What are the important elements involved in the three crises in the history of the development of mathematics

What are the important elements involved in the three crises in the history of the development of mathematics are as follows: mathematical concepts such as irrational numbers, calculus and sets.

Crisis 1, Hippasus (a native of Mittabanden, around 470 BC) discovered that the hypotenuse (i.e., the second power root of 2) of an isosceles right triangle with a waist of 1 can never be expressed in terms of the ratio of the simplest integers (the ratio of the incommensurable degrees), and thus discovered the first irrational number, overthrowing Pythagoras's famous theory. It is rumored that the Pythagoreans were at sea at the time, but threw Hippas into the sea because of this discovery.

Crisis two, the rationality of calculus was seriously questioned and came close to overturning the entire theory of calculus. Crisis three, Russell's paradox: S consists of the set of everything that is not an element of itself, so does S belong to S? In layman's terms, Ming said one day, "I'm lying!" Ask Ming whether he is lying or telling the truth. The scary thing about Russell's paradox is that it doesn't involve high knowledge of sets like the maximal ordinal paradox or the maximal cardinal paradox; it's simple, yet it can easily destroy set theory.

After the crisis arose, mathematicians came up with their own solutions. It was hoped that the paradox could be ruled out by revamping Cantor's set theory and by placing restrictions on the definition of sets, which required the establishment of new principles. These principles had to be narrow enough to ensure the exclusion of all paradoxes; on the other hand, they had to be sufficiently broad so that everything of value in Cantor's set theory would be preserved.

In 1908, Zermelo based this principle of his own on the first system of axiomatized set theory, which was later improved by other mathematicians and became known as the ZF system. This axiomatized set system largely made up for the shortcomings of Cantor's plain set theory. In addition to the ZF system, there are various other axiomatic systems of set theory, such as the NBG system proposed by Neumann and others.