1734, British Archbishop Becquerel published Analytic Scholars or To an Unbelieving Mathematician. Among them, whether the object, principle and inference of modern analysis are clearer in concept or more obvious in reasoning than the mystery and dogma of religion severely criticized the calculus theory at that time. He said that Newton first thought infinitesimal was not zero, and then made it equal to zero, which violated the antinomy, and the number of streams obtained was actually 0/0, because "you got the correct result by double mistakes, although it was unscientific", because the mistakes compensated each other. It is called "Becquerel Paradox" in the history of mathematics. The discovery of this paradox caused some confusion at that time, which led to the second crisis in the history of mathematics and the debate on the basic theory of calculus for more than 200 years.

Becker's attack on infinitesimal aims at demonstrating religious theology, but as the "Becker paradox" itself, it is a question of thinking method. Because mathematics should think according to the non-contradictory law of formal logic, it cannot be admitted that it is not equal to zero or zero in the same thinking process. But the movement of things takes its end point as the limit, and the result of movement is equal to zero in quantity and not equal to zero in starting point. These are two aspects of the movement of things and should not be included in the same thinking process. If we link them mechanically, it will inevitably lead to a paradox in thinking. The cause of Becquerel's paradox lies in the contradiction between the dialectical nature of infinitesimal quantity and the formal characteristics of mathematical methods. The product of the second mathematical crisis-the rigidity of the basic theory of analysis and the establishment of set theory.

After the "Becquerel Paradox" was put forward, many famous mathematicians studied and explored from different angles, trying to rebuild calculus on a reliable basis. French mathematician Cauchy is a master of mathematical analysis. Through the analysis course (182 1), the lecture on infinitesimal computation (1823) and the application of infinitesimal computation in geometry (1826), Cauchy established a modern limit-oriented society. But Cauchy's system still needs to be improved. For example, his language about limit is still vague, relying on things that are intuitive in motion and geometry; Lack of real number theory. The German mathematician Wilstrass is one of the main founders of the foundation of mathematical analysis. He improved the methods of Porzano, Abel and Cauchy, described a series of important concepts such as limit, continuity, derivative and integral in calculus with "ε-δ" method for the first time, and established a strict system of the subject. The proposal and application of "ε-δ" method in calculus marks the completion of calculus operation. In order to establish the basic theorem of limit theory, many mathematicians began to define irrational numbers strictly. In 1860, Weierstrass proposed to define irrational numbers by adding bounded sequence; 1872 Dai Dejin proposed to define irrational numbers by division. 1883, Cantor proposed to define irrational numbers with basic sequence; Wait a minute. These definitions profoundly reveal the essence of irrational numbers from different aspects, thus establishing a strict real number theory, completely eliminating hippasus's paradox, establishing a limit theory based on the strict real number theory, and then leading to the birth of set theory.