The first crisis was that Hibbas discovered that the hypotenuse of an isosceles right triangle with waist 1 could never be expressed by the simplest integer ratio (incommensurability ratio), thus discovering the first irrational number and overthrowing Pythagoras' famous theory. It is said that Pythagoras was at sea at that time, but because of this discovery, Herb was thrown into the sea. The second crisis is that the rationality of calculus is seriously questioned, which almost subverts the whole calculus theory.
The third crisis is Russell's paradox: S is composed of all elements that do not belong to itself, so does S belong to S? In layman's terms, one day Xiaoming said, "I'm lying!" "Ask xiao Ming is lying or telling the truth. The terrible thing about Russell's paradox is that it doesn't involve the profound knowledge of sets like the maximum ordinal paradox or the maximum cardinal paradox. It is simple, but it can easily destroy set theory.
Solve:
After the crisis, mathematicians put forward their own solutions. 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, Tzemero put forward the first axiomatic set theory system according to his own principles, which was later improved by other mathematicians and called ZF system. This axiomatic set theory system makes up for 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 paradox in set theory was successfully eliminated, thus the third mathematical crisis was successfully solved.
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.