Von Neumann is one of the most important mathematicians in the 20th century. He has made outstanding contributions to both pure mathematics and applied mathematics. His work can be roughly divided into two periods: before 1940, he mainly studied pure mathematics: he put forward a simple and clear ordinal number theory in mathematical logic, and made a new axiomatization of set theory, in which set and class were clearly distinguished; Later, he studied the spectral theory of linear self-adjoint operators on Hilbert space, thus laying the mathematical foundation of quantum mechanics; From 65438 to 0930, he proved that the average ergodic theorem opened up a new field of ergodic theory; In 1933, he solved Hilbert's fifth problem by using compact groups. In addition, he also made pioneering contributions in the fields of measure theory, lattice theory and continuous geometry. From 1936 to 1943, he cooperated with Murray to establish the operator ring theory, which is now called von Neumann algebra.