Хилбертови проблеми — разлика између измена

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Хилбертови проблеми, то су 23 проблема, од којих је тринаест поставио математичар Давид Хилберт да би на Другом међународном конгресу математичара у Паризу, 8. августа 1900. године било додато још десет, овде број 1, 2, 6, 7, 8, 13, 16, 19, 21, и 22. Неки од ових проблема су заправо подручја за истраживање, а заједно са осталима били су пример нарастања читавих дисциплина, временом, из малих „проблема“. Потпуна листа од 23 проблема објављена је касније, посебно у преводу на енглески 1902. од стране Mary Frances Winston NewsonМери Френсис Винстон Њусон у Bulletin of the American Mathematical Society.^[1]

Игнорабимус

Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms.^[2] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.^[а]

However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.^[б]^[в]

Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.^[г] He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus" (statement whose truth can never be known).^[д] It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is not the integer solution, but (in a certain sense) the ability to discern in a specific way whether a solution exists.

On the other hand, the status of the first and second problems is even more complicated: there is not any clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.^[ђ]

Двадесет четврти проблем

Хилберт је првобитно укључио 24 проблема на своју листу, али је одлучио да не укључи један од њих на објављену листу. „24. проблем“ (у теорији доказа, о критеријуму једноставности и општим методама) поново је открио немачки историчар Ридигер Тиле 2000. године у Хилбертовим оригиналним белешкама у рукопису.^[5]

Проблеми

Канторов проблем кардиналног броја континуума.^[6]
Конзистентност аксиома аритметике.
Једнакост запремина два тетраедра једнаких база и висина.
Проблем праве линије као најкраћег растојања између две тачке.
Концепт Лијевих група непрекидних трансформација, без претпоставке диференцијабилности.
Математички третман аксиома физике. Може ли се физика аксиоматизовати?^[7]^[8]
Ирационалност и трансцедентност извесних бројева, oblika $a^{b}$ , нпр. $2^{\sqrt {2}}$ ,; $e^{\pi }$ .
Проблем простих бројева, Риманова хипотеза.
Општи доказ теорема реципрочности теорије бројева.
Опште решење Диофантове једначине.

Квадратна форма произвољног целобројног алгебарског поља.^[9]
Кронекерова теорема, конструкција холоморфне функције.^[10]
Немогућност решења опште једначине 7-ог степена функцијама са само два аргумента.^[11]^[12]^[13]^[14]
Проблем коначности извесних функција.
Строго заснивање Шубертовог непребројивог рачуна (Schubert).
Проблем топологије алгебарских кривих и површи.
Репрезентација кончане форме квадрата.
Изградња простора из конгруентног полиедра.
Јесу ли решења проблема варијација увек аналитичка?
Општи проблем граничне вредности.

Доказ егзистенције решења линеарне диференцијалне једначине за монодромску групу.
Униформизација аналитичких релација помоћу аутоморфних функција.
Даљи развој метода рачуна варијација.

Напомене

^ See Nagel and Newman revised by Hofstadter (2001, p. 107),^[3] footnote 37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page 109^[3] and was not modified there by Hofstadter (p. 108).^[3]
^ Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p. 198–199).^[4] Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).^[4]
^ Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p. 217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p. vii).^[4]
^ This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the Law of Excluded Middle be employed in proofs. See much more at Brouwer–Hilbert controversy.
^ "This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." (Hilbert, 1902, p. 445.)
^ Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of being mirrored inside Principia Mathematica (footnote 39, page 109). The authors conclude that the prospect "is most unlikely."^[3]

Референце

^ Hilbert, David (1902). „Mathematical Problems”. Bulletin of the American Mathematical Society. 8 (10): 437—479. doi:10.1090/S0002-9904-1902-00923-3 . Earlier publications (in the original German) appeared in Hilbert, David (1900). „Mathematische Probleme”. Göttinger Nachrichten: 253—297. and Hilbert, David (1901). „[no title cited]”. Archiv der Mathematik und Physik. 3. 1: 44—63, 213—237.
^ van Heijenoort, Jean, ур. (1976) [1966]. From Frege to Gödel: A source book in mathematical logic, 1879–1931 ((pbk.) изд.). Cambridge MA: Harvard University Press. стр. 464ff. ISBN 978-0-674-32449-7.
A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational "crisis" that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).
^ ^а ^б ^в ^г Nagel, Ernest; Newman, James R. (2001). Hofstadter, Douglas R., ур. Gödel's Proof. New York, NY: New York University Press. ISBN 978-0-8147-5816-8.
^ ^а ^б ^в Reid, Constance (1996). Hilbert. New York, NY: Springer-Verlag. ISBN 978-0387946740.
^ Thiele, Rüdiger (јануар 2003). „Hilbert's twenty-fourth problem” (PDF). American Mathematical Monthly. 110: 1—24. S2CID 123061382. doi:10.1080/00029890.2003.11919933.
^ Cantor, Georg (1878). „Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Mathematik. 1878 (84): 242—258. doi:10.1515/crll.1878.84.242.
^ Corry, L. (1997). „David Hilbert and the axiomatization of physics (1894–1905)”. Arch. Hist. Exact Sci. 51 (2): 83—198. S2CID 122709777. doi:10.1007/BF00375141.
^ Gorban, A.N.; Karlin, I. (2014). „Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations”. Bulletin of the American Mathematical Society. 51 (2): 186—246. arXiv:1310.0406 . doi:10.1090/S0273-0979-2013-01439-3 .
^ Hazewinkel, Michiel (2009). Handbook of Algebra. 6. Elsevier. стр. 69. ISBN 978-0080932811.
^ Houston-Edwards, Kelsey. „Mathematicians Find Long-Sought Building Blocks for Special Polynomials”.
^ Abhyankar, Shreeram S. „Hilbert's Thirteenth Problem” (PDF).
^ Vitushkin, A.G. „On Hilbert's thirteenth problem and related questions” (PDF).
^ Chebotarev, N.G., On certain questions of the problem of resolvents
^ Hilbert, David (1927). „Über die Gleichung neunten Grades”. Math. Ann. 97: 243—250. S2CID 179178089. doi:10.1007/BF01447867.

Литература

Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford, UK: Oxford University Press. ISBN 978-0-19-850651-5.
Yandell, Benjamin H. (2002). The Honors Class: Hilbert's problems and their solvers. Wellesley, MA: A.K. Peters. ISBN 978-1-56881-141-3.
Thiele, Rüdiger (2005). „On Hilbert and his twenty-four problems”. Ур.: Van Brummelen, Glen. Mathematics and the Historian's Craft: The Kenneth O. May lectures. CMS Books in Mathematics / Ouvrages de Mathématiques de la SMC. 21. стр. 243—295. ISBN 978-0-387-25284-1.
Dawson, John W. Jr. (1997). Logical Dilemmas: The life and work of Kurt Gödel. A.K. Peters.
A wealth of information relevant to Hilbert's "program" and Gödel's impact on the Second Question, the impact of Arend Heyting's and Brouwer's Intuitionism on Hilbert's philosophy.
Browder, Felix E., ур. (1976). „Mathematical Developments Arising from Hilbert Problems”. Proceedings of Symposia in Pure Mathematics XXVIII. American Mathematical Society.
A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments.
Matiyasevich, Yuri (1993). Hilbert's Tenth Problem. Cambridge, MA: MIT Press. ISBN 978-0262132954.
An account at the undergraduate level by the mathematician who completed the solution of the problem.
Cohen, Paul Joseph (2008). Set theory and the continuum hypothesis. Mineola, New York City: Dover Publications. ISBN 978-0-486-46921-8.
Dales, H.G.; Woodin, W.H. (1987). An Introduction to Independence for Analysts. Cambridge.
Enderton, Herbert (1977). Elements of Set Theory. Academic Press.
Gödel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in Mathematical Developments Arising from Hilbert's Problems, Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1
McGough, Nancy. „The Continuum Hypothesis”.
Wolchover, Natalie (15. 7. 2021). „How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer”.

Спољашње везе

Hazewinkel Michiel, ур. (2001). „Hilbert problems”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
„Original text of Hilbert's talk, in German.”. Архивирано из оригинала 2012-02-05. г. Приступљено 2005-02-05.
„David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900.” (PDF).
Mathematical Problems public domain audiobook at LibriVox

[4] See Nagel and Newman revised by Hofstadter (2001, p. 107),^[3] footnote 37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page 109^[3] and was not modified there by Hofstadter (p. 108).^[3]

[6] Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p. 198–199).^[4] Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).^[4]

[7] Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p. 217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p. vii).^[4]

[8] This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the Law of Excluded Middle be employed in proofs. See much more at Brouwer–Hilbert controversy.

[9] "This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus." (Hilbert, 1902, p. 445.)

[10] Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of being mirrored inside Principia Mathematica (footnote 39, page 109). The authors conclude that the prospect "is most unlikely."^[3]

[Hilbert_1902-1] Hilbert, David (1902). „Mathematical Problems”. Bulletin of the American Mathematical Society. 8 (10): 437—479. doi:10.1090/S0002-9904-1902-00923-3 . Earlier publications (in the original German) appeared in Hilbert, David (1900). „Mathematische Probleme”. Göttinger Nachrichten: 253—297. and Hilbert, David (1901). „[no title cited]”. Archiv der Mathematik und Physik. 3. 1: 44—63, 213—237.

[2] van Heijenoort, Jean, ур. (1976) [1966]. From Frege to Gödel: A source book in mathematical logic, 1879–1931 ((pbk.) изд.). Cambridge MA: Harvard University Press. стр. 464ff. ISBN 978-0-674-32449-7.
A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational "crisis" that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).

[Hofstadter_2001-3] а ^б ^в ^г Nagel, Ernest; Newman, James R. (2001). Hofstadter, Douglas R., ур. Gödel's Proof. New York, NY: New York University Press. ISBN 978-0-8147-5816-8.

[Reid_1996-5] а ^б ^в Reid, Constance (1996). Hilbert. New York, NY: Springer-Verlag. ISBN 978-0387946740.

[11] Thiele, Rüdiger (јануар 2003). „Hilbert's twenty-fourth problem” (PDF). American Mathematical Monthly. 110: 1—24. S2CID 123061382. doi:10.1080/00029890.2003.11919933.

[Cantor1878-12] Cantor, Georg (1878). „Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Mathematik. 1878 (84): 242—258. doi:10.1515/crll.1878.84.242.

[13] Corry, L. (1997). „David Hilbert and the axiomatization of physics (1894–1905)”. Arch. Hist. Exact Sci. 51 (2): 83—198. S2CID 122709777. doi:10.1007/BF00375141.

[14] Gorban, A.N.; Karlin, I. (2014). „Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations”. Bulletin of the American Mathematical Society. 51 (2): 186—246. arXiv:1310.0406 . doi:10.1090/S0273-0979-2013-01439-3 .

[15] Hazewinkel, Michiel (2009). Handbook of Algebra. 6. Elsevier. стр. 69. ISBN 978-0080932811.

[16] Houston-Edwards, Kelsey. „Mathematicians Find Long-Sought Building Blocks for Special Polynomials”.

[17] Abhyankar, Shreeram S. „Hilbert's Thirteenth Problem” (PDF).

[18] Vitushkin, A.G. „On Hilbert's thirteenth problem and related questions” (PDF).

[19] Chebotarev, N.G., On certain questions of the problem of resolvents

[20] Hilbert, David (1927). „Über die Gleichung neunten Grades”. Math. Ann. 97: 243—250. S2CID 179178089. doi:10.1007/BF01447867.

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@@ Ред 1: / Ред 1: @@
+{{Short description|Двадесет три задатка из математике објављена 1900}}
-'''Хилбертови проблеми''', то су 23 [[проблем]]а, од којих је тринаест поставио [[математичар]] Давид [[Давид Хилберт|Хилберт]] да би на Другом међународном конгресу математичара у [[Париз]]у, 8. августа 1900. године било додато још десет, овде број 1, 2, 6, 7, 8, 13, 16, 19, 21, и 22. Неки од ових проблема су заправо подручја за истраживање, а заједно са осталима били су пример нарастања читавих дисциплина, временом, из малих „проблема“.
+[[File:Hilbert.jpg|thumb|[[Давид Хилберт]]]]
-# [[Георг Кантор|Канторов]] проблем [[кардиналан број|кардиналног]] броја [[континуум (математика)|континуума]].
+'''Хилбертови проблеми''', то су 23 [[проблем]]а, од којих је тринаест поставио [[математичар]] Давид [[Давид Хилберт|Хилберт]] да би на Другом међународном конгресу математичара у [[Париз]]у, 8. августа 1900. године било додато још десет, овде број 1, 2, 6, 7, 8, 13, 16, 19, 21, и 22. Неки од ових проблема су заправо подручја за истраживање, а заједно са осталима били су пример нарастања читавих дисциплина, временом, из малих „проблема“. Потпуна листа од 23 проблема објављена је касније, посебно у преводу на енглески 1902. од стране [[Mary Frances Winston NewsonМери Френсис Винстон Њусон]] у ''[[Bulletin of the American Mathematical Society]]''.<ref name=Hilbert_1902>{{cite journal |last=Hilbert |first=David |title=Mathematical Problems |url=https://www.ams.org/journals/bull/1902-08-10/home.html |journal=[[Bulletin of the American Mathematical Society]] |volume=8 |issue=10 |year=1902 |pages=437–479|doi=10.1090/S0002-9904-1902-00923-3 |doi-access=free }} Earlier publications (in the original German) appeared in {{cite journal |last=Hilbert |first=David |title=Mathematische Probleme | url=https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002498863&physid=phys271#navi |journal=Göttinger Nachrichten |year=1900 |pages=253–297}} and {{cite journal |last=Hilbert |first=David |title=[no title cited] |journal=Archiv der Mathematik und Physik |series=3 |volume=1 |year=1901 |pages=44–63, 213–237}}</ref>
+== Игнорабимус ==
+{{рут}}
+Following [[Gottlob Frege]] and [[Bertrand Russell]], Hilbert sought to define mathematics logically using the method of [[formal system]]s, i.e., [[finitism|finitistic]] [[Mathematical proof|proofs]] from an agreed-upon set of [[Axiom|axioms]].<ref>{{cite book |pages=464ff |editor-first=Jean |editor-last=van Heijenoort |year=1976 |orig-year=1966 |title=From Frege to Gödel: A source book in mathematical logic, 1879–1931 |publisher=Harvard University Press |location=Cambridge MA |isbn=978-0-674-32449-7 |edition=(pbk.)}}<br />''A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational "crisis" that was on-going at the time (translated into English), appears as Hilbert's'' 'The Foundations of Mathematics' (1927).</ref> One of the main goals of [[Hilbert's program]] was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.{{refn|group=lower-alpha|See Nagel and Newman revised by Hofstadter (2001, p.&nbsp;107),<ref name=Hofstadter_2001>{{cite book |author1=Nagel, Ernest |author2=Newman, James R. |editor=Hofstadter, Douglas R. |editor-link=Douglas Hofstadter |year=2001 |title=Gödel's Proof |publisher=New York University Press |location=New York, NY |isbn=978-0-8147-5816-8}}</ref> footnote&nbsp;37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page&nbsp;109<ref name=Hofstadter_2001/> and was not modified there by Hofstadter (p.&nbsp;108).<ref name=Hofstadter_2001/>}}
+However, [[Gödel's second incompleteness theorem]] gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible.  Hilbert lived for 12&nbsp;years after [[Kurt Gödel]] published his theorem, but does not seem to have written any formal response to Gödel's work.{{refn|group=lower-alpha|Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p.&nbsp;198–199).<ref name=Reid_1996>{{cite book |first=Constance |last=Reid |year=1996 |title=Hilbert |publisher=Springer-Verlag |location=New York, NY |isbn=978-0387946740 |url-access=registration |url=https://archive.org/details/hilbert0000reid }}</ref> Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p.&nbsp;199).<ref name=Reid_1996/>}}{{refn|group=lower-alpha|Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p.&nbsp;217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p.&nbsp;vii).<ref name=Reid_1996/>}}
+Hilbert's tenth problem does not ask whether there exists an [[algorithm]] for deciding the solvability of [[Diophantine equations]], but rather asks for the ''construction'' of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in [[Rational integer|rational integers]]." That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
+In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.<ref group=lower-alpha>This issue that finds its beginnings in the "foundational crisis" of the early 20th&nbsp;century, in particular the controversy about under what circumstances could the [[Law of Excluded Middle]] be employed in proofs. See much more at [[Brouwer–Hilbert controversy]].</ref> He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "[[Ignoramus et ignorabimus|ignorabimus]]" (statement whose truth can never be known).<ref group=lower-alpha>"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ''ignorabimus.''" (Hilbert, 1902, p.&nbsp;445.)</ref>  It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is not the integer solution, but (in a certain sense) the ability to discern in a specific way whether a solution exists.
+On the other hand, the status of the first and second problems is even more complicated: there is not any clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.{{refn|group=lower-alpha|Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as ''Principia Mathematica'' is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is ''not'' capable of being mirrored inside ''Principia Mathematica'' (footnote&nbsp;39, page&nbsp;109). The authors conclude that the prospect "is most unlikely."<ref name=Hofstadter_2001/>}}
+== Двадесет четврти проблем ==
+{{Main article|Хилбертов двадесет четврти проблем}}
+Хилберт је првобитно укључио 24 проблема на своју листу, али је одлучио да не укључи један од њих на објављену листу. „24. проблем“ (у [[proof theory|теорији доказа]], о критеријуму [[simplicity|једноставности]] и општим методама) поново је открио немачки историчар [[Rüdiger Thiele|Ридигер Тиле]] 2000. године у Хилбертовим оригиналним белешкама у рукопису.<ref>{{cite journal |url=http://www.maa.org/news/Thiele.pdf |title=Hilbert's twenty-fourth problem |first=Rüdiger |last=Thiele |journal=American Mathematical Monthly |date=January 2003|volume=110 |pages=1–24 |doi=10.1080/00029890.2003.11919933 |s2cid=123061382 }}</ref>
+== Проблеми ==
+# [[Георг Кантор|Канторов]] проблем [[кардиналан број|кардиналног]] броја [[континуум (математика)|континуума]].<ref name=Cantor1878>{{Cite journal |last=Cantor |first=Georg  |author-link=Georg Cantor |year=1878 |title=Ein Beitrag zur Mannigfaltigkeitslehre |journal=[[Journal für die Reine und Angewandte Mathematik]] |volume=1878  |issue=84  |pages=242–258 |doi=10.1515/crll.1878.84.242 |url=http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15 }}</ref>
 # Конзистентност [[аксиома]] [[аритметика|аритметике]].
 # Једнакост запремина два [[Тетраедар|тетраедра]] једнаких база и висина.
 # Проблем [[права (линија)|праве]] линије као најкраћег растојања између две тачке.
 # Концепт Лијевих група непрекидних [[трансформација]], без претпоставке [[извод|диференцијабилности]].
+# Математички третман аксиома физике. Може ли се [[физика]] аксиоматизовати?<ref>{{cite journal |last1=Corry |first1=L. |year=1997 |title=David Hilbert and the axiomatization of physics (1894–1905) |journal=Arch. Hist. Exact Sci. |volume=51 |issue=2 |pages=83–198 |doi=10.1007/BF00375141|s2cid=122709777 }}</ref><ref>{{cite journal |last1=Gorban |first1=A.N. |author-link=Alexander Nikolaevich Gorban |last2=Karlin |first2=I. |year=2014 |title=Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations |journal=Bulletin of the American Mathematical Society |volume=51 |issue=2 |pages=186–246 |arxiv=1310.0406 |doi=10.1090/S0273-0979-2013-01439-3| doi-access= free}}</ref>
-# Математички третман аксиома физике. Може ли се [[физика]] аксиоматизовати?
 # [[Ирационалност]] и [[трансцедентност]] извесних бројева, oblika <math>a^b</math>, нпр. <math>2^\sqrt{2}</math>,; <math>e^\pi</math>.
 # Проблем [[Прост број|простих]] бројева, [[Риманова хипотеза]].
 # Општи доказ [[теорема]] реципрочности [[Теорија бројева|теорије бројева]].
 # Опште решење [[Диофантове једначине]].
-# Квадратна форма произвољног целобројног [[поље (математика)|алгебарског поља]].
+# Квадратна форма произвољног целобројног [[поље (математика)|алгебарског поља]].<ref>{{cite book|first=Michiel|last=Hazewinkel|date= 2009|title= Handbook of Algebra|publisher=Elsevier|page=69|isbn=978-0080932811|volume=6}}</ref>
-# [[Кронекерова теорема]], конструкција [[Холоморфизам|холоморфне]] функције.
+# [[Кронекерова теорема]], конструкција [[Холоморфизам|холоморфне]] функције.<ref>{{cite web|url=https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/|first=Kelsey|last=Houston-Edwards|title=Mathematicians Find Long-Sought Building Blocks for Special Polynomials}}</ref>
-# Немогућност решења опште једначине 7-ог степена функцијама са само два аргумента.
+# Немогућност решења опште једначине 7-ог степена функцијама са само два аргумента.<ref>{{cite web |url=http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_1-11.pdf |first=Shreeram S. |last=Abhyankar |title=Hilbert's Thirteenth Problem}}</ref><ref>{{cite web |url=http://www.iop.org/EJ/article/0036-0279/59/1/R03/RMS_59_1_R03.pdf?request-id=ef17fbdb-1a1c-4250-ae5f-0e1885b837fa |first=A.G. |last=Vitushkin |title=On Hilbert's thirteenth problem and related questions}}</ref><ref>{{citation |first=N.G. |last=Chebotarev |title=On certain questions of the problem of resolvents}}</ref><ref>{{cite journal |first=David |last=Hilbert |title=Über die Gleichung neunten Grades |journal=Math. Ann. |volume=97 |year=1927 |pages=243–250|doi=10.1007/BF01447867 |s2cid=179178089 }}</ref>
 # Проблем [[коначност]]и извесних функција.
 # Строго заснивање Шубертовог непребројивог рачуна (''-{Schubert}-'').
@@ Ред 20: / Ред 44: @@
 # Јесу ли решења проблема варијација увек аналитичка?
 # Општи проблем [[Гранична вредност|граничне вредности]].
 # Доказ егзистенције решења [[Линеарна диференцијална једначина|линеарне диференцијалне једначине]] за монодромску групу.
 # Униформизација аналитичких релација помоћу [[Аутоморфизам|аутоморфних]] функција.
 # Даљи развој метода [[Рачун варијација|рачуна варијација]].
+== Напомене ==
+{{notelist|30em}}
+== Референце ==
+{{Reflist|}}
+== Литература ==
+{{refbegin|colwidth=30em}}
+* {{cite book |last=Gray |first=Jeremy J. |author-link=Jeremy Gray |year=2000 |title=The Hilbert Challenge |publisher=[[Oxford University Press]] |location=Oxford, UK |isbn=978-0-19-850651-5}}
+* {{cite book |author=Yandell, Benjamin H. |author-link=Benjamin Yandell |year=2002 |title=The Honors Class: Hilbert's problems and their solvers |publisher=A.K. Peters |location=Wellesley, MA |isbn=978-1-56881-141-3 |url-access=registration |url=https://archive.org/details/honorsclasshilbe0000yand }}
+* {{cite book |last=Thiele |first=Rüdiger |chapter=On Hilbert and his twenty-four problems |title=Mathematics and the Historian's Craft: The Kenneth O. May lectures |pages=243–295 |isbn=978-0-387-25284-1 |editor-last=Van Brummelen |editor-first=Glen |year=2005 |series=[[Canadian Mathematical Society|CMS]] Books in Mathematics / Ouvrages de Mathématiques de la SMC |volume=21|title-link=Kenneth May }}
+* {{cite book |author-last=Dawson |author-first=John W. Jr. |title=Logical Dilemmas: The life and work of Kurt Gödel |publisher=A.K. Peters |year=1997}}<br />''A wealth of information relevant to Hilbert's "program" and [[Gödel]]'s impact on the Second Question, the impact of [[Arend Heyting]]'s and [[L. E. J. Brouwer|Brouwer]]'s [[Intuitionism]] on Hilbert's philosophy.''
+*  {{cite book |editor-first=Felix E. |editor-last=Browder |editor-link=Felix Browder |contribution=Mathematical Developments Arising from Hilbert Problems |title=Proceedings of Symposia in Pure Mathematics XXVIII |year=1976 |publisher=American Mathematical Society}}<br />''A collection of survey essays by experts devoted to each of the 23&nbsp;problems emphasizing current developments.''
+* {{cite book |first=Yuri |last=Matiyasevich |title=Hilbert's Tenth Problem |publisher=MIT Press |location=Cambridge, MA |year=1993 |isbn=978-0262132954}}<br />''An account at the undergraduate level by the mathematician who completed the solution of the problem.''
+* {{Cite book |last=Cohen |first=Paul Joseph |author-link=Paul Cohen (mathematician) |date=2008 |title=Set theory and the continuum hypothesis |location=Mineola, New York City |publisher=Dover Publications |isbn=978-0-486-46921-8  }}
+* {{Cite book |last1=Dales |first1=H.G. |last2=Woodin |first2=W.H. |date= 1987 |title=An Introduction to Independence for Analysts |publisher=Cambridge }}
+* {{Cite book |last=Enderton |first=Herbert |date=1977 |title=Elements of Set Theory |publisher=Academic Press }}
+* Gödel, K.: ''What is Cantor's Continuum Problem?'', reprinted in Benacerraf and Putnam's collection ''Philosophy of Mathematics'', 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
+* Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in ''Mathematical Developments Arising from Hilbert's Problems,'' Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp.&nbsp;81–92. {{ISBN|0-8218-1428-1}}
+* {{Cite web |author=McGough, Nancy |title=The Continuum Hypothesis |url=http://www.ii.com/math/ch/ }}
+* {{Cite web |author=Wolchover, Natalie |title=How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer |date=15 July 2021 |url=https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/ }}
+{{div col end}}
+== Спољашње везе ==
+{{Wikisource|Mathematical Problems}}
+* {{springer|title=Hilbert problems|id=p/h120080}}
+* {{cite web |url=http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html |title=Original text of Hilbert's talk, in German. |access-date=2005-02-05 |archive-url=https://web.archive.org/web/20120205025851/http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html |archive-date=2012-02-05 |url-status=dead }}
+* {{cite web |url=https://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00881-8/S0273-0979-00-00881-8.pdf |title=David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900.}}
+* {{librivox book | title=Mathematical Problems| author=Hilbert}}
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