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Ricardo Da Silva - One of the best experts on this subject based on the ideXlab platform.
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los teoremas de incompletitud de godel teoria de conjuntos y el programa de David Hilbert
Episteme, 2014Co-Authors: Ricardo Da SilvaAbstract:Kurt Godel demostro en 1931, que para todo sistema formal Z recursivo lo suficientemente potente como para derivar los axiomas de Peanoy que ademas se suponga como consistente, se tiene que en el sistema hay proposiciones indecidibles, es decir, el sistema no es completo. Por otra parte, Godel probo que si el sistema Z es consistente entonces no se puedederivar en Z una proposicion que afirme la consistencia de Z. Estos resultadosson los que se conocen como Primer Teorema de Incompletitud Godel y Segundo Teorema de Incompletitud de Godel. Dichos resultados tienen un gran impacto sobre la investigacion de los fundamentos de la matematica quevenia gestandose en los primeros treinta anos del siglo pasado, y tiene ademasconsecuencias sobre la filosofia de la matematica de dicha epoca. Este articulo se encuentra estructurado en tres partes: En una primera parte nos ocupamos de la formulacion de los Teoremas de incompletitud y las ideas principalesde su demostracion en cada caso. Seguidamente mostraremos una aplicacion del Segundo Teorema de Incompletitud en la teoria de conjuntos referente a los cardinales inaccesibles. Por ultimo, desarrollaremos las consecuencias filosoficas que los Teoremas de incompletitud de Godel tienen sobre el proyecto meta-matematico de David Hilbert. Abstract: Kurt Godel proved in 1931 that for any formal recursive system Z powerful enough to derive the Peano axioms and also supposed to be consistent, we have that in the System there are undecidable propositions, i.e., the system is not complete. Moreover, Godel proved that if the Z system is consistent then it can not derive in Z a proposition asserting the consistency of Z. These results are known as Godel's First Incompleteness Theorem and Godel's Second Incompleteness Theorem. Such results have a great impact on the investigation of the foundations of mathematics that had been developing in the first thirty years of the last century, and it, furthermore, has implications for philosophy of mathematics of that time. This article is structured in three parts: In the first part we deal with the formulation of the incompleteness theorems and the main ideas of its proof in each case. Then, we will show an application of the Second Incompleteness Theorem in set theory concerning inaccessible cardinal. Finally, we will develop the philosophical consequences that Godel's incompleteness theorems have on the meta-mathematical project that David Hilbert proposed.
Da Silva Ricardo - One of the best experts on this subject based on the ideXlab platform.
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LOS TEOREMAS DE INCOMPLETITUD DE GÖDEL, TEORÍA DE CONJUNTOS Y EL PROGRAMA DE David Hilbert
Episteme NS, 2014Co-Authors: Da Silva RicardoAbstract:Kurt Gödel demostró en 1931, que para todo sistema formal Z recursivo lo suficientemente potente como para derivar los axiomas de Peanoy que además se suponga como consistente, se tiene que en el sistema hay proposiciones indecidibles, es decir, el sistema no es completo. Por otra parte, Gödel probó que si el sistema Z es consistente entonces no se puedederivar en Z una proposición que afirme la consistencia de Z. Estos resultadosson los que se conocen como Primer Teorema de Incompletitud Gödel y Segundo Teorema de Incompletitud de Gödel. Dichos resultados tienen un gran impacto sobre la investigación de los fundamentos de la matemática quevenía gestándose en los primeros treinta años del siglo pasado, y tiene ademásconsecuencias sobre la filosofía de la matemática de dicha época. Este artículo se encuentra estructurado en tres partes: En una primera parte nos ocupamos de la formulación de los Teoremas de incompletitud y las ideas principalesde su demostración en cada caso. Seguidamente mostraremos una aplicación del Segundo Teorema de Incompletitud en la teoría de conjuntos referente a los cardinales inaccesibles. Por último, desarrollaremos las consecuencias filosóficas que los Teoremas de incompletitud de Gödel tienen sobre el proyecto meta-matemático de David Hilbert. Abstract: Kurt Gödel proved in 1931 that for any formal recursive system Z powerful enough to derive the Peano axioms and also supposed to be consistent, we have that in the System there are undecidable propositions, i.e., the system is not complete. Moreover, Gödel proved that if the Z system is consistent then it can not derive in Z a proposition asserting the consistency of Z. These results are known as Gödel's First Incompleteness Theorem and Gödel's Second Incompleteness Theorem. Such results have a great impact on the investigation of the foundations of mathematics that had been developing in the first thirty years of the last century, and it, furthermore, has implications for philosophy of mathematics of that time. This article is structured in three parts: In the first part we deal with the formulation of the incompleteness theorems and the main ideas of its proof in each case. Then, we will show an application of the Second Incompleteness Theorem in set theory concerning inaccessible cardinal. Finally, we will develop the philosophical consequences that Gödel's incompleteness theorems have on the meta-mathematical project that David Hilbert proposed
Leo Corry - One of the best experts on this subject based on the ideXlab platform.
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Mie’s Electromagnetic Theory of Matter and the Background to Hilbert’s Unified Foundations of Physics
Foundations of Mathematics and Physics One Century After Hilbert, 2018Co-Authors: Leo CorryAbstract:On November 20, 1915, David Hilbert delivered a talk in Gottingen, presenting his new axiomatic derivation of the “basic equations of physics”. This talk is often remembered because, allegedly, Hilbert presented in them, five days prior to Einstein, the correct, generally-covariant equations of gravitation that lie at the heart of the general theory of relativity (GTR).
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David Hilbert and the Axiomatization of Physics (1898–1918) - David Hilbert and the Axiomatization of Physics (1898-1918)
Archimedes, 2004Co-Authors: Leo CorryAbstract:Liburu hau David Hilbert matematikariak fi sikaren axiomatizazioaz eginiko hausnarketez zien tziaren historialari batek eginiko hainbat ikerketa urteren emai tza da. Israelgo Tel-Aviv Uniber tsitatearen baitakoa da liburu honen egilea den Leo Corry irakaslearen ohiko lantokia, Zien tziaren Historia eta Filosofi arako Cohn Institutua. Aurreko hamar urteetan, Berlingo Max-Planck Institut fur Wissenschaf tsgeschichte delakoan eta MITeko Teknologiaren eta Zien tziaren Historiarako Dibner Institutuan egon da egilea, ikerketa-proiektuari hasiera eta garapena emateko. Liburuaren ai tzin-solasean eta ondorengo esker onean topa daitezke liburu hau mamitu duten haziak: iturri nagusi tzat Hilberten ondarea gorde tzen duen Nachlass David Hilbert liburutegia, beste liburutegi ba tzuen artean, eta batez ere bertan gorde tzen diren Gottingengo Hilberten klase irekietako ikasleen apunteak, orain arte bazter u tziak eta Hilbert beraren esanetan diziplinarteko irismen zabaleko pen tsaketa metodo fun tsezkoak zi tzaizkionak.
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David Hilbert and the axiomatization of physics 1898 1918
2004Co-Authors: Leo CorryAbstract:Liburu hau David Hilbert matematikariak fi sikaren axiomatizazioaz eginiko hausnarketez zien tziaren historialari batek eginiko hainbat ikerketa urteren emai tza da. Israelgo Tel-Aviv Uniber tsitatearen baitakoa da liburu honen egilea den Leo Corry irakaslearen ohiko lantokia, Zien tziaren Historia eta Filosofi arako Cohn Institutua. Aurreko hamar urteetan, Berlingo Max-Planck Institut fur Wissenschaf tsgeschichte delakoan eta MITeko Teknologiaren eta Zien tziaren Historiarako Dibner Institutuan egon da egilea, ikerketa-proiektuari hasiera eta garapena emateko. Liburuaren ai tzin-solasean eta ondorengo esker onean topa daitezke liburu hau mamitu duten haziak: iturri nagusi tzat Hilberten ondarea gorde tzen duen Nachlass David Hilbert liburutegia, beste liburutegi ba tzuen artean, eta batez ere bertan gorde tzen diren Gottingengo Hilberten klase irekietako ikasleen apunteak, orain arte bazter u tziak eta Hilbert beraren esanetan diziplinarteko irismen zabaleko pen tsaketa metodo fun tsezkoak zi tzaizkionak.
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David Hilbert: Algebra and Axiomatics
Modern Algebra and the Rise of Mathematical Structures, 2004Co-Authors: Leo CorryAbstract:Few accounts of the development of particular mathematical disciplines around the turn of the century can be complete without analyzing Hilbert’s contribution to them. Algebra, and the particular account presented here, are no exception to this rule.1 David Hilbert (1862-1943) was the leading mathematician of his era, and the mathematical institute in Gottingen—first under the leadership of Felix Klein (1849-1925) and later on under Hilbert—became the world center of mathematics until the rise of Nazism in Germany.2 Dedekind also spent his early career in Gottingen, many years before Hilbert’s arrival there. Later on, Emmy Noether—invited to Gottingen by Hilbert in 1915—developed her own algebraic work at the same place.
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David Hilbert y su filosofia empiricista de la geometria
Boletín de la Asociación Matemática Venezolana, 2002Co-Authors: Leo CorryAbstract:David Hilbert (1862-1943) fue sin duda uno de los matematicos mas influyentes de principios del siglo veinte. Su obra se extendio sobre numerosos y diversos campos de investigacion abarcando desde la teoŕia de los invariantes algebraicos, hasta las ecuaciones integrales, pasando por los fundamentos de la geometŕia, de la fisica y de la logica. Junto con Henri Poincare (1854-1912), Hilbert ha sido reconocido como el ultimo gran universalista de las matematicas. A partir de 1918 el principal campo de investigacion de Hilbert fue el de los fundamentos de la aritmetica, campo que cultivo junto con sus ultimos grandes colaboradores, Paul Bernays (1888-1977) y Wilhelm Ackermann (1896-1962). Como parte de esta actividad, Hilbert se vio envuelto en una serie de vividas discusiones con importantes matematicos que representaban visiones opuestas a la suya. Estas discusiones llegaron a conocerse en la historia de la disciplina como “la crisis de fundamentos, y los principales puntos de vista expuestos en ella se denominaron intuicionismo, logicismo y formalismo. Hilbert fue el mas prominente matematico entre aquellos que defendieron el punto de vista formalista. La posicion intuicionista fue propugnada inicialmente por el holandes Luitzen Egbertus Brouwer (1881-1966). Brouwer habia realizado una destacada labor de investigacion como topologo, pero su verdadero interes se concentraba, desde muy temprano en su carrera, en la pregunta de los fundamentos de la matematica. Siguiendo una ĺinea de pensamiento originada por el matematico berlines Leopold Kronecker (1823-1891), Brouwer sostenia una filosofia de las matematicas que se oponia al uso del infinito actual, y en particular al uso indiscriminado de la gran gama de cardinales infinitos recientemente dados a conocer por la teoŕia de los conjuntos desarrollada a fines del siglo diecinueve por Georg Cantor (1845-1918) y Richard Dedekind (1831-1916). Matematicos como Kronecker y Brouwer consideraban que el unico tipo de infinito que podŕia
Reinhard Kahle - One of the best experts on this subject based on the ideXlab platform.
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Μηδεὶς ἀγεωμέτρητοϛ εἰσίτω
Logic Epistemology and the Unity of Science, 2018Co-Authors: Reinhard KahleAbstract:This paper provides a discussion to which extent the Mathematician David Hilbert could or should be considered as a Philosopher, too. In the first part, we discuss some aspects of the relation of Mathematicians and Philosophers. In the second part we give an analysis of David Hilbert as Philosopher.
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Poincaré in Göttingen
The Western Ontario Series in Philosophy of Science, 2014Co-Authors: Reinhard KahleAbstract:In this paper we discuss the relation between Henri Poincare and the Gottingen mathematician David Hilbert , in particular, in connection with Poincare’s visit to Gottingen in 1909.
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David Hilbert and Principia Mathematica
The Palgrave Centenary Companion to Principia Mathematica, 2013Co-Authors: Reinhard KahleAbstract:After the failure of Frege’s Grundgesetze (1903a), due to Russell’s paradox, it was the Principia Mathematica of Whitehead and Russell which first successfully developed mathematics within a logical framework. As such it attracted the attention of David Hilbert and his school. For the reception of the first edition of Principia in Gottingen, one has to consider four aspects: (a) the context in which Principia was studied in Gottingen, (b) Heinrich Behmann’s PhD thesis on Principia, (c) Paul Bernays’s reaction to Principia, and (d) the fate of logicism.
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a heranca de David Hilbert na filosofia da matematica
Boletim da Sociedade Portuguesa de Matemática, 2011Co-Authors: Reinhard KahleAbstract:Apresentamos algumas linhas gerais do projecto de investigacao A Heranca de Hilbert na Filosofia da Matematica, financiado pela FCT/MCTES, PTDC/FIL-FCI/109991/2009. O nosso objectivo e reavaliar as ideas de David Hilbert que contribuiram—e contribuem—para o desenvolvimento da filosofia da matematica. Por um lado, a historia do programa de Hilbert e um successo, apesar dos resultados de Godel. Gerhard Gentzen foi o primeiro que mostrou como podemos demonstrar a consistencia (relativa) de sistemas matematicos formais. Ainda hoje, o estudo da consistencia relativa e uma parte importante da investigacao em logica matematica. Por outro lado, muitos topicos da actual filosofia da matematica contem ideias de Hilbert, nao observadas ou ignoradas.
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Hilbert's Paradox
Historia Mathematica, 2002Co-Authors: Volker Peckhaus, Reinhard KahleAbstract:Abstract In this paper Hilbert's paradox is for the first time published completely. It was discovered by David Hilbert while he was struggling with Cantor's set theory. According to Hilbert, it initiated Ernst Zermelo's version of the Zermelo–Russell paradox. It is the paradox of all sets derived from addition (union) and self-mapping. It is similar to Cantor's paradox of the set of all cardinals, but, following Hilbert, of “purely mathematical nature”, because an open reference to Cantor's cardinal and ordinal arithmetic is avoided. © 2002 Elsevier Science (USA). In diesem Aufsatz wird erstmals die Hilbertsche Antinomie vollstandig publiziert. David Hilbert hat sie wahrend seiner Auseinandersetzungen mit der Cantorschen Mengenlehre gefunden. Seinen Angaben zufolge wurde Ernst Zermelo durch sie zu seiner Version der Zermelo–Russellschen Antinomie angeregt. Es handelt sich um die Antinomie der Menge aller durch Addition (Vereinigung) und Selbstbelegung erzeugbaren Mengen. Sie ahnelt der Cantorschen Antinomie der Menge aller Kardinalzahlen, ist aber, so Hilbert, “rein mathematisch,” da in ihr ein offensichtlicher Bezug zur Cantorschen Kardinal- und Ordinalzahlarithmetik vermieden wird. © 2002 Elsevier Science (USA). MSC 2000 subject classifications: 01A55, 01A60, 03-03, 03A05, 03E30.
Mark Ravaglia - One of the best experts on this subject based on the ideXlab platform.
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David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark
2018Co-Authors: Wilfried Sieg, Mark RavagliaAbstract:Wilfred Sieg and Mark Ravaglia. David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark
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David Hilbert and paul bernays grundlagen der mathematik first edition 1934 1939
Landmark Writings in Western Mathematics 1640-1940, 2005Co-Authors: Wilfried Sieg, Mark RavagliaAbstract:Publisher Summary In the book Grundlagen Der Mathematik, Hilbert and Bernays systematically present their proof-theoretic investigations and a wide range of current results, such as Herbrand's theorems and Godel's incompleteness theorems. Some specialized topics are also discussed, such as the development of mathematical analysis and the unsolvability of the decision problem. Hilbert changed his basic attitude towards consistency proofs only around 1903 after the discovery of the elementary contradiction of Russell and Zermelo, which convinced him that there was a deep problem. The formalism F and the numbering are required to satisfy roughly two representability conditions: primitive recursive arithmetic is “contained in” F; and the syntactic properties and relations of F's expressions, as well as the processes that can be carried out on such expressions, are given by primitive recursive predicates and functions. Finally, it is to consider the stimulus his approach and questions provided to contemporaries outside the Hilbert school. There is no foundational enterprise with a more profound and far-reaching effect on the emergence and development of mathematical logic.
