This paper gives a new definition of set and introduces the concept of elastic set with variable extension. Without any unprovable or unreliable proposition, hypothesis or conjecture, it eliminates Russell's paradox, Cantor's paradox, Galileo's paradox and Infinite Hotel's paradox. Reveals the essence of infinityand gives the expression of the number of elements of infinite set, Corrects the error that an infinite set can correspond to its proper subset one by one. At the same time, the hypothesis that there is an unique natural numbers set that has included all natural numbers has been refuted. It is pointed that the diagonal Argument and Cantor theorem have not proved that real numbers can not be one-to-one corresponded to natural numbers. Since the set is variable, we can also use the variable things such as variables or functions as the elements of the set.There are so many mistakes in the most strict mathematics, let alone the humanities? It shows that the wisdom of modern Westerners, who lack the gene of cultural diversity, is far less than that of the ancient Greek era, or even degenerated to the barbaric era. No wonder a good world would fight for some differences that may not be meaningful, and even foolishly walked to the edge of nuclear war!