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 mathematician 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 Cohen proved that the continuum hypothesis and ZF axiom are independent of each other. Therefore, the continuum hypothesis cannot be proved right or wrong by the accepted ZF axiom. Hilbert's first problem has been solved in this sense.

(2) There is no contradiction in arithmetic axioms.

The contradiction of Euclidean geometry can be summed up as the contradiction of arithmetic kilometers. Hilbert once put forward the proof theory method of formalism plan to prove it. In 193 1, Godel published the incompleteness theorem to deny it. 1936 G?nc (g G?nc, 1909+0909? 1945) proves the non-contradiction of arithmetic axioms under the condition of using transfinite induction.

(3) The volumes of two tetrahedrons with equal bases and equal heights are equal.

The meaning of the problem is that there are two tetrahedrons with equal bases, which cannot be decomposed into finite small tetrahedrons, so that the two groups of tetrahedrons are congruent. Dean proved that such two tetrahedrons do exist (1900).

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

The second question is too general. There are many geometries that satisfy this property, so some restrictions are required. 1973, the Soviet mathematician Pogrelov announced that this problem was solved under the condition of symmetrical distance.

(5) Lie concept of continuous transformation group, the function defining this group is not assumed to be differentiable.

This problem is simply called the analytic property of continuous groups, that is, must every local Euclidean group be a Lie group? Through the efforts of von Neumann (compact group: 1933), Bandelli-Qin (commutative group: 1939) and Chegu (solvable group: 194 1), in 65438.

(6) Axiomatization of physics

Hilbert suggested that all physics be deduced by the axiomatic method of mathematics, first of all probability theory and mechanics. 1933, the Soviet mathematician Andrei Kolmogorov axiomatized probability theory. Later, he achieved great success in quantum mechanics and quantum field theory. However, many people doubt whether physics can be completely axiomatized.

(7) The transcendence of some figures

The problem requires proof that if it is algebraic number and irrational number, it must be transcendental number or at least irrational number (such as sum). 1934 the Soviet mathematician a.o. Gelfond proved that this is right. 1935, the German mathematician Schneider also solved this problem independently.

(8) Prime number problem

Prime number is an ancient research field. Hilbert mentioned Riemann conjecture, Goldbach conjecture and twin prime numbers here.

Riemann conjecture is still unsolved. Goldbach conjecture was not finally solved, but China and Chen Jingrun took the lead in solving it. At present, the best result of twin prime numbers also belongs to Chen Jingrun.

(9) Prove the most general law of reciprocity in any number field.

This problem has been basically solved by German mathematician E. Artin (1927), but the category theory is still developing.

Solvability of (10) Diophantine Equation

Finding the integer root of the integral coefficient equation is called Diophantine (about 2 10? 290, an ancient Greek mathematician) equation is solvable. Hilbert proposed whether the solvability of a Diophantine equation can be judged by a general algorithm composed of finite steps. Around 1950, American mathematicians such as Davis, Putnam and Robinson made key breakthroughs. 1970, Sevik of the Soviet Union finally proved that the answer to the question 10 was negative. Despite its negative results, it has produced a series of valuable by-products, many of which are closely related to computer science.

Quadratic form of (1 1) arbitrary algebraic coefficient

Germans Hasse and Siegel made important achievements in the 1920s. In 1960s, Weil in France made new progress.

(12) generalizes Kroneck theorem on Abelian field to arbitrary algebraic rational field.

This problem has only some sporadic results and is far from being completely solved.

(13) the impossibility of solving general seventh-order equations with binary functions

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 about to be solved. Soviet mathematician v·I· Arnold solved the case of continuous function (1957). The generalization of Vituskin to continuously differentiable function in 1964. If the analytic function is found, the problem has not been solved.

Proof of finiteness of (14) some complete function systems

This is related to algebraic invariants. Japanese mathematician Masayoshi Nagata gave a beautiful counterexample (1959).

The Strict Basis of (15) 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 closely related to algebraic geometry. But the strict foundation has not been established.

Topological problems of (16) algebraic curves and surfaces

This problem is divided into two parts. The first half involves the maximum number of closed bifurcation curves in algebraic curves. In the second half, we need to discuss the maximum number and relative position of limit cycles, where x and y are n-degree polynomials of x and y, Petrovski of the Soviet Union? Academician has proved that the number of limit cycles does not exceed 3. In 1979, China's Shi Songling and Wang cited four counterexamples of limit cycles respectively.

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

For all arrays (x 1, …, xn), polynomials with real coefficients n are always greater than or equal to 0. Can it be written as sum of squares? 1927, Atin proved that this is correct.

(18) Constructing space with congruent polyhedron

German mathematicians Bieber Bach (19 10) and Reinhardt (1928) gave some answers.

Is the solution of (19) regular variational problem necessarily analytical?

There is little research on this issue. S Bernstein and Petrovsky have got some results.

(20) General boundary value problems

This problem has made rapid progress and has become a major branch of mathematics. Research is still going on.

Proof of Existence of Solutions for (2 1) Linear Differential Equations with Specific Single-valued Groups

By Hilbert himself (1905) and Lerner (H R? Hrl)( 1957), p.d. Ligne (1970), etc.

(22) The univalence of analytic functions composed of automorphism functions.

It involves the difficult Riemann surface theory. 1907, P Koebe has made an important breakthrough, but other aspects have not been solved.

(23) The further development of variational method

This is not a clear mathematical problem, but a general view of variational method. Variational method has made great progress in the 20th century.

It is not difficult to see from the above introduction that Hilbert's question is quite difficult, and many ordinary people simply can't understand it. It is the difficulties that attract people with lofty ideals to work hard. But it is not inaccessible, thus providing a scientific hunting ground for people to finally gain something. It is no accident that people have been paying attention to the study of Hilbert problem for 80 years. Of course, the predictions can't all conform to the later development. The breadth and depth of mathematics development in the 20th century far exceeded the expectations at the beginning of this century. Algebraic topology, abstract algebra, functional analysis, multiple complex variable functions and many other theoretical disciplines. None of them are within the 23 questions, not to mention applied mathematics related to applications and computational mathematics and computer science developed with the emergence of computers.

The great mathematician H Weyl once said in the eulogy of Hilbert's death: "Hilbert is like a piper in variegated clothes. His sweet flute lured so many mice and followed him into the deep river of mathematics. " For aspiring people, these 23 questions are such a sweet flute, and we still seem to hear its call. Happily, mathematicians in China made some contributions to Question 8 and 16.