Around 1900, there are three famous paradoxes in mathematical set theory, and Barber paradox is a popular expression of Russell paradox. In addition, there are Cantor Paradox and Bly-Forcy Paradox. These paradoxes, especially Russell's paradox, caused great shock in mathematics and logic at that time. Triggered the third crisis of mathematics.

paradox

Let's first understand what paradox is. Paradox comes from the Greek word "para+dokein", which means "think more". The meaning of this word is rich, including all mathematical conclusions that contradict human intuition and daily experience, and those conclusions will surprise us. Paradox is a contradictory proposition. That is, if this proposition is admitted, it can be inferred that its negative proposition is established; On the other hand, if we admit the negative proposition of this proposition, we can deduce that this proposition is true. If the admission is true, after a series of correct reasoning, the conclusion is false; If you admit that it is false, after a series of correct reasoning, it is true. There are many famous paradoxes at home and abroad, which have impacted the foundation of logic and mathematics, stimulated people's knowledge and precise thinking, and attracted the attention of many thinkers and enthusiasts throughout the ages. Solving paradoxes requires creative thinking, and the solution of paradoxes can often bring people new ideas. Paradox has three main forms. 1. An assertion seems to be definitely wrong, but it is actually right (paradox). 2. An assertion seems to be definitely right, but it is actually wrong (specious theory). A series of reasoning seems impeccable, but it leads to logical contradictions.

Russell paradox definition:

M: all sets containing the set itself;

N: all sets that do not contain the set itself;

Q: N∈M or n?

If N ∈M, it means that n has the characteristics of m. According to the definition of m, n contains the set itself.

But this contradicts the definition of n; If N ∈N, it means that n has its own characteristics, which are different from those of n.

Defining contradictions; But m+n traverses all set domains, so n is not an empty set.

Thus, the paradox has arisen.

Examples of Russell Paradox:

There is a story in the world literary masterpiece Don Quixote:

Sancho Panza, the servant of Don Quixote, ran to an island and became the king of the island. He made a strange law: everyone who arrives on this island must answer a question: "What are you doing here?" If the answer is right, let him go to the island to play, if the answer is wrong, hang him. For everyone who comes to the island, they will either have fun or be hanged. How many people dare to risk their lives to play on this island? One day, a bold man came. Ask him this question as usual, and the man's answer is: "I'm here to hang myself." Will Sancho Panza let him play on the island or hang him? If he should be allowed to play on the island, this is inconsistent with what he said about being hanged, that is, what he said about being hanged is wrong. Since he is wrong, he should be hanged. But what if Sancho Panza wanted to hang him? At this time, what he said "to be hanged" was true and correct. Since he answered correctly, he should not be hanged, but should be allowed to play on the island. The king of this island found that his laws could not be enforced, because they would be destroyed anyway. He thought and thought, and finally let the guard let him go and declared the law invalid. This is another paradox.

The paradox put forward by the famous mathematician Bertrand Russell (Russel, 1872- 1970) is similar:

There is a barber in a certain city. His advertisement reads: "My haircut skills are superb and the whole city is famous. I will shave all the people in this city who don't shave themselves. I will only shave these people. I would like to extend a warm welcome to everyone! " When people come to him to shave, they naturally don't shave themselves. One day, however, the barber saw in the mirror that his beard had grown. He instinctively grabbed the razor. Do you think he can shave himself? If he doesn't shave himself, then he belongs to the "person who doesn't shave himself" and he has to shave himself. What if he shaved himself? He belongs to the "person who shaves himself" and should not shave himself.

Barber paradox and Russell paradox are equivalent. Because, if everyone is regarded as a set, then the elements of this set are defined as the objects that this person shaves. Then, the barber claimed that his element was all the collections in the village that did not belong to him, and all the collections in the village that did not belong to him. So does he belong to himself? Thus, Russell's paradox is obtained from Barber's paradox. The same is true of reverse transformation.

affect

/kloc-In the second half of the 9th century, Cantor founded the famous set theory, which was severely criticized by many people when it was first produced. But soon this groundbreaking achievement was accepted by mathematicians and won wide and high praise. Mathematicians found that starting from natural numbers and Cantor's set theory, the whole mathematical building could be established. Therefore, set theory has become the cornerstone of modern mathematics. The discovery that "all mathematical achievements can be based on set theory" intoxicated mathematicians. 1900, at the international congress of mathematicians, poincare, a famous French mathematician, declared cheerfully: "… with the help of the concept of set theory, we can build the whole mathematical building … today, we can say that we have reached absolute strictness …"

However, the good times did not last long. 1903, a shocking news came out: set theory is flawed! This is the famous Russell paradox put forward by British mathematician Russell. Russell's paradox leads to the crisis of set theory. Very simple and easy to understand, only involving the most basic things in set theory. So Russell's paradox caused a great shock in mathematics and logic at that time when it was put forward. Frith, a famous German logician, received Russell's letter about this paradox when his Foundation of Set Theory was completed for printing. He immediately found that a series of achievements he had been busy with for a long time were all messed up by this paradox. He can only write at the end of the book: "The worst thing for a scientist is to find that the foundation of his work has collapsed when his work is about to be completed."

1874, the German mathematician Cantor founded the set theory, which soon penetrated into most branches and became their foundation. By the end of 19, almost all mathematics was based on set theory. At this time, some contradictory results appeared in set theory, especially the paradox reflected in the barber's story put forward by Russell in 1902, which is extremely simple and easy to understand. In this way, the foundation of mathematics has been shaken passively, which is the so-called third "mathematical crisis".

After the publication of Russell Paradox, a series of paradoxes (later classified as so-called semantic paradoxes) were discovered: 1, Richard Paradox 2, Perry Paradox 3. The paradox of Green and Nelson.

solve

Russell paradox puts forward that mathematicians put forward their own solutions after the crisis. I hope to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles. "These principles must be narrow enough to ensure that all contradictions are eliminated; On the other hand, it must be broad enough so that all valuable contents in Cantor's set theory can be preserved. " 1908, zemelo put forward the first axiomatic set theory system on the basis of his own principles. Later, this axiomatic set theory system made up the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems in set theory, such as NBG system proposed by Neumann et al. The establishment of axiomatic set system successfully eliminated the paradox in set theory, thus successfully solving the third mathematical crisis. On the other hand, Russell's paradox has a far-reaching influence on mathematics. It puts the basic problems of mathematics in front of mathematicians for the first time with the most urgent needs, and guides mathematicians to study the basic problems of mathematics. The further development of this aspect has profoundly affected the whole mathematics. For example, the debate on the basis of mathematics has formed three famous schools of mathematics in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.

The above briefly introduces three mathematical crises and experiences caused by paradox in the history of mathematics, from which we can easily see that paradox has greatly promoted the development of mathematics. Some people say that "asking a question is half the solution", and the paradox is exactly what mathematicians can't avoid. It said to the mathematician, "solve me, or I will swallow your system!" " As Hilbert pointed out in On Infinity: "It must be admitted that in the face of these paradoxes, the current situation we are in cannot be tolerated for a long time. People imagine that in mathematics, a model called reliability and truth value, the conceptual structure and reasoning methods that everyone has learned, taught and applied will lead to unreasonable results. If even mathematical thinking fails, where should we look for reliability and authenticity? "The emergence of paradox forces mathematicians to devote their greatest enthusiasm to solving it. In the process of solving the paradox, various theories came into being: the first mathematical crisis led to the birth of axiomatic geometry and logic; The second mathematical crisis promoted the perfection of the basic theory of analysis and the establishment of set theory; The third mathematical crisis promoted the development of mathematical logic and the emergence of a number of modern mathematics. Mathematics has developed vigorously from this, which may be the significance of mathematical paradox, and Russell paradox has played an important role in it.

Reason cannot answer questions about itself, which was discovered in Kant's time. There are irreparable loopholes in logic, but it is the only way for people to know the world. In the end, you will find that either you deny rationality or you deny faith. Because the so-called struggle between idealism and materialism is a pure rational science based on such an incomplete logical system. Since reason cannot judge itself, the choice of position cannot be based on reason, thus becoming an essential superstition. Of course, if you insist that your position is in line with the so-called science or practice, then in fact you are neither a materialist nor an idealist, but in essence you are just a kind of pan-empiricism or pan-logicism. Of course, the logicism here is certainly not Russell's, but just an image point.