1874, Cantor speculated that there was no other cardinality between countable set cardinality and real set cardinality, that is, the famous continuum hypothesis. 1938, Austrian mathematical logician Godel living in the United States proved that there is no contradiction between the continuum hypothesis and the axiomatic system of ZF set theory. 1963, American mathematician P.Choen proved that the continuum hypothesis and ZF axiom are independent of each other. Therefore, the continuum hypothesis cannot be proved by ZF axiom. In this sense, the problem has been solved.
(2) Arithmetic axiom system is not contradictory.
The contradiction of Euclidean geometry can be summed up as the contradiction of arithmetic axioms. Hilbert once put forward the method of proving formalism plan, but Godel's incompleteness theorem published in 193 1 denied it. Gnc(G. genta en,1909-1945)1936 proved the non-contradiction of the arithmetic axiomatic system by means of transfinite induction.
(3) It is impossible to prove that two tetrahedrons with equal base and equal height are equal in volume only according to the contract axiom.
The significance of the problem is that there are two tetrahedrons with equal height, which cannot be decomposed into finite small tetrahedrons, so that the congruence of the two tetrahedrons (M. DEHN) has been solved in 1900.
(4) Take a straight line as the shortest distance between two points.
This question is rather general. There are many geometries that satisfy this property, so some restrictions are required. 1973, the Soviet mathematician Bo gref announced that this problem was solved under the condition of symmetrical distance.
(5) Conditions for topology to be a Lie group (topological group).
This problem is simply called the analytic property of continuous groups, that is, whether every local Euclidean group must be a Lie group. 1952 was solved by Gleason, Montgomery and Zipin. 1953, Hidehiko Yamanaka of Japan got a completely positive result.
(6) Axiomatization of physics, which plays an important role in mathematics.
1933, the Soviet mathematician Andrei Andrey Kolmogorov axiomatized probability theory. Later, he succeeded in quantum mechanics and quantum field theory. However, many people have doubts about whether all branches of physics can be fully axiomatized.
(7) Proof of transcendence of some numbers.
It is proved that if α is algebraic number and β is algebraic number of irrational number, then α β must be transcendental number or at least irrational number (for example, 2√2 and eπ). Gelfond of the Soviet Union (1929) and Schneider and Siegel of Germany (1935) independently proved its correctness. But the theory of transcendental number is far from complete. At present, there is no unified method to determine whether a given number exceeds the number.
(8) The distribution of prime numbers, especially for Riemann conjecture, Goldbach conjecture and twin prime numbers.
Prime number is a very old research field. Hilbert mentioned Riemann conjecture, Goldbach conjecture and twin prime numbers here. Riemann conjecture is still unsolved. Goldbach conjecture and twin prime numbers have not been finally solved, and the best result belongs to China mathematician Chen Jingrun.
(9) Proof of the general law of reciprocity in arbitrary number field.
192 1 was basically solved by Kenji Takagi of Japan, and 1927 was basically solved by E.Artin of Germany. However, category theory is still developing.
(10) Can we judge whether an indefinite equation has a rational integer solution by finite steps?
Finding the integer root of the integral coefficient equation is called Diophantine (about 2 10-290, an ancient Greek mathematician) equation solvable. Around 1950, American mathematicians such as Davis, Putnam and Robinson made key breakthroughs. In 1970, Baker and Feros made positive conclusions about the equation with two unknowns. 1970. The Soviet mathematician Marty Sevic finally proved that, on the whole, the answer is negative. Although the result is negative, it has produced a series of valuable by-products, many of which are closely related to computer science.
Quadratic theory in (1 1) algebraic number field.
German mathematicians Hassel and Siegel made important achievements in the 1920s. In 1960s, French mathematician A.Weil made new progress.
Composition of (12) class domain.
That is, Kroneck's theorem on Abelian field is extended to any algebraic rational field. This problem has only some sporadic results and is far from being completely solved.