Introduction
Paradox production --- Third mathematics crisis
1897, Falli reveals the first paradox in the collection . Two years later, the controversy found very similar paradox. In 1902, Russell discovered a paradox, which in addition to the concept of not involving the collection concept itself. Russell paradise has been popular in a variety of forms. The most famous of these is that Russell is given in 1919, which involves the dilemma of a village. The hairdresser announced such a principle: he gave all people who didn't give yourself a face, and only gave some people such as the village. When people try to answer the following questions, they recognize the paradox nature of this situation: "Do you give yourself a face?" If he doesn't give yourself, then he should shave his own by principle; if he gives yourself Shaving, then he does not meet his principles.
Russell paradise accurate expression:
If there is a collection a = {x | x∉ a}, whether the x∈a is established? If it is established, X∈A is not satisfied with the characteristic nature of A. If it is not established, A is satisfied with the characteristics.
Russell Paradox makes the entire mathematics building. No wonder that Frege After receiving the letter from Russell, he wrote at the end of the "Basic Law" of the "Arithmetic Basic Law", and a scientist would not encounter more embarrassed things than this. When the work is completed, its foundation has fallen. When the book is waiting to be printed, a letter from Mr. Russell is placed in this situation. " So I ended the hardships of nearly 12 years.
Recognizes infinite collection, recognizes the infinite base, as if all disasters come out, this is the essence of the third mathematics crisis. Although the paradox can be eliminated, the contradiction can be resolved, but the determinism of mathematics is lost step by step. The big bunch of modern aircraft aggregation is difficult to say, but they can't eliminate them, they have a flesh with the whole mathematics. Therefore, the surface of the third crisis is solved, and it is essentially more deeply in other forms.
Background
The third mathematics crisis is generated in the end of the 19th century and early 20th century. At that time, it was a period of unprecedented prosperation of mathematics. The first is the logic's mathematical, which has prompted the academics of the JPC.
The collection of Concentrates in 1970s is the basis of modern mathematics, and it is also a direct source of crisis. At the end of the nineteenth century, Dadejin and Pikiano routines the arithmetic and realistic theory and promoted the axioming movement. The maximum achievement of the axioming movement is the axioming of Hilbert in 1899 for the first-class geometric.
Definition
In order to clear the dragon debut of the third mathematical crisis, we must first explain what is a mathematical crisis. Generally speaking, the crisis is an intensive, non-resolved contradiction. From philosophical, the contradiction is ubiquitous, inevitable, even if it is undoubtedly known as mathematics.
Many contradictions in mathematics, such as positive and negative, addition and subtraction, differential and integral, alignment and non-irrelevant, real numbers, etc. However, there are many profound contradictions throughout the mathematical development, such as poor and endless, continuous and discrete, and even existence and constructing, logic, intuitive, specific objects, and abstract objects, concepts and calculations. In the history of the overall mathematical development, it runs through the contradiction struggle and resolution. When the contradiction is intensified to the basis involving the entire mathematics, a mathematical crisis is produced.
Contradiction, the crisis is solved, often brings new content, new progress, and even revolutionary changes, which also reflects the basic power of the contradiction is the basic power of the development of things. principle. The history of the overall mathematics is the history of contradictory struggles. The result of struggle is the development of mathematics.
Study
Humans earliest aware of the natural number. Experience from the introduction of zero and negative numbers: either introduce these numbers, or a large number of subtraction will not pass; Similarly, the introduction fraction makes multiplication with counterputting - division, otherwise many practical problems cannot be resolved. But then there will be such a problem, all the amounts can be used to use rational numbers? So the number of unreasonable numbers has led to the first mathematical crisis, and the resolution of the crisis also enables logic development and geometry.
The solution of the equation has caused the appearance of imaginary, and the imaginary is considered "not" from the beginning. However, this untrue number can solve the problem that the implementation cannot solve, thus fighting for himself.
The development of geometry develops from the union of the geometric world of European to various non-European geometry. In the 19th century, many problems that cannot be solved with traditional methods. If the above five times or more algebra equations cannot pass the addition, minus, multiply, divide, division, and the roots; ancient Greek geometry three major problems, That is, three-level dismisions, double cubes, and the circle cannot be solved by conjunction, and the rule will be solved.
These negative results indicate the limitations of the traditional method, but also reflect the in-depth of human understanding. This discovery brings great impact to these disciplines, almost completely changes their direction. For example, in the development of the album, the roots of the solution of the solution have become an analysis and calculation of mathematics. In the third mathematical crisis, this situation has also appeared many times, especially the incompleteness of the formal system, including integer arithmetic, and many problems have greatly improved people's understanding, and also promotes the logic of the proactive logic. Great development.
Development
This contradiction, the development of the crisis, changing the face, and even causing revolution, in history in mathematics development. The second mathematics crisis is caused by an infinite amount of contradiction, it reflects the limited and endless contradictions of mathematics. Mathematics has also been used through the calculation method, and the analysis method is clear and logically strictly contradictory in application and concept. In this regard, pay more attention to practical mathematicians blind application. More pay more attention to strict mathematicians and philosophers put forward criticism. Only after these two aspects have been coordinated, contradictions can be solved. The lane operator calculation and δ function also repeated this process, starting to be a formal calculation, arbitrary application until Schwarz lays a strict system of generalized function.
Crisis
For the third mathematical crisis, some people think is just the crisis of mathematics, and have nothing to do with mathematics. This view is a one-sided. It is true that the problem involves the logic and aggregation theory, but it also involves infinity collection at the beginning, and modern mathematics can say inch, if you get out of infinity collection, you can say inch. Because if only consider a limited set or at most, the most mathematics will not exist. And even if these limited mathematics content, there are many problems to involve inferior methods, such as solving many of the problems in the number of questions. From this point of view, the third mathematics crisis is a deep mathematical crisis.
Solve
Mathematics, by excluding the existence of such a collection by means of a collection of constructor.
For example, in the ZF axiom system (also known as the ZFC system) proposed by Zermelo and France (also known as ZFC system), a collection of conditions (simply Said, there is an empty set [empty collection axiom]; there is a power set [power collection axiom] for each collection; all the collection of each collection also forms a collection [and agreed axiom]; each collection satisfies a condition Element constitutes subset [subset axiom]; a "defined domain" "function" exists "value domain" [replacement axiom], etc.), which cannot define a collection in the paradox.
The third mathematics crisis is resolved.