19 In the 1970s, German mathematician G. Cantor gave a relatively complete set theory and studied the ordinal number and cardinality of infinite sets. At the beginning of 20th century, Russell's paradox pointed out the contradiction of Cantor's set theory. In order to overcome the paradox, people try to axiomatize the set theory and constrain the set with axioms.

The first commonly used axiomatic system is the ZF system proposed by Zemelo and frenkel. There is only one illogical binary relation symbol ∈ in this system. The illogical axioms are zermelo-fraenkel, empty set axiom, disordered pair axiom, union axiom, power set axiom, infinite axiom, separation axiom mode, replacement axiom mode and regular axiom. If axiom of choice is added, the ZFC system will be formed. Using axioms, we can define empty sets, ordered pairs, relations, functions and other sets, and also give concepts such as ordered relations, well-ordered relations, ordinal numbers, cardinality, natural numbers, integers and real numbers.

Through metalanguage, we can also understand the compatibility and independence of axioms in axiomatic systems. For example, Cohen founded the coercive method in axiomatic set theory in 1960, and used it to prove that ZFC and continuum assume that CH is independent. Axiomatic set theory develops rapidly, new axioms and methods such as Martin's axiom and suslin's hypothesis are widely used, and the research on combinatorial set theory, descriptive set theory, large cardinality and coercive method is also developing.