Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/25). Includes bibliographical references.

Thesis (Ph.D.) - Carleton University, 2005. / Includes bibliographical references (p. 341-348). Also available in electronic format on the Internet.

Neste trabalho investigamos problemas sobre reflexão de funções cardinais, fazendo uso de técnicas como submodelos elementares e Teoria PCF. Mostramos que o grau de Lindelöf reflete todos os cardinais fortemente inacessíveis e que um exemplo de espaço onde a mesma função cardinal não reflita um cardinal fracamente inacessível requer a existência de 0#. Além disso, estendemos um resultado de reflexão do caráter, de espaços Lindelöf para espaços linearmente Lindelöf, obtendo novas equivalências com a Hipótese do Contínuo (CH). Obtivemos ainda várias respostas parciais para problemas clássicos deste tópico de pesquisa. / This work investigates problems about reflection of cardinal functions, using techniques such as elementary submodels and PCF Theory. We show that the Lindelöf degree reflects all the strongly inaccessible cardinals and that a example of a space in which the same cardinal function does not reflect a weakly inaccessible cardinal requires \"0# exists\". Furthermore, we extend a result of reflection of the character from Lindelöf spaces to linearly Lindelöf spaces, obtaining new equivalences with the Continuum Hypothesis (CH). We also obtained several partial answers to classic problems of this research topic.

Neste trabalho investigamos problemas sobre reflexão de funções cardinais, fazendo uso de técnicas como submodelos elementares e Teoria PCF. Mostramos que o grau de Lindelöf reflete todos os cardinais fortemente inacessíveis e que um exemplo de espaço onde a mesma função cardinal não reflita um cardinal fracamente inacessível requer a existência de 0#. Além disso, estendemos um resultado de reflexão do caráter, de espaços Lindelöf para espaços linearmente Lindelöf, obtendo novas equivalências com a Hipótese do Contínuo (CH). Obtivemos ainda várias respostas parciais para problemas clássicos deste tópico de pesquisa. / This work investigates problems about reflection of cardinal functions, using techniques such as elementary submodels and PCF Theory. We show that the Lindelöf degree reflects all the strongly inaccessible cardinals and that a example of a space in which the same cardinal function does not reflect a weakly inaccessible cardinal requires \"0# exists\". Furthermore, we extend a result of reflection of the character from Lindelöf spaces to linearly Lindelöf spaces, obtaining new equivalences with the Continuum Hypothesis (CH). We also obtained several partial answers to classic problems of this research topic.

This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages. / Denna uppsats behandlar grundläggande mängdlära och inriktar sig sedan på kardinaliteter för oändliga mängder. Bland de resultat som redovisas finns bland annat resultatet som säger att linjen R och planet R2 innehåller precis lika många punkter. Då mängdläran beskrivs av ett formellt språk så innehåller uppsatsen en bilaga om formella språk.

<p>This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages.</p> / <p>Denna uppsats behandlar grundläggande mängdlära och inriktar sig sedan på kardinaliteter för oändliga mängder. Bland de resultat som redovisas finns bland annat resultatet som säger att linjen R och planet R2 innehåller precis lika många punkter. Då mängdläran beskrivs av ett formellt språk så innehåller uppsatsen en bilaga om formella språk.</p>

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-08-27T12:52:40Z No. of bitstreams: 2 Dissertação - Rogério J de M Júnior.pdf: 1412276 bytes, checksum: 362c58043c52c9e05db80b80055fe0f2 (MD5) Ficha.pdf: 6202 bytes, checksum: 7bc34b7b59769c34edf90b86a7983117 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-08-28T20:05:14Z (GMT) No. of bitstreams: 2 Dissertação - Rogério J de M Júnior.pdf: 1412276 bytes, checksum: 362c58043c52c9e05db80b80055fe0f2 (MD5) Ficha.pdf: 6202 bytes, checksum: 7bc34b7b59769c34edf90b86a7983117 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-08-28T20:24:20Z (GMT) No. of bitstreams: 2 Dissertação - Rogério J de M Júnior.pdf: 1412276 bytes, checksum: 362c58043c52c9e05db80b80055fe0f2 (MD5) Ficha.pdf: 6202 bytes, checksum: 7bc34b7b59769c34edf90b86a7983117 (MD5) / Made available in DSpace on 2015-08-28T20:24:20Z (GMT). No. of bitstreams: 2 Dissertação - Rogério J de M Júnior.pdf: 1412276 bytes, checksum: 362c58043c52c9e05db80b80055fe0f2 (MD5) Ficha.pdf: 6202 bytes, checksum: 7bc34b7b59769c34edf90b86a7983117 (MD5) Previous issue date: 2015-05-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation we discuss briefly some issues quickly treated during the undergraduate course such as countable and uncountable sets, cardinality and other related subjects. We will present a brief historical review of the facts that gave rise to these problems, as well as people who have developed knowledge on these issues. The purpose of this report is succinctly present a direction to the Basic Education teachers for their classes, giving the opportunity to teachers to have more confidence when working with numerical sets and functions on these sets. It will also be used as a motivational element to the theoretical approach, or this associated with the problems that gave rise to such issues, both for teachers, and for students and scholars interested, because these are curious and intriguing subjects for those which enjoy studying mathematics of such subjects that are, of some kind, advanced or abstract. Among others, we can assign the comparison of cardinality of infinite sets, demonstrating that sets of racional numbers and the algebraic numbers are countable, and the real numbers and the transcendental numbers are uncountable, and besides, we show the cardinality of other interesting sets that are of great value to research in modern mathematics. Thus we think we are contributing to the improvement of teachers and students of Basic Education. / Neste trabalho abordaremos alguns assuntos tratados brevemente durante o curso de graduação tais como enumerabilidade e não enumerabilidade de conjuntos, cardinalidade e outros assuntos correlatos. Apresentaremos um pequeno aparato histórico que deram origem a esses problemas, assim como as pessoas que lançaram conhecimento sobre tais temas. O objetivo é apresentar sucintamente aos professores do ensino básico suporte para as aulas, dando a oportunidade do professor ter mais segurança quando trabalhar com conjuntos numéricos. Também servirá como elemento motivacional tanto para professores como para os alunos interessados, pois trata de assuntos curiosos e atiçadores para quem gosta de estudar matemática, como comparar a cardinalidade de conjuntos infinitos, a infinidade de números transcendentes e sua dificuldade de determiná-los e outros assuntos que são de grande riqueza de pesquisa na matemática moderna. Dessa forma pensamos estar contribuindo para o aperfeiçoamento de professores e alunos do ensino básico.