Хилбертови проблеми

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По време на втория математически конгрес, проведен в Париж през 1900, Давид Хилберт представя 23 нерешени от математиците проблема. Според него, те ще обозначат целите на математиците през ХХ век.

Нека вземем за пример Първият Хилбертов проблем. Какво представлява той? Всъщност е доказване хипотезата на Кантор за континиума. Пол Кохен базирайки се на трудовете на Гъодел през 1963 г. доказва нерешимоста на хипотезата.

По-нататък следва таблица изброяваща проблемите, както и тяхното състояние - решеност, нерешеност, полу- или частична решеност за момента.


[редактиране] Таблица на 23-те проблема

23-те проблема на Хилберт са:

Проблем Кратко обяснение Пложение
1ви Хипотеза за континиума (тоест няма множество, чиято големина е стриктно помежду целите и реалните числа) Доказано за невъзможно да се докаже или опровергае според Теория на множествата на Зермело-Франкел. Няма консесус дали това е решение на проблема.[1]
2ри Prove that the axioms of arithmetic are consistent (тоест, че аритметиката е формална система, която не доказва противоречие). Частично решен: Some hold it has been shown impossible to establish in a consistent, finitistic axiomatic system [2] - However, Gentzen proved in 1936 that consistency of arithmetic followed from the well-foundedness of the ordinal ε0, a fact amenable to combinatorial intuition.
3ти Can two tetrahedra be proved to have equal volume (under certain assumptions)? Може ли да се докаже, че два тетраедъра имат един и същ обем? (при определени допускания) Решен. Резултат: не, proved using Dehn invariants
4ти Construct all metrics where lines are geodesics. Too vague[3] to be stated решен или не.
5ти Are continuous groups automatically differential groups? Решен
6ти Axiomatize all of physics Нерешен. Нематематически
7ми Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? Решен. Резултат: да, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem
8ми The Riemann hypothesis (the real part of any non-trivial zero of the Riemann zeta function is ½) and Goldbach's conjecture (every even number greater than 2 can be written as the sum of two prime numbers). Открит[4]
9ти Find most general law of the reciprocity theorem in any algebraic number field Частично решен[5]
10ти Determination of the solvability of a Diophantine equation Решен. Резултат: не, Matiyasevich's theorem implies that this is impossible
11ти Solving quadratic forms with algebraic numerical coefficients. Частично решен
12ти Extend Kronecker's theorem on abelian extensions of the rational numbers to any base number field. Открит
13ти Solve all 7-th degree equations using functions of two parameters. Решен
14ти Proof of the крайност of certain complete systems of functions. Решен. Резултат: не, generally, due to counterexample
15ти Rigorous foundation of Schubert's enumerative calculus. Частично решен
16ти Topology of algebraic curves and surfaces. Открит
17ти Expression of definite rational function as quotient of sums of squares Решен. Резултат: An upper limit was established for the number of square terms necessary
18ти Is there a non-regular, space-filling polyhedron? What is the densest sphere packing? Решен[6]
19ти Are the solutions of Lagrangians always analytic? Решен. Резултат: да
20ти Do all variational problems with certain boundary conditions have solutions? Решен. A significant area of research throughout the 20th century, culminating in solutions for the non-linear case.
21ви Proof of the existence of linear differential equations having a prescribed monodromic group Решен. Резултат: Да or не, depending on more exact formulations of the problem
22ри Uniformization of analytic relations by means of automorphic functions Решен
23ти Further development of the calculus of variations Решен

[редактиране] Бележки

  1. Cohen's independence резултат, showing the continuum hypothesis to be independent of ZFC (Теория на множествата на Зермело-Франкел, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved. One contemporary view is that it may be the case that теория на множествата should have additional axioms, capable of settling the problem.
  2. A matter of opinion, не shared by all. Gentzen's резултат shows rather precisely how much needs to be assumed to prove that Peano arithmetic is consistent. It is widely held that Gödel's second incompleteness theorem shows that няма finitistic proof that PA is consistent (though Gödel himself disclaimed this inference [this needs a better reference-- but cf Dawson p.71ff "...Gödel too [like Hilbert] believed that никой mathematical problems lay beyond the reach of human reason. Yet his резуртати showed that the program that Hilbert had proposed to validate that belief -- his proof theory -- не може да бъде carried through as Hilbert had envisioned" (p.71) See also p. 98ff for more discussion of 'finite procedure').
  3. According to Rowe & Gray (see reference below), most of the problems have been solved. Някои не са били пълно/достатъчно добре дефинирани, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
  4. Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
  5. Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
  6. Rowe & Gray also list the 18th problem as "открит" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).