The second mathematical crisis stems from the use of calculus tools. Because the definition is not strict and the concept of infinitesimal is controversial, the real number theory and limit theory are finally established, which makes the mathematical analysis have a strict foundation.
The third mathematical crisis is about set theory, which is the famous Russell paradox. The definition of set is attacked and finally solved by different axiomatic systems, which makes the subject of mathematical logic develop.
Three mathematical crises in history have brought great troubles to people. The emergence of the crisis made people realize the defects of the existing theory. The appearance of paradox in science often indicates that human understanding will enter a new stage, so paradox is the product of scientific development and one of the sources of scientific development. The first mathematical crisis made people discover irrational numbers, established a complete real number theory, and Euclidean geometry came into being, establishing a geometric axiom system. The emergence of the second mathematical crisis directly led to the emergence and perfection of the three theories of limit theory, real number theory and set theory, which made calculus on a solid and perfect basis. The third mathematical crisis made set theory a complete axiomatic system of set theory (ZFC system), which promoted the basic research of mathematics and the modernity of mathematical logic.