Collection Theory is a basic branch of mathematics, accounting for an extremely unique position in mathematics, and its basic concept has penetrated into all areas of mathematics. If modern mathematics is compared to a brilliant building, it can be said that the collection is that it is the cornerstone of this building, thereby visible it in mathematics. Its founder Condur also is known as one of the practitioners who have the most influence of mathematics development in the 20th century. Let's go to explore the unique and important mathematical theory of this unique math, chasing the twists and turns it walked.
Birth of the aggregation
The collection theory is found in the late 19th century, famous mathematician Condur. A new branch appeared in the seventeenth century mathematics: calculus. This new discipline has got a rapid development and a fruitful result. Its promotion speed will not be asked to check and consolidate its theoretical basis. At the beginning of the 19th century, many urgent issues have been resolved, and there has been a movement of reconstruction mathematics. It is in this movement that Condur has begun to explore the real number of points that have never hitted, which is the beginning of the research. By 1874 Condal began to generally propose the concept of "collection". He is the definition of the collection is: combining some of the determined things (whether specific or abstract) things, as a whole, is called a collection, which is called the set of elements. In the first day of the first day of the intention of the integrated theory of thought in the letter to Dadekin, December 7, 1873. Condal's immortal merits, in the middle school mathematics, it is only the most basic knowledge of aggregation theory. During the study, students may feel that everything is natural and simple. It is not possible to imagine that it is fiercely opposed by the birthday, and it is not the same as Cantone's merit. Former Soviet mathematician Kirmo Golf evaluated Condal's work and said: "The immortal performance of Conal is that he moves into endless adventures." Thus only when we understand what Cantone has made some conclusions in infinite research, it will truly understand the value of his work and the sound of many opposition.
Mathematics and infinite have no solution, but in the infinite road, it is filled with traps. For this reason, in the course of mathematical development, mathematician always looks endless with a suspicion, and avoids this concept as much as possible. But trying to grasp the infinite Condal but bravely embarked on this trap. He introduced the word infinite set into mathematics, which entered a unknown virgin, opening up a wonderful new world. The study of infinite set has made him open "unlimited" mathematical Pandora box. Let's take a look at what he releases after the box is open.
"We put the collection of all the natural numbers as a natural number set, expressed in the letter N." After learning the chapter of the collection, the students should not feel unfamiliar with this sentence. But the students can't think of this year when Cantale is doing this year when accepting this sentence. In this case, the mathematicians just extends unlimited as they always extend, and a changed thing is explained. Infinity is always in the structure, it will never complete, it is potential, not really. This kind of endorsement is known in mathematics. The Mathematics Prince of the 18th Century Hold this view. In his words, it is "... I oppose the infrequent number as an entity, this is never allowed in mathematics. The so-called endless, just a way to speak ..." and when Cantone regards all the natural numbers When a collection, he used the unlimited overall as a structure to be completed, so he affirmed the endless endless endorsement, this concept is called real infinite ideas. Due to the unlimited thoughts of unlimited ideas have been fully victorious in the basis of the calculus, Cantone's unlimited idea was ignorant of some mathematicians in some mathematicians. However, Condur did not stop this, he continued to be inferior in a completely unprecedented manner. He has further concluded a series of conclusions on the basis of unlimited concept, and founded exciting and meaningful theory. This theory makes people really entered an unpredictable strange world.
It is best to show his originality of his number of infinite set elements. He proposed a number of infinite set elements with one-to-one correspondence. He called the elements to build a corresponding collection, and his own concept is an equal potential. Since an infinite set can be established with its true subset -, for example, students can find a corresponding relationship between the natural numerical set and the piped number set - that is, the infinite set can be with its true subset, etc. Potential, that is, the same number. This is contradictory with the traditional concept "all of the part". And Cantale believes this is just an infinite feature. In this sense, the natural numerical set is the same number with the piped number set, and he refers to it as an agreed. It is also easy to prove that the rational number of columns and natural numerical sets, so the rational quantity set is also a number of sets. Later, when he proved that the algebra [Note] Collection is also a set of aggregate, a natural idea is that the infinity is clear, all of which is a number set. It is unexpectedly that he proves that the real set of real sets in 1873 is greater than the number of natural numbers. This not only means the number of unreasonable numbers than the number, and it is clear that the large number of algebraic Note is only a seas in the sea than the number of records, as someone describes: "The number of algex embellished is like night sky. Star; the silent night sky is composed of transcendence numbers. "And when he got this conclusion, people can find only one or two. What is a shocking result! However, things did not end. Once the magic box is open, it cannot be combined again, and it is no longer limited to the number of unexbeatable monsters. From the above conclusion, Cantalence aware of the difference between infinity, with different orders of magnitude, can be divided into different levels. The next step he has to do is to prove that there is still an infinite level between all infinite sets. He has achieved success and has a complete sequence for all kinds of infinity, and he is called "super limit" according to the infiniteness of infinity. He used the first letter "Alav" in the Hebele alphabet to express the excessive elves, and finally he established an unlimited so-called Arev spectrum which can be unlimited. In this way, he created a new super-limiting theory that depicts a complete picture of an infinite kingdom. It can be imagined that we still feel that there are some whimsical conclusions on how to vibrate mathematicians at the time. It is not exaggerated that Conor's infinite theory has caused the opposition to the endless hustle and bustle. They called yelling against his theory. Some people laugh at the collection theory is a "disease". Some people ridicule the limit is "the fog in the fog", called "Cantone entered the overrun of hell". As a new year of traditional concept, because he has created a new field, it was proposed that his theory was fiercely refuted. When looking back at this history, maybe we can think of him against him is a kind of praise that really has its own original fruit.
At the beginning of the integration, the intense opposition to many mathematicians, Condal himself once became the fierce victim. In the violent attack and excessive use of brain, he has a schizophrenia, which is trapped in a spirit crashed several times. However, the collection theory has experienced more than 20 years before and after, and finally it has been recognized by the world. By the 20th century episode has been agreed with mathematicians. Mathematics can be built for all mathematics results in the foreground on the basis of collection. They optimistically think that from the arithmetic system, with the concept of aggregate theory, they can build the entire mathematics building. At the Second International Mathematics Conference in 1900, the famous mathematician Pangola announced that "..." Mathematics has been arithmetic. Today, we can say that it is absolutely strict. "But this kind of self-service The emotions did not last long. Soon, the collection theory is a vulnerability message quickly spread through the mathematics. This is the Russell paradox derived from Russeu in 1902. Russell constructs a collection R that does not belong to itself (ie, does not contain itself as an element). Now ask whether R is R? If R belongs to R, R satisfies the definition of R, so R should not belong to it, that is, R does not belong to R; on the other hand, if r does not belong to R, R does not meet the definition of R, so R should belong to itself, ie R belongs to R. In this way, there is a contradiction in any case. This only involved in the collection and the paradox belonging to the two most basic concepts is so simply, and there is no room for the collection of a collection of vulnerabilities. Absolute strict mathematics fall into contradictory contradictions. This is the third mathematics crisis in mathematical history. After the crisis is generated, many polyers are invested in the work of resolving the crisis. In 1908, Cori Luo put forward the aggregation of the axioms, and then improved the form of non-contradictory collection on the axiom system, referred to as the ZF axiom system. The original intuitive collection concept is established on a strict system, which avoids the set of condensed Conale founded by the paradox. The Commercial Type of Affairs is the strict treatment of simplicity aggregation. It retains the valuable results of simplicity collection and eliminates the paradox it may exist, and thus solves the third mathematics crisis. The establishment of the aircraftization aggregation, marking a passionate victory expressed by the famous mathematician Hutbert, and he screamed: no one can drive us from Cantor to create a paradise created by us.
Since the contestation of Conale, the time has been over a hundred years. In this period, mathematics has undergone great changes, including the emergence of further development of the above classics, etc. . And all of this is not open to Cantonese's pioneering work. Therefore, when he looks back at Condur's contribution, we can still quote the evaluation of the famous mathematician on his collection as our summary.
The most profound insight, it is the best work of mathematics genius, one of the highest achievements of human pure intelligence activities. Over-limit arithmetic is the most amazing product of mathematics ideas, one of the most beautiful performances of human activities in purely rational. This achievement may be the greatest job of this era. Conale's infinite set is one of the most disturbing original contributions to mathematics in the past two hundred and fifty years.
Note:
The root of the whole factor is one dollar N times equation, the number of times. If all the rational numbers are algebraes. A large number of unreasonable numbers is also an algebra. Such as the root number 2. Because it is the root of equation X2-2 = 0. The number of independent numbers is called transcendence. In contrast, transcendence is hard to get. The first transcendence was given by Liu Wei in 1844. About π is a proven to be over-numbered in Condur's study after ten years.
