Russell Paradox: Let the property P(x) mean "X does not belong to X". Now suppose that a class A is determined by the property p-that is, "A={x|x? A} .

Then the question comes: Is it true that A belongs to A? First of all, if A belongs to A, then A is an element of A, then A has the property P, which shows that A does not belong to A; Secondly, if A does not belong to A, that is to say, A has the property P, and A consists of all classes with the property P, then A belongs to A. ..

There are some popular descriptions about Russell Paradox, such as Barber Paradox and Bibliography Paradox.

Barber paradox:

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.