Students who graduated from junior high school and entered senior high school invariably found that the first mathematical concept they learned was: set. This mathematical theory of research set is properly called set theory in modern mathematics. It is a basic branch of mathematics and occupies an extremely unique position in mathematics, and its basic concepts have penetrated into all fields of mathematics. If modern mathematics is compared to a magnificent building, it can be said that set theory is the cornerstone of this building, which shows its importance in mathematics. Its founder, Cantor, is also regarded as one of the most influential scholars in the 20th century for his achievements in set theory. Let's explore the ins and outs of this unique and important mathematical theory and trace its tortuous course. The Birth of Set Theory Set theory was founded by the famous German mathematician Cantor at the end of 19. /kloc-in the 0/7th century, a new branch of mathematics appeared: calculus. In the next one or two hundred years, this brand-new discipline has achieved rapid development and fruitful results. The speed of its advancement makes it too late for us to check and consolidate its theoretical foundation. /kloc-at the beginning of the 0/9th century, after many urgent problems were solved, there was a movement to rebuild the mathematical foundation. It was in this movement that Cantor began to explore the set of real numbers that he had never touched before, which was the beginning of the study of set theory. In 1874, Cantor began to put forward the general concept of "set". His definition of set is: a number of definite and differentiated (whether concrete or abstract) things are combined and regarded as a whole, which is called a set, and each thing is called an element of the set. People regard1February 7, 873, the day when Cantor first put forward the idea of set theory in his letter to Dai Dejin, as the birth date of set theory. Cantor's Immortal Contribution What we learned in middle school mathematics is only the most basic knowledge of set theory. In the process of learning, students may feel that everything is natural and simple, and they can't imagine the scene of fierce opposition on the day of their birth, nor can they appreciate Cantor's achievements. Commenting on Cantor's work, Kolmo Golov, a mathematician of the former Soviet Union, said: "Cantor's immortal achievement lies in his stride towards infinite adventure". Therefore, only when we know what conclusions Cantor has drawn from endless research can we truly understand the value of his work and the sources of many objections. Mathematics has an indissoluble bond with infinity, but the road of studying infinity is full of pitfalls. Because of this, in the process of mathematical development, mathematicians always treat infinity with suspicion and try to avoid this concept as much as possible. However, Cantor, who tried to grasp the infinity, bravely embarked on the road of no return full of traps. He introduced the word infinite set into mathematics, thus entering a virgin land and opening up a wonderful new world. The study of infinite sets enabled him to open Pandora's box in mathematics. Let's see what he released when the box was opened. "We call the set of all natural numbers natural number set for short, which is represented by the letter n". After learning this chapter of compilation, students should be familiar with this sentence. However, when students accept this sentence, they can't imagine that Cantor was doing a job of updating the concept of infinity when he did so. Prior to this, mathematicians only regarded infinity as an extension, change and growth thing to explain. Infinity is always under construction and can never be completed. It is potential, not reality. This concept of infinity is mathematically called potential infinity. Gauss, the prince of mathematics in the eighteenth century, held this view. In his words, it is "... I object to taking infinity as an entity, which is absolutely not allowed in mathematics." The so-called infinity is just a statement ... "And when Cantor regarded all natural numbers as a set, he regarded the infinite whole as a complete thing, so he affirmed that infinity is a complete whole, which is called real infinite thought in mathematics. Since the idea of infinite potential has won a comprehensive victory in the basic reconstruction of calculus, it is not surprising that Cantor's idea of real infinity was criticized and attacked by some mathematicians at that time. However, Cantor did not stop there. He continued to explore infinity in an unprecedented way. On the basis of the concept of real infinity, he further drew a series of conclusions and founded an exciting and far-reaching theory. This theory makes people really enter an elusive and unfamiliar infinite world. What best reflects his originality is his research on the number of elements in an infinite set. He proposed a one-to-one correspondence criterion to compare the number of elements in an infinite set. He called the set that can establish one-to-one correspondence between elements the same number, and used his own concept to mean equipotential. Because an infinite set can establish a one-to-one correspondence with its proper subset-for example, students can easily find a one-to-one correspondence between a natural number set and a positive even number set-that is, an infinite set can be equipotential with its proper subset, that is, it has the same number. This is in contradiction with the traditional concept that "the whole is greater than the parts". And Cantor thinks this is precisely the characteristic of Infinite Set. In this sense, natural number set and even set have the same number, which he called countable set. It is also easy to prove that the set of rational numbers and natural number set are equipotential, so the set of rational numbers is countable. Later, when he proved that algebraic number sets are also countable sets, it was natural to think that infinite sets are uniformly countable sets. Unexpectedly, in 1873, he proved that the potential of real number set is greater than that of natural number set. This not only means that irrational numbers are far more than rational numbers, but also obviously huge algebraic numbers are just a drop in the ocean compared with transcendental numbers, as someone described: "Algebraic numbers dotted on a plane are like stars in the night sky; The thick night sky is made up of transcendental numbers. " When he came to this conclusion, people could only find one or two transcendental numbers. What a shocking result! However, things are not over. Once the box is opened, it can't be closed again, and what is released in the box is no longer limited to countable sets, an infinite monster. From the above conclusion, Cantor realized that there are differences between infinite sets, which have different orders of magnitude and can be divided into different levels. His next job is to prove that there are infinite levels in all infinite sets. He succeeded. According to the theory that there are infinite kinds of infinity, he established a complete sequence of different infinity, which he called "remainder". He used the first letter of the Hebrew alphabet "Alef" to represent the super-limited elves, and finally he established the so-called Alef pedigree about infinity, which can be extended indefinitely. In this way, he founded a new theory of remainder and painted a complete picture of an infinite kingdom. It is conceivable that this conclusion, which still makes us feel a little whimsical, will shake the hearts of mathematicians at that time. It is no exaggeration to say that Cantor's theory of infinity has caused endless noise from opponents. They loudly opposed his theory. Some people mock that set theory is a kind of "disease", while others mock that surplus is a "fog in a fog", saying that "Cantor has entered a hell of surplus". As a great innovation of traditional ideas, it is normal for his theory to be strongly criticized, because he has created a brand-new field and raised and answered questions that no one thought of before. When reviewing this period of history, perhaps we can regard the opposition to him as a praise for his truly original achievements. When the axiomatic set theory was put forward, it was strongly opposed by many mathematicians, and Cantor himself was once a victim of this fierce debate. Under the fierce attack and excessive thinking, he got schizophrenia and had several mental breakdowns. However, after more than 20 years, the set theory has finally been recognized by the world. By the beginning of the twentieth century, set theory had been recognized by mathematicians. Mathematicians are intoxicated with the prospect that all mathematical achievements can be based on set theory. They are optimistic that the whole building of mathematics can be built from the system of arithmetic axioms and with the help of the concept of set theory. At the Second International Mathematical Congress in 1900, the famous mathematician Poincare happily announced that "... mathematics has been arithmetically. Today, we can say that we have reached absolute strictness. " However, this complacency did not last long. Soon, the news that set theory is flawed quickly spread throughout the mathematical world. This is the Russell paradox in 1902. Russell constructs a set R that does not belong to itself (that is, does not contain itself as an element). Now ask whether R belongs to R? If R belongs to R, then R satisfies the definition of R, so R should not belong to itself, that is, R does not belong to R; On the other hand, if R does not belong to R, then R does not meet the definition of R, so R should belong to itself, that is, R belongs to R ... In this way, there is contradiction in any case. This paradox, which only involves set and belongs to the two most basic concepts, is simple and clear, and there is no room for defending the loopholes of set theory. Absolute rigorous mathematics is contradictory. This is the third mathematical crisis in the history of mathematics. After the crisis, many mathematicians devoted themselves to solving it. 1908, zemelo put forward axiomatic set theory, and then improved it, forming an axiomatic system of set theory without contradiction, called ZF axiomatic system for short. The original intuitive concept of set is based on strict axioms, thus avoiding paradox. This is the second stage of the development of set theory: axiomatic set theory. Correspondingly, the set theory founded by Cantor before 1908 is called naive set theory. Axiomatic set theory is a strict treatment of naive set theory. It retains the precious achievements of naive set theory and eliminates its possible paradoxes, thus successfully solving the third mathematical crisis. The establishment of axiomatic set theory marks the victory of a passion expressed by the famous mathematician Hilbert. He shouted: No one can drive us out of the paradise created for us by Cantor. It has been more than one hundred years since Cantor put forward the set theory. During this period, great changes have taken place in mathematics, including the appearance of fuzzy set theory, which further developed the above classical set theory. All this is inseparable from Cantor's pioneering work. Therefore, when we look back at Cantor's contribution, we can still quote the evaluation of his set theory by famous mathematicians at that time as our summary. It is the deepest insight into infinity, the best work of mathematical genius and one of the highest achievements of human pure intellectual activities. Transcendental arithmetic is the most amazing product of mathematical thought and one of the most beautiful manifestations of human activities in the category of pure reason. This achievement may be the greatest work that can be boasted of this era. Cantor's infinite set theory is one of the most disturbing original contributions to mathematics in the past 2500 years. Note: The root of a univariate equation with integer coefficients of degree n is called algebraic number. If all rational numbers are algebraic numbers. Many irrational numbers are also algebraic numbers. For example, root number 2. Because it is the root of the equation x2-2=0. Numbers that are not algebraic numbers in real numbers are called transcendental numbers. In contrast, it is hard to get beyond the number. The first transcendental number is given by joseph liouville in 1844. The proof that π is a transcendental number did not come out until ten years after Cantor's research.