position. This article is extremely important to Cantor. 1883, Cantor published it as a monograph with the title "Basis of Set Theory".
The publication of Basics of Set Theory is a milestone in Cantor's mathematical research. Its main achievement is the introduction of super-finite numbers as the independence and expansion of the natural number system. Cantor clearly realized that what he had done was a bold initiative. "I know very well that doing so will put me in a position opposite to the traditional infinite concept and the nature of natural numbers in mathematics, but I firmly believe that the super-poor number will eventually be recognized as the simplest, most appropriate and natural extension of the logarithmic concept." The Basis of Set Theory is a systematic exposition of Cantor's early set theory and the beginning of his special contribution with far-reaching influence.
Cantor published two decisive papers in 1895 and 1897 respectively. In this paper, he changed the early method of defining (ordered) numbers by axioms and adopted set as the basic concept. He gave the definitions of transcendental cardinal number and transcendental ordinal number, and introduced their symbols. Arrange them in a "sequence" according to their potential; Their addition, multiplication and multiplication are specified. So far, what Cantor can do about the theory of overrun cardinal number and overrun ordinal number has been completed. But the internal contradiction of set theory began to be exposed. Cantor himself first discovered the internal contradiction of set theory. In his articles from 65438 to 0895, he left two unresolved questions: one is the continuum hypothesis; The other is the comparability of all ultra-poor bases. Although he thinks that infinite cardinality has a minimum number but no maximum number, he does not clearly describe its contradiction. Until Russell published his famous paradox in 1903. The internal contradiction of set theory is prominent, which has become the starting point of set theory and basic research of mathematics in the 20th century.
4. Different comments on Cantor's set theory.
Cantor's set theory is the most revolutionary theory in mathematics. He dealt with the most difficult object in mathematics-infinite set. Therefore, his development path is naturally uneven. He abandoned all experience and intuition and demonstrated with thorough theory, so his conclusion is highly surprising, incredible, true and beyond doubt. There is no bolder idea and step in the history of mathematics than Cantor. Therefore, it is inevitable to be opposed by traditional ideas.
The recognized proof of the existence of19th century is constructive. If you want to prove that something exists, you must concretize it. Therefore, people can only draw conclusions from specific numbers or shapes step by step through limited steps. As for "infinity", many people think it is a world beyond people's cognitive ability. Not to mention whether it counts, it is difficult to determine whether it exists. Cantor actually counted them "randomly", compared their sizes, and imagined that there was an infinite set with no maximum cardinality ... This was naturally opposed and reprimanded.
The fiercest opponent of set theory is Kroneck, who thinks that only the number theory and algebra he studied are the most reliable. Because natural numbers are created by God, and the rest are human works. He expressed strong opposition to Cantor's research objects and argumentation methods. As Berlin was the center of mathematics at that time, Kroneck was the leader of Berlin School, which greatly hindered the development of Cantor and his set theory. Will, another German perceptionist, thinks that Cantor's classification of infinity is adding fog to the fog. Poincare, an authoritative figure in French mathematics, predicted that "later generations will regard (Cantor's) set theory as a disease" and so on. Because of the difficulties brought by the infinite concept mathematics in the past two thousand years, and because of the authoritative position of the opposition, Cantor's achievements have not been properly evaluated, but have been rejected. 189 1 year, after the death of Kroneck, the situation in Cantor began to improve.
On the other hand, many great mathematicians support Cantor's set theory. In addition to Dydykin, Swedish mathematician Midach-Levreux reprinted Cantor's paper on set theory in French in his own international mathematical magazine, which greatly promoted the international spread of set theory. 1897, at the first international congress of mathematicians, Huo Weici summarized the latest progress of analytic functions and expounded Cantor's contribution to set theory. Three years later, Hilbert, who fought bravely to defend set theory, further emphasized the importance of Cantor's work at the Second International Mathematical Congress. He listed the continuum hypothesis as the first of 23 major mathematical problems to be solved in the early 20th century. Hilbert declared, "No one can drive us out of the paradise created for us by Cantor." Especially since 190 1 produced Lebesgue integral and Lebesgue measure theory enriched set theory, set theory was recognized and Cantor's work was highly praised. 1904 When the Third International Mathematical Congress was held, "modern mathematics cannot be without set theory" has become everyone's view. Cantor's reputation is universally recognized.
5. The significance of set theory
Set theory is an important basic theory in modern mathematics. Its concepts and methods have penetrated into many branches of mathematics, such as algebra, topology and analysis, and some natural science departments, such as physics and particle mechanics, which have provided basic methods for these disciplines and changed their faces. It can almost be said that it is difficult to have a profound understanding of modern mathematics without the viewpoint of set theory. Therefore, the establishment of set theory is not only of great significance to the study of mathematical foundation, but also has a far-reaching impact on the development of modern mathematics.
Cantor suffered all his life. He and his set theory were brutally attacked for ten years. Although Cantor once lost interest in mathematics and turned to philosophy and literature, he could not give up set theory. Cantor can firmly defend the theory of superheterodyne set despite the opposition of many mathematicians, philosophers and even theologians, which is inseparable from his scientist temperament and personality. Cantor's personality is greatly influenced by his father. His father George Waldemar Cantor grew up under the influence of evangelical Protestantism. He is a shrewd businessman, smart and talented. His deep religious belief and strong sense of mission always bring him courage and confidence. It is this firm and optimistic belief that makes Cantor go to the road of mathematicians without hesitation and truly succeed.
Today, set theory has become the foundation of the whole mathematical building, and Cantor has thus become one of the greatest mathematicians at the turn of the century.