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La Teor´Ia De Conjuntos En El Periodo Entreguerras

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La Teor´Ia De Conjuntos En El Periodo Entreguerras La teor´ıade conjuntos en el periodo Entreguerras: la internacionalizaci´onde la matem´atica polaca a trav´esde Fundamenta Mathematicae ySierpi´nski Tesis doctoral de Andr´esChaves Beltr´an Director: Dr. Jos´eFerreir´os Dom´ınguez Tutor: Dr. Xavier Roqu´eRodr´ıguez Centre D’ Hist`oria de la Ci`encia Universitat Autonoma de Barcelona Septiembre de 2014 Agradezco a: La Universidad de Nari˜no por patrocinar mis estudios de doctorado, sin este acto de confianza hacia mi por parte de la UDENAR habr´ıa sido poco pro- bable que me aventurara a realizar un doctorado. Al profesor Jos´eFerreir´os Dom´ınguez, por acceder a dirigir esta tesis y por permitirme conocer parte de su bagage intelectual. A Lucho Recalde por seguir brind´andome su experiencia acad´emica. Al Master y el Doctorado en Historia de la Ciencia por mostrarme enormes posibilidades de estudio. Amifamiliaporsoportarmiscambiosde´animoduranteestosa˜nos. A Nancy, por tantas cosas. Resumen En esta tesis se abordan algunos aspectos del desarrollo hist´orico de la teor´ıa de conjuntos desde su nacimiento en la d´ecada de 1870 hasta el inicio de la Segunda Guerra Mundial en 1939, enfatizando en el periodo 1920-1939 concretamente en Varsovia, en la revista Fundamenta Mathematicae yenla obra del matem´aticoWaclaw Sierpi´nski. En este sentido, se hace un estudio hist´orico de la teor´ıa de conjuntos en la escuela de Varsovia en el periodo Entreguerras, estudiando los aspectos que se conjugaron para que la comunidad matem´atica en Polonia adquiriera una posici´on de reconocimiento a nivel internacional. Entre estos aspectos est´an el surgimiento de la revista Fundamenta Mathematicae como ´organo de di- fusi´onde los matem´aticos de la escuela de Varsovia, la cual se especializ´oen teor´ıade conjuntos y ramas afines a ´esta, tambi´en se estudia la obra de Sier- pi´nski en relaci´ona la teor´ıade conjuntos. Abstract In this thesis some aspects of the historical development of set theory are tackled, since its birth in the 1870s until the beginning of the second World war in 1939, making stress in Varsovia, in the period from 1920 to 1939, in the journal Fundamenta Mathematicae and in the Sierpi´nski’swork. In this sense, a historical study of set theory in the Varsovia’s school during the interwar period is made, assessing the facts that brought the mathematical community in Poland to acquire international recognition. Between them, the birth of the journal Fundamenta Mathematicae as an entity for spreading the work of mathematicians of the Varsovia’s school, which become expert in set theory and similar branchs. It is also studied the Sierpinski’s work in regard to set theory. ´Indice general 1. Introducci´on 9 2. La categor´ıadisciplinar de la teor´ıade conjuntos 19 2.1. Clasificaci´ondisciplinar en matem´aticas. 20 2.2. Estado de las clasificaciones disciplinares en matem´aticas a principios de Siglo XX: Hilbert, Jahrbuch, Zentralblatt y Congresos Internacionales . 25 2.2.1. Los23problemasdeHilbert . 27 2.2.2. Jahrbuch Uber¨ die Fortschritte der Mathematik .... 29 2.2.3. Zentralblatt f¨urMathematik und ihre Grenzgebiete ... 34 2.2.4. Congresos Internacionales de Matem´aticas . 38 2.3. Conclusiones y comentarios . 45 3. Expansi´ony enfoques de la teor´ıade conjuntos 51 3.1. Primer momento: Cantor y Dedekind . 52 3.2. Segundo momento: Italianos (Peano, Bettazzi, Volterra, Levi) yHilbert.............................. 58 3.3. Tercer momento . 61 3.3.1. Par´ıs:Borel, Baire, Lebesgue . 61 3.3.2. Alemania: Hilbert, Zermelo, Haussdor↵ ......... 67 3.3.3. Inglaterra . 72 3.4. Cuarto momento . 74 3.4.1. Fraenkel, G¨odel, Von Neumann, Bernays . 74 3.4.2. Mosc´u: Luzin, Alexandro↵ ................ 79 3.5. Conclusiones y Comentarios . 83 5 4. Las matem´aticas en Polonia 89 4.1. Estado de las matem´aticas en Polonia antes de la Primera Guerra Mundial . 90 4.2. Escuela Matem´aticade Varsovia . 95 4.2.1. Origen de la escuela . 96 4.2.2. Fundamenta Mathematicae ................104 4.3. Relaci´onentre l´ogica,matem´aticasy filosof´ıaen la escuela de Varsovia ..............................111 4.3.1. Jan Lukasiewicz . 115 4.3.2. Alfred Tarski . 117 4.3.3. Andrzej Mostowski . 120 4.4. Escuela de Le´opolis . 123 4.4.1. Stefan Banach . 125 4.4.2. El cuaderno escoc´es. 129 4.5. Otros centros matem´aticosen Polonia . 132 4.6. La Sociedad Polaca de Matem´aticas. 134 4.7. Biograf´ıade Sierpi´nski . 139 4.8. Conclusiones y comentarios . 145 5. Algunos art´ıculosrelevantes 153 5.1. La teor´ıade conjuntos y sus aplicaciones en Sierpi´nski . 154 5.1.1. Teor´ıageneral de conjuntos . 154 5.1.2. Conjuntos anal´ıticos y proyectivos . 156 5.1.3. Medida ..........................158 5.1.4. Topolog´ıaGeneral . 160 5.1.5. Funciones de variable real . 163 5.2. Una selecci´on m´as aguda . 165 5.2.1. L’ axiome de M. Zermelo et son rˆoledans la Th´eorie des Ensembles et l’ Analyse . 165 5.2.2. Otros art´ıculosde Sierpi´nski. 171 5.2.3. Algunos art´ıculos en FM hasta 1939 . 177 5.3. Conclusiones y comentarios . 181 6. Sobre la teor´ıadescriptiva de conjuntos 187 6.1. Teor´ıadescriptiva de conjuntos en los textos . 189 6.1.1. Textos de Hausdor↵ y de Kuratowski . 190 6.1.2. Textos de Luzin y de Sierpi´nski . 193 6.2. Or´ıgenesde la teor´ıadescriptiva de conjuntos . 198 6.2.1. Propiedades de regularidad . 199 6.2.2. Borelianos y clases de Baire . 202 6.2.3. Los conjuntos anal´ıticosy los coanal´ıticos . 209 6.2.4. La jerarqu´ıaproyectiva . 214 6.3. Algunos aspectos t´ecnicos y filos´oficos de la obra de Luzin . 219 6.3.1. Dominio fundamental . 221 6.3.2. El problema de la existencia . 223 6.3.3. Operaciones negativas-virtualidades . 228 6.4. Conclusiones y comentarios . 230 6.5. Anexo: La etapa post-cl´asica de la teor´ıa descriptiva de conjuntos. Resultados principales . 238 7. Conclusiones generales 243 A. Clasificaci´ondisciplinar MSC10 251 B. Axiomas ZFE 253 C. Lista de problemas de Hilbert 255 D. Indice del Jahrbuch 257 E. Sobre las necesidades de las matem´aticas en Polonia 263 F. Art´ıculosde Sierpi´nski en FM 273 F.1. Clasificaci´on de art´ıculos . 274 F.2. Resultados de la clasificaci´on . 295 7 8 Cap´ıtulo1 Introducci´on Esta investigaci´on se enmarca en el estudio de la historia de la teor´ıa de conjuntos, en concreto, se examina la concepci´on y los usos que una co- munidad matem´atica emergente, como la polaca, ten´ıa de esta rama de las matem´aticas. Tambi´en se aborda la conformaci´ony la internacionalizaci´on de la comunidad polaca de matem´aticos, enfatizando en la importancia de la revista Fundamenta Mathematicae como la primera revista especializada en una tem´atica de las matem´aticas como lo es la teor´ıade conjuntos, y en la figura y obra de Wac law Sierpi´nski, uno de los fundadores de esta revista. La teor´ıa de conjuntos ha jugado un papel b´asico en las matem´aticas contempor´aneas, m´as a´un en el an´alisis matem´atico y sus ramas cercanas, ya que por ejemplo, el an´alisis se fundamenta en la estructuraci´on de la l´ınea recta, lo que permite determinar la idea moderna de funci´on, que es indispensable para todas las ramas de las matem´aticas. En [De La Pava, 2010, p. 65] se plantea la importancia de la teor´ıade conjuntos de la siguiente forma: De hecho, la relaci´onentre la teor´ıa de funciones, que al- canz´otambi´ensu independencia del an´alisis, y la teor´ıa de con- juntos ha llegado a ser tan estrecha que hoy en d´ıa dif´ıcilmente la primera puede dar un paso sin apoyarse en la segunda. La mayor´ıa de textos modernos sobre teor´ıade funciones inician su presenta- ci´oncon uno o varios cap´ıtulos dedicados a la teor´ıa de conjuntos. M´asa´un, los libros de texto de an´alisis, topolog´ıa o ´algebra, 9 tambi´en tienen en sus preliminares o en el ap´endice un cap´ıtulo dedicado a la teor´ıa de conjuntos. En particular, el concepto de biyecci´on,como instrumento para medir el tama˜node los conjuntos, ha sido trascendental en casi todas las ´areas de las matem´aticas. Dentro de los conceptos m´as extraordinarios que se han producido en la teor´ıa de funciones, a trav´esde m´etodos conjuntistas, est´anel de integral de Lebesgue y el concepto de medida. La presentaci´onaxiom´atica de la integral de Lebesgue se debe a los desarrollos de Cantor. En ese aspecto tambi´ense puede traer a colaci´onla visi´onmoderna del gru- po Nicolas Bourbaki, el cual ve la teor´ıa de conjuntos como base de todas las matem´aticas, donde el primero de los diez vol´umenenes de su tratado Elementos de matem´atica se llama teor´ıa de conjuntos, lo cual se relaciona con el enfoque conjuntista que se desprende de la obra de Dedekind y que se plantea en parte del cap´ıtulo2. Es pertinente aclarar que actualmente la teor´ıa de conjuntos tiene un estado de independencia que no tuvo a principios del Siglo XX, ya que para ese entonces sus fronteras eran difusas y sus contenidos se solapaban con otras ramas que ahora son independientes. Estas ramas son la teor´ıa general de con- juntos, la teor´ıa de funciones, la topolog´ıa, la teor´ıa de la medida, la teor´ıa descriptiva de conjuntos y la l´ogica.
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