1874, Cantor speculated that there was no other cardinality between countable set cardinality and real set cardinality, that is, the famous continuum hypothesis. 1938, an Austrian mathematical logician living in the United States, Godel proved that there is no contradiction between the continuum hypothesis and the axiomatic system of ZF set theory. 1963, the American mathematician P.Choen proved the continuum hypothesis.

(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 proof theory method of formalism plan, and the Godel's incompleteness theorem published in 193 1 denied it. Gentan,1909-1945) 5438+0936.

(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) The shortest distance between two points is a straight line.

This question is 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, 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 doubt whether all branches of physics can be 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, Schneider of Germany and Siegel of Germany were independent.

(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 their best achievement 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 E.Artin of Germany in 1927, and the 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) and the equation is solvable. 1950 or so, American mathematicians Davis, Putnam, Robinson, etc. A key breakthrough was made .36438+0970. Baker and Feros reached a positive conclusion for the equation with two unknowns. 1970. the Soviet mathematician Marty Sevik finally proved that the answer was no. Although negative results have been obtained, a series of valuable by-products have been produced, 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, and French mathematician A. Weil made new progress in the 1960s.

Composition of (12) class domain.

That is to say, Kroneck's theorem in Abelian field is extended to any algebraic rational number field. This problem has only some sporadic results, far from being completely solved.

The impossibility of (13) combination of binary continuous functions to solve the seventh general algebraic equation.

The root of equation x7+ax3+bx2+cx+ 1=0 depends on three parameters A, B and C; X=x(a，b，c)。 Can this function be represented by a binary function? This problem is almost solved. 1957, the Soviet mathematician Arnold proved that any continuous real function f(x 1, x2, x3) on [0, 1] can be written in the form of ∑ hi (ξ i (x 1, x2). Here hi and ξi are continuous real functions. Andre Andrey Kolmogorov proved that f(x 1, x2, x3) can be written in the form of ∑ hi (ξ I1(x1)+ξ I2 (x2)+ξ i3 (x3)) (I = 65438).

The finite proof of (14) some complete function systems.

That is, polynomial fi (I = 1, ..., Xn), where r is the negative solution of this problem related to algebraic invariants by rational functions f (X 1, ..., XM) and f1959.

(15) Establish the foundation of algebraic geometry.

Dutch mathematicians Vander Waals Deng 1938 to 1940 and Wei Yi 1950 have solved the problem.

(15) Note 1 The strict foundation of Schubert counting calculus.

A typical problem is that there are four straight lines in three-dimensional space. How many straight lines can intersect all four? Schubert gave an intuitive solution. Hilbert asked to generalize the problem and give a strict basis. Now there are some computable methods, which are closely related to algebraic geometry. But the strict foundation has not been established.

Topological research on (16) algebraic curves and surfaces.

The first half of this problem involves the maximum number of closed bifurcation curves in algebraic curves. In the second half, it is required to discuss the maximum number N(n) of limit cycles of dx/dy=Y/X and their relative positions, where x and y are polynomials of degree n of x and y. For the case of n=2 (i.e. quadratic system), in 1934, Froxianer obtains that n (2) ≥ is in 1952. 1955, Podlovschi of the Soviet Union declared that n(2)≤3, which was a shocking result. With regard to the relative position, China mathematicians and Ye proved in 1957 that (E2) does not exceed two strings. In.5438+0957, China mathematicians Qin Yuanxun and Pu Fujin gave concrete examples, and the equation with n = 2 has at least three limit cycle sequences. 19438+0978 Under the guidance of Qin Yuanxun and Hua, Shi Songling and Wang respectively gave at least four concrete examples of limit cycles. 19438+0983, Qin Yuanxun further proved that a quadratic system has at most four limit cycles, and it is (6978.

The square sum representation of (17) semi-positive definite form.

The rational function f (x 1, ..., xn) is for any array (x 1, ..., xn). Are you sure that f can be written as the sum of squares of rational functions? 1927 Atin has been definitely solved.

(18) Construct space with congruent polyhedron.

German mathematician Bieber Bach made a partial solution in 19 10, and Reinhardt made a partial solution in 1928.

(19) Is the solution of the regular variational problem always an analytic function?

German mathematician Bernstein (1929) and Soviet mathematician Petrovsky (1939) have solved this problem.

(20) Study the general boundary value problem.

This problem has developed rapidly and has become a big branch of mathematics.

(2 1) Proof of the existence of solutions for Fuchs-like linear differential equations with given singularities and single-valued groups.

This problem belongs to the large-scale theory of linear ordinary differential equations. Hilbert himself achieved important results in 1905 and H.Rohrl in 1957 respectively. Deligne, a French mathematician, made outstanding contributions in 1970.

(22) Automorphic single-valued analytic function.

This problem involves the difficult Riemann surface theory. In 1907, P.Koebe solved a variant and made an important breakthrough in the research of this problem. Other aspects have not been solved.

(23) Carry out the research of variational method.

This is not a clear mathematical problem. Variational method has made great progress in the 20th century.

It can be seen that Hilbert's problem is quite difficult, attracting people with lofty ideals to work hard.