使用者:Bieraaa/Hilbert
以下列出希爾伯特的23個問題。欲了解更多信息，請閱讀表格第一排條目鏈接。
| 問題 | 簡述 | 狀態 | 解決年份 |
|---|---|---|---|
| 第1題 | 連續統假設，即不存在一個基數絕對大於可列集而絕對小於實數集的集合。 | 美國數學家保羅·柯恩證明了連續統假設無法從帶選擇公理的策梅洛-弗蘭克爾集合論（ZFC）中得到證明或反證，也就是說，連續統假設成立與否無法由ZFC確定。對於該結論是否可以算作對希爾伯特原始問題的答覆，不存在共識。 | 1940, 1963 |
| 第2題 | 算術公理之相容性 | 哥德爾於1931年發表第二不完備定理，證明了算術的相容性不可能從算術本身中得到。而根岑在1936年則證明了算術的相容性取決於序數 ε₀ 的良基性。對於該結論在何種意義上可以算作對該問題的回答，不存在共識。一些數學家認為哥德爾給出了否定的回答，但另一些則認為根岑給出了肯定的回答。 | 1931, 1936 |
| 第3題 | 已知兩個多面體有相同體積，能否把其中一個多面體分割成有限塊再將之結合成另一個？ | 已解決，答案是不能。希爾伯特的學生馬克斯·德恩以一反例給出了證明，後將其使用的證明方法稱為德恩不變量。 | 1900 |
| 第4題 | 建立所有度量空間使得所有線段為測地線。 | 英國數學家Jeremy John Gray認為，大部分相關問題都已經得到解決，沒解決的部分也取得了相當大的進展。然而希爾伯特對於這個問題的定義過於含糊，故不能確定是否解決。 | – |
| 第5題 | 所有連續群是否皆為可微群？ | 1953年日本數學家山邊英彥已得到完全肯定的結果。但是，該問題還可以解讀為希爾伯特－史密斯猜想的表述，而這個問題目前仍未解決。 | 1953? |
| 6th | Mathematical treatment of the axioms of physics | Partially resolved depending on how the original statement is interpreted.[1] In particular, in a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for foundation of statistical physics and (ii) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua." Kolmogorov's axiomatics (1933) is now accepted as standard. There is some success on the way from the "atomistic view to the laws of motion of continua."[2] | 1933–2002? |
| 7th | Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? | Resolved. Result: yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem. | 1934 |
| 8th | The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture | Unresolved. | – |
| 9th | Find the most general law of the reciprocity theorem in any algebraic number field. | Partially resolved.[n 1] | – |
| 10th | Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. | Resolved. Result: impossible, Matiyasevich's theorem implies that there is no such algorithm. | 1970 |
| 11th | Solving quadratic forms with algebraic numerical coefficients. | Partially resolved.[3] | – |
| 12th | Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. | Unresolved. | – |
| 13th | Solve 7-th degree equation using algebraic (variant: continuous) functions of two parameters. | The problem was partially solved by Vladimir Arnold based on work by Andrei Kolmogorov.[n 2] | 1957 |
| 14th | Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? | Resolved. Result: no, a counterexample was constructed by Masayoshi Nagata. | 1959 |
| 15th | Rigorous foundation of Schubert's enumerative calculus. | Partially resolved. | – |
| 16th | Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. | Unresolved, even for algebraic curves of degree 8. | – |
| 17th | Express a nonnegative rational function as quotient of sums of squares. | Resolved. Result: yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary. | 1927 |
| 18th | (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? |
(a) Resolved. Result: yes (by Karl Reinhardt). (b) Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.[n 3] |
(a) 1928 (b) 1998 |
| 19th | Are the solutions of regular problems in the calculus of variations always necessarily analytic? | Resolved. Result: yes, proven by Ennio de Giorgi and, independently and using different methods, by John Forbes Nash. | 1957 |
| 20th | Do all variational problems with certain boundary conditions have solutions? | Resolved. A significant topic of research throughout the 20th century, culminating in solutions for the non-linear case. | ? |
| 21st | Proof of the existence of linear differential equations having a prescribed monodromic group | Partially resolved. Result: Yes, no, open depending on more exact formulations of the problem. | ? |
| 22nd | Uniformization of analytic relations by means of automorphic functions | Resolved. | ? |
| 23rd | Further development of the calculus of variations | Too vague to be stated resolved or not. | – |
- ^ Corry, L. David Hilbert and the axiomatization of physics (1894–1905). Arch. Hist. Exact Sci. 1997, 51 (2): 83–198. doi:10.1007/BF00375141.
- ^ Gorban, A.N.; Karlin, I. Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations (PDF). Bull. Amer. Math. Soc. 2014, 51 (2): 186–246. doi:10.1090/S0273-0979-2013-01439-3.
- ^ Hazewinkel, Michiel. Handbook of Algebra 6. Elsevier. 2009: 69. ISBN 0080932819.
- ^ D. Hilbert, "¨Uber die Gleichung neunten Grades", Math. Ann. 97 (1927), 243–250
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