In order to understand the basis of mathematics, mathematical logic and set theory are developed. German mathematician Georg Cantor (1845- 19 18) initiated the set theory, boldly marched into infinity, provided a solid foundation for all branches of mathematics, and its own content was quite rich, put forward the existence of real infinity, and made inestimable contributions to the future development of mathematics. Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Even the mathematician Pi Aucar, who is famous for his "profound and creative", compared set theory to an interesting "morbid situation", and even his teacher Kroneck hit back at Cantor as a "mental derangement" and "walked into a hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said: "My theory is as firm as a rock, and anyone who opposes it will shoot himself in the foot." He also pointed out: "The essence of mathematics lies in its freedom, and it is not bound by traditional ideas." This argument lasted for ten years. Cantor suffered from schizophrenia at 1884 because of frequent depression, and finally died in a mental hospital.

However, after all, history has fairly evaluated his creation. Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. At the beginning of the 20th century, Hilbert, the greatest mathematician in the world, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Russell praised Cantor's works as "the greatest works that can be boasted in this era".

Mathematical logic focuses on putting mathematics on a solid axiomatic framework and studying the results of this framework. It is the birthplace of Godel's second incomplete theorem, which is perhaps the most widely spread achievement in logic-there is always a true theorem that cannot be proved. Modern logic is divided into recursion theory, model theory and proof theory, which are closely related to theoretical computer science.