The paradox of irrational numbers, Pythagoras put forward the view that "everything is a number", the element of number is the element of everything, and everything in the world can't be expressed by numbers, and numbers themselves are the order of the world. A disciple of the Pythagorean school, Hibersos, found that the diagonal of a square is incommensurable with the length of one side, which is quite different from the Pythagorean school's philosophy of "everything counts" (referring to rational numbers). This is the first mathematical crisis, from which human beings discovered irrational numbers. 2. Zeno Paradox "Dichotomy": An object moving to its destination must first pass through the midpoint of the journey, but to pass this point, it must first pass through the 1/4 point of the journey, and so on. The conclusion is that infinity is an endless process and movement is impossible. "Achilles (the running hero in Homer's epic) can't catch up with the tortoise": Achilles always reaches the starting point of the tortoise first, so the tortoise must always run ahead. This argument is the same as the dichotomy paradox, except that it is not necessary to divide the required distance equally again and again. "The arrow doesn't move": It means that the arrow must be in a certain position at any time during the movement, so it is stationary, so it can't be moving. This is the second mathematical crisis, which has perfected the definitions of limit and calculus. 3. The popular version of Russell's paradox is that a barber in a small town releases grandiloquence: "I only shave all the people in the city who don't shave." But the question is: Should the barber shave himself? If he shaves himself, don't shave only those who don't shave themselves, as he said. But if he doesn't shave himself, he should shave himself, as he said, "shave everyone who doesn't shave himself in the city." This is the third mathematical crisis, which finally formed the present axiomatic set theory.

Satisfied, please adopt.