Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with particular attention paid to the early twentieth century search to secure the foundations of mathematics and a detailed look at contemporary Hungarian mathematics. After doing so, I conclude that the scope of mathematics is not what one might expect but that it’s still absolutely worth doing and appreciating.

Orientador: Maria Aparecida Viggiani Bicudo / Banca: Fábio Maia Bertato / Banca: Irineu Bicudo / Banca: Orlando de Andrade Figueiredo / Banca: Gustavo Barbosa / Resumo: Apresentamos nesta tese uma proposta de inserção do tema teorema da incompletude de Gödel em cursos de Licenciatura em Matemática. A interrogação norteadora foi: como sentidos e significados do teorema da incompletude de Gödel podem ser atualizados em cursos de Licenciatura em Matemática? Na busca de elaborarmos uma resposta para essa questão, apresentamos o cenário matemático presente à época do surgimento deste teorema, expondo-o como a resposta negativa para o projeto do Formalismo que objetivava formalizar toda a Matemática a partir da aritmética de Peano. Além disso, trazemos no contexto, as outras duas correntes filosóficas, Logicismo e Intuicionismo, e os motivos que impossibilitaram o completamento de seus projetos, que semelhantemente ao Formalismo buscaram fundamentar a Matemática sob outras bases, a saber, a Lógica e os constructos finitistas, respectivamente. Assim, explicitamos que teorema da incompletude de Gödel aparece oferecendo resposta negativa à questão da consistência da aritmética, que era um problema para a Matemática na época, estabelecendo uma barreira intransponível para a demonstração dessa consistência, da qual dependia o sucesso do Formalismo e, consequentemente, a fundamentação completa da Matemática no ideal dos formalistas. Num segundo momento, focamos na demonstração deste teorema expondo-a em duas versões distintas, que para nós se nos mostraram apropriadas para serem trabalhadas em cursos de Licenciatura em Matemática. Uma, como possibilidad... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this thesis we present a proposal to insert Gödel's incompleteness theorem in Mathematics Education undergraduate courses. The main research question guiding this investigation is: How can the senses and meanings of Gödel's incompleteness theorem be updated in Mathematics Education undergraduate courses? In answering the research question, we start by presenting the mathematical scenario from the time when the theorem emerged; this scenario proposed a negative response to the project of Formalism, which aimed to formalize all Mathematics based upon Peano's arithmetic. We also describe Logicism and Intuitionism, focusing on reasons that prevented the completion of these two projects which, in similarly to Formalism, were sought to support mathematics under other bases of Logic and finitists constructs. Gödel's incompleteness theorem, which offers a negative answer to the issue of arithmetic consistency, was a problem for Mathematics at that time, as the Mathematical field was passing though the challenge of demonstrating its consistency by depending upon the success of Formalism and upon the Mathematics' rationale grounded in formalists' ideal. We present the proof of Gödel's theorem by focusing on its two different versions, both being accessible and appropriate to be explored in Mathematics Education undergraduate courses. In the first one, the reader will have a chance to follow the details of the proof as developed by Gödel in 1931. The intention here is to expose Gödel' ideas used at the time, as well as to clarify understanding of the proof. In the second one, the reader will be familiarized with another proof that validates the incompleteness theorem, presenting it in its formal version. The intention here is to highlight Gödel's numbering experience and the construction of undecidable sentence, and to present the formal ... (Complete abstract electronic access below) / Doutor

A presente dissertação consiste em um estudo de apresentações da demonstração do Segundo Teorema de Incompletude de Gödel. Considera, com especial atenção, aquelas feitas por Shoefield no Mathematical Logic e por Hilbert e Bernays no Grundlagen der Mathematik. Como resultado, obtém-se uma análise das condições de derivabilidade e considerações sobre como é possível demonstrá-las.

In the last few years there has been a large effort for analysing the computational properties of reasoning in fuzzy Description Logics. This has led to a number of papers studying the complexity of these logics, depending on their chosen semantics. Surprisingly, despite being arguably the simplest form of fuzzy semantics, not much is known about the complexity of reasoning in fuzzy DLs w.r.t. witnessed models over the Gödel t-norm. We show that in the logic G-IALC, reasoning cannot be restricted to finitely valued models in general. Despite this negative result, we also show that all the standard reasoning problems can be solved in this logic in exponential time, matching the complexity of reasoning in classical ALC.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-13T14:26:30Z No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) / Made available in DSpace on 2017-09-13T14:26:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) Previous issue date: 2017-02-22 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this thesis we deal with G odel-type Universes in the context of modi ed gravity, in particular, Chern-Simons modi ed gravity and Brans-Dicke theory with cosmological constant (BD- ). The G odel-type metrics have been intensively discussed in the General Relativity (GR) over years. It is known that such a metrics present Closed Time-like Curves (CTC's), in other words, the G odel-type metrics are themselves an example of the causality violation. Our goal is verify the consistency of the G odel-type metrics within the Chern-Simons modi ed gravity in both: non-dynamical and dynamical formulations. In the non-dynamical framework, we show that is possible a vacuum solution in contrast to GR. Another essentially new result that we get is the presence of causal solutions for a well-motivated matter source, in general, the solutions have no analogue in GR. Moreover, the vacuum solution represents the limiting case separating the completely causal and non-causal regions, such a property re ects the topological features of the Chern-Simons theory. The primordial distinguishing feature between the Chern-Simons modi ed gravity and GR solutions is the presence of the breaking Lorentz symmetry. It turns out this breaking opens up a range of new solutions. We show that the non-trivial Chern-Simons solutions in the non-dynamical framework is accompanied by rst-order corrections of the Lorentz-violating parameter. Furthermore, in the dynamical framework the geometric parameters are also a ected by second-order corrections of the Lorentzviolating parameter. We also investigated the G odel-type metrics in BD- model. We obtain a vacuum solution which is completely causal, m2 = 4!2, where for ~! ! 1 one recovers the GR with a scalar eld and cosmological constant. It is worth calling attention to the role of the cosmological constant that is fundamental in this context. / Nesta tese tratamos os Universos tipo Code' no contexto da gravidade modificada, em particular, na gravidade modificada de Chern-Simons e na teoria de Brans-Dicke com constante cosmolOgica (BD-A). As metricas tipo Code' vem sendo intensamente discuti­das na Relatividade Geral (RG) ao longo dos anon. Sabe-se que tail metricas apresentam Curvas tipo Tempo Fechadas (CTC's), ou seja, as prOprias metricas tipo Code' sao um exemplo da violagao de causalidade. Nosso objetivo é verificar a consistencia das metricas do tipo Code' dentro da gravidade modificada de Chern-Simons em ambas as formula-goes: nao dinamica e dinamica. Na formulagao nao-dinamica, mostramos que é possivel uma solugao de vacuo diferentemente da RG. Outro resultado essencialmente novo que obtemos é a presenga de solugoes causais para uma fonte de materia bem motivada, em geral, as solugoes nao tem analog° na RG. Alem disso, a solugao de vacuo representa o caso limite que separa as regioes completamente causal e nao causal, tal propriedade re­flete as caracteristicas topolOgicas da teoria de Chern-Simons. A caracteristica que difere fundamentalmente as solugoes da gravidade modificada de Chern-Simons e as da RG é a presenga da quebra da simetria de Lorentz. Acontece que essa quebra abre um leque de novas solugoes. Mostramos que as solugoes nao triviais de Chern-Simons na formulagao nao dinamica sao acompanhadas por corregoes de primeira ordem do parametro de viola­gao de Lorentz. Alem disso, na formulagao dinamica os parametros geometricos tambem sao afetados por corregoes de segunda ordem do parametro de violagao de Lorentz. Investigamos tambem as metricas tipo Code' no modelo BD-A. Obtemos uma solugao de vacuo completamente causal, m2 = 4w2, onde para cD -+ oo recupera-se a GR com um campo escalar e constante cosmolOgica. Vale a pena chamar a atengao para o papel da constante cosmolOgica que é fundamental neste contexto.

A presente dissertação consiste em um estudo de apresentações da demonstração do Segundo Teorema de Incompletude de Gödel. Considera, com especial atenção, aquelas feitas por Shoefield no Mathematical Logic e por Hilbert e Bernays no Grundlagen der Mathematik. Como resultado, obtém-se uma análise das condições de derivabilidade e considerações sobre como é possível demonstrá-las.

Notre travail vise à étudier les Max-Phil, textes inédits de Kurt Gödel, dans lesquels il développe sa pensée philosophique. Nous nous intéressons plus spécifiquement à la question du réalisme conceptuel, position déjà défendue dans ses écrits publiés selon laquelle les concepts existent indépendamment de nos définitions et constructions. L’objectif est de montrer qu’une interprétation cohérente de ces textes encore peu connus est possible. Pour ce faire, nous proposons une interprétation de certains passages, interprétation hypothétique mais susceptible d’apporter de nouveaux éléments à des questions laissées sans réponse par les textes publiés, telles celles relatives au réalisme conceptuel. Cette dernière position ne peut être comprise que par un éclairage de la notion de concept chez Gödel. Les concepts sont des entités logiques objectives, au cœur du projet d’une théorie des concepts conçue comme une logique intentionnelle et inspirée de la scientia generalis de Leibniz. L’analyse des Max-Phil souligne que la notion de concept et la primauté du réalisme conceptuel sur le réalisme mathématique ne peuvent se comprendre qu’à la lumière du cadre métaphysique que se donne Gödel, à savoir d’une monadologie d’inspiration leibnizienne. Les Max-Phil offrent ainsi des indices sur la façon dont Gödel reprend et modifie la monadologie de Leibniz, afin, notamment, d’y inscrire les concepts. L’examen de ce cadre métaphysique tend également à éclaircir les rapports entre les concepts objectifs, les concepts subjectifs (tels que nous les connaissons), et les symboles (par lesquels nous exprimons les concepts), mais aussi les rapports entre logique et mathématiques. / Our work aims at studying the unpublished texts of Kurt Gödel, known as the Max-Phil, in which the author develops his philosophical thought. This study follows the specific issue of conceptual realism which is adopted by Gödel in his published texts (during his lifetime or posthumously), and according to which concepts are independent of our definitions and constructions. We want to show that a consistent interpretation of the Max-Phil is possible. To do so, we propose an interpretation of some excerpts, which, even if it is only hypothetical, can give new elements in order to answer open questions of the published texts, e.g. questions about conceptual realism. This last position is not understandable without explaining Gödel’s notion of concept. For him, concepts are logical and objective entities, and they are at the core of a theory of concepts, which is conceived as an intensional logic, following Leibniz’s scientia generalis. The analysis of the Max-Phil underlines that we can understand the notion of concept and the primacy of conceptual realism over mathematical realism only in the light of Gödel’s metaphysical frame, i.e. of a monadology inspired by Leibniz. Thus the Max-Phil shows how Gödel reinvestigates Leibnizian monadology, and offers some clues on the modifications he makes on it in order to include concepts. The examination of this metaphysical frame tends to elucidate the relationships between objective concepts, subjective concepts (as we know them) and symbols (through which we express concepts), and also the relationship between logic and mathematics.

In a famous paper of 1931, Gödel proved that any formalization of elementary Arithmetic is incomplete, in the sense that it contains statements which are neither provable nor disprovable. Some two years before this, Presburger proved that a mutilated system of Arithmetic, employing only addition but not multiplication, is complete. This essay is partly an exposition of a system such as Presburger's, and partly an attempt to gain insight into the source of the incompleteness of Arithmetic, by linking Presburger's result with Gödel's.

FARIAS, Pablo Mayckon Silva. Um estudo sobre as origens da Lógica Matemática e os limites da sua aplicabilidade à formalização da Matemática. 2007. 110 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2007. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-12T14:54:53Z No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-20T13:48:23Z (GMT) No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) / Made available in DSpace on 2016-07-20T13:48:23Z (GMT). No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) Previous issue date: 2007 / This work is a study about the origins of Mathematical Logic and the limits of its applicability to the formal development of Mathematics. Firstly, Dedekind’s arithmetical theory is presented, which was the first theory to provide a precise definition for natural numbers and to demonstrate relying on it all facts commonly known about them. Peano’s axiomatization for Arithmetic is also presented, which in a sense simplified Dedekind’s theory. Then, Frege’s Begriffsschrift is presented, the formal language from which modern Logic originated, and in it are represented Frege’s basic definitions concerning the notion of number. Afterwards, a summary of important topics on the foundations of Mathematics from the first three decades of the twentieth century is presented, beginning with the paradoxes in Set Theory and ending with Hilbert’s formalist doctrine. At last, are presented, in general terms, Gödel’s incompleteness. theorems and Turing’s computability concept, which provided precise answers to the two most important points in Hilbert’s program, to wit, a direct proof of consistency for Arithmetic and the decision problem, respectively. Keywords: 1. Mathematical Logic 2. Foundations of Mathematics 3. Gödel’s incompleteness theorems / Este trabalho é um estudo sobre as origens da Lógica Matemática e os limites da sua aplicabilidade ao desenvolvimento formal da Matemática. Primeiramente, é apresentada a teoria aritmética de Dedekind, a primeira teoria a fornecer uma definição precisa para os números naturais e com base nela demonstrar todos os fatos comumente conhecidos a seu respeito. É também apresentada a axiomatização da Aritmética feita por Peano, que de certa forma simplificou a teoria de Dedekind. Em seguida, é apresentada a ome{german}{Begriffsschrift} de Frege, a linguagem formal que deu origem à Lógica moderna, e nela são representadas as definições básicas de Frege a respeito da noção de número. Posteriormente, é apresentado um resumo de questões importantes em fundamentos da Matemática durante as primeiras três décadas do século XX, iniciando com os paradoxos na Teoria dos Conjuntos e terminando com a doutrina formalista de Hilbert. Por fim, são apresentados, em linhas gerais, os teoremas de incompletude de Gödel e o conceito de computabilidade de Turing, que apresentaram respostas precisas às duas mais importantes questões do programa de Hilbert, a saber, uma prova direta de consistência para a Aritmética e o problema da decisão, respectivamente.