Continuum hypothesis, a mathematical hypothesis about continuum potential. Usually written as CH. Usually, the set of real numbers, that is, the set of points on a straight line, is called continuum, and the potential (size) of continuum is recorded as C 1. For more than 2000 years, people have always thought that any two infinite sets are the same size. It was not until 189 1 year that G. Cantor proved that the power set of any set (that is, the set composed of all its subsets) is greater than the potential of this set that people realized that infinite sets can also be compared in size. Natural number set is the smallest infinite set, and natural number set's potential is recorded as Alev zero. Cantor proved that the potential of continuum is equal to that of natural number set's power set. Is there an infinite set whose potential is greater than that of natural number set and less than that of continuum? This problem is called the continuum problem. Cantor conjectures that the answer to this question is no, that is, the continuum potential is the smallest infinite potential among the potentials greater than natural number set potential, which is recorded as c1; The potential of natural number set is recorded as C0. This conjecture is called the continuum hypothesis. In 1938, K. Godel proved that CH is in harmony with ZF axiomatic system (see axiomatic set theory), and in 1963, P. J. Cohen proved that CH is independent of ZF axiomatic system, so it is impossible to judge whether it is true or false. Therefore, in the ZF axiomatic system, it is impossible for CH to judge whether it is true or false. However, in the 2 1 century, the previous conclusions began to be passively shaken again.

Cantor proved that the cardinality of the continuum is equal to the cardinality of the natural number set power set, and recorded it as 2s╲s0. Cantor also arranged the infinite cardinality as s╲s0, s╲s 1, ... south ... where A is an arbitrary ordinal number, Cantor guessed 2s ╲ sa = s ╲. Generally speaking, for any ordinal number A, it is concluded that 2s╲sa=s╲sa+ 1 holds, which is the so-called generalized continuum hypothesis (GCH for short). In 1938, Godel proved that CH and ZFC are relatively coordinated, and in 1963, Cohen proved that CH is independent of ZFC. The results of Godel and Cohen show that CH is undecidable for ZFC. This is one of the greatest advances in set theory in the 1960s.