1. Hippasu (a native of Mitterrand, 5th century BC) discovered that the hypotenuse of an isosceles right triangle with waist 1 (i.e. the root number 2) can never be expressed by the simplest integer ratio (incommensurable ratio), thus discovering the first irrational number and overthrowing Pythagoras' famous theory. According to legend, Pythagoras were at sea at that time, but because of this discovery, Hebers was thrown into the sea.

Solution:

1. Burnett explained Zhi Nuo's "dichotomy": it is impossible to pass through an infinite number of points in a limited time. Before you complete the course, you must walk half of the given distance, and for this reason, you must walk half of it, and so on until infinity.

Aristotle criticized Zhi Nuo for making a mistake here: "He argued that it is impossible for a thing to pass through infinite things in a limited time, or to contact infinite things separately. It should be noted that length and time are said to be" infinite "with two meanings.

Generally speaking, all continuous things are said to be "infinite" with two meanings: either divided infinity or extended infinity. Therefore, on the one hand, things can't be in contact with infinite things in a limited time.

on the other hand, it can be in contact with things that are divided into infinity, because time itself is divided into infinity. Therefore, passing through an infinite thing is carried out in an infinite time instead of a limited time, and contact with infinite things is carried out in an infinite number instead of a limited number. ?

2. Aristotle pointed out that this argument is the same as the previous dichotomy, and the conclusion of this argument is that slow runners cannot be caught up.

Therefore, the solution to this argument must be the same method. It is wrong to think that something that is ahead in sports can't be caught up, because it can't be caught up during its leading time, but it can be caught up if Zhi Nuo allows it to cross the prescribed limited distance. ?

3. Aristotle thinks that Zhi Nuo's statement is wrong, because time is not composed of inseparable present, just as any other quantity is not composed of inseparable parts. Aristotle believes that this conclusion is caused by taking time as' now'. If we are not sure of this premise, this conclusion will not appear.

4. Aristotle thinks that the mistake here is that he regards the time it takes for a moving object to pass another moving object as the time it takes to pass a stationary object of the same size at the same speed, but in fact the two are not equal.

second, the rationality of calculus has been seriously questioned, which almost overturns the whole calculus theory.

Solution: After Cauchy (the ending person of calculus) defined the infinitesimal by the limit method, the calculus theory was developed and perfected, thus making the mathematics building more brilliant and beautiful!

Third, Russell Paradox: S is composed of all elements that are not its own, so does S include S? In popular terms, Xiao Ming said one day, "I am lying!" " Ask Xiaoming whether he is lying or telling the truth. The terrible thing about Russell's paradox is that it doesn't involve advanced knowledge of sets like the paradox of maximum ordinal number or the paradox of maximum cardinal number. It is simple, but it can easily destroy the set theory!

Solve

1. Eliminate the paradox. After the crisis, mathematicians have put forward their own solutions. It is hoped that Cantor's set theory can be reformed and the paradox can be eliminated 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 sufficiently broad so that all valuable contents in Cantor's set theory can be preserved. "

in p>198, zemero put forward the first axiomatic set theory system based on his own principle, which was later improved by other mathematicians and called ZF system. This axiomatic set system makes up the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems of set theory, such as NBG system proposed by Neumann and others.

2. Axiomatic set system successfully eliminated the paradox in set theory, thus successfully solving the third mathematical crisis. But on the other hand, Russell's paradox has a more profound influence on mathematics. It makes the basic problems of mathematics put in front of mathematicians for the first time with the most urgent need, which leads mathematicians to study the basic problems of mathematics.

and the further development of this aspect has deeply influenced the whole mathematics. For example, around the dispute on the basis of mathematics, three famous schools of mathematics have been formed in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.

Extended information:

In the axiomatic system of classes, some basic concepts are undefined, and we can only explain them from their objective meanings, but such explanations only help to understand these concepts.

Any object studied in mathematics is called a class. The concept of class is unlimited. There may be a relationship between classes called belonging. Class A belongs to class B, and class A is also called an element of class B (called meta for short).

we can understand a class as a whole consisting of several elements. Whether a class is an element of another class is completely certain, which is the certainty of class elements. If class a is not an element of class b, it is said that a does not belong to B.

Reference: Baidu Encyclopedia-the third mathematical crisis

Reference: Baidu Encyclopedia-the second mathematical crisis

Reference: Baidu Encyclopedia-the first mathematical crisis

Reference: Baidu Encyclopedia-the three major mathematical crises.