Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1996. / Includes bibliographical references (p. 121).

In this thesis, we present a number of results in set theory, particularly in the areas of forcing, large cardinals, and combinatorial set theory. Chapter 2 concerns covering matrices, combinatorial structures introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of this proof and subsequent work with Sharon, Viale isolated two reflection principles, CP and S, which can hold of covering matrices. We investigate covering matrices for which CP and S fail and prove some results about the connections between such covering matrices and various square principles. In Chapter 3, motivated by the results of Chapter 2, we introduce a number of square principles intermediate between the classical and (+). We provide a detailed picture of the implications and independence results which exist between these principles when is regular. In Chapter 4, we address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik-Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik-Sharon model and various other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales. In Chapter 5, we prove that, assuming large cardinals, it is consistent that there are many singular cardinals such that every stationary subset of + reflects but there are stationary subsets of + that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth and is joint work with James Cummings. In Chapter 6, we extend a result of Gitik, Kanovei, and Koepke regarding intermediate models of Prikry-generic forcing extensions to Radin generic forcing extensions. Specifically, we characterize intermediate models of forcing extensions by Radin forcing at a large cardinal using measure sequences of length less than. In the final brief chapter, we prove some results about iterations of w1-Cohen forcing with w1-support, answering a question of Justin Moore.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 99-102) and index.

This thesis examines the interactions between the continuum function and large cardinals. It is know, by a result of Easton, that the continuum function on regular cardinals has great freedom in ZFC. However, large cardinals lay additional constraints to possible behaviour of the continuum function. We focus on weakly compact and measurable cardinal to point out the differences in interactions with the continuum function between various types of large cardinals. We also study the case of indescribable cardinals for the comparison, and the results lead us to conclude that it is not easy to pinpoint the reason for these differences. 1

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.

We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker.