Three crises in mathematics involve irrational numbers, calculus and set.

1, Crisis 1, hippasus (a native of Mitterrand, about 470 BC) found that the hypotenuse of an isosceles right triangle with a waist of 1 (that is, the square root of 2) could never be expressed by the simplest integer ratio (incommensurable ratio), thus discovering the first irrational number and overthrowing Pythagoras.

2. Crisis 2. The rationality of calculus is seriously questioned, and the whole calculus theory is almost overthrown.

3. Crisis 3, Russell 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.

Extended data:

Eliminate paradox

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.

Axiomatic set system

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.