The Lebesgue Measure and Large Cardinals Jacinto Carlos Ferreira Monteiro de Freitas França Thesis to obtain the Master of Science Degree in Mathematics and Applications Supervisor: Prof. António Marques Fernandes Examination Committee Chairperson: Prof. Maria Cristina Sales Viana Serôdio Sernadas Supervisor: Prof. António Marques Fernandes Members of the Committee: Prof. Luís Pereira Prof. Fernando Ferreira July 2017 ii Dedicated to my Father and to the Three Gems iii iv Acknowledgments This work would not have been possible without the support of generous people during the research and writing process and throughout my years at Instituto Superior T´ecnico. First I would like to express my gratitude to my supervisor, Prof. Ant´onioMarques Fernandes, for his guidance and expertise during the research and writing of this thesis. In addition to the countless technical corrections, and to the generosity with his time, his advice and thoughtful discussions revealed numerous times the layers of subtlety and complexity in the field of set theory that I had not noticed. I would like to thank Prof. Ana Leonor Silvestre for her support during the MSc. program. I would also like to thank Prof. Jo˜aoRasga for the high quality of his supervision of my BSc. final project. I am grateful to Prof. Paulo Pinto and to Prof. Cristina Cˆamarafor their encouragement and support of my academic aspirations. I am also grateful to Dr. Carlos Filipe for the decisive impact of his help in my life. And I am indebted for the second opportunity that Instituto Superior T´ecnicogave me, and for the high quality of the education provided by this institution and its faculty members. I am thankful to my family for their encouragement, support, and advice during the difficult periods of my life. My deepest gratitude is to my father, Jo˜aoFran¸ca,for the immeasurable help he gave me throughout my life and for the strength of character he so often revealed. Finally, I would like to thank the Triple Gem for providing the invaluable instructions that allowed me to surpass the enormous difficulties that characterized my earlier life. v vi Resumo Um dos factores que motivaram grandes avan¸cosna teoria dos conjuntos foi a medida de Lebesgue. A medida de Lebesgue ´ea componente fundamental da teoria da integra¸c˜aode Lebesgue, que ´euma gener- aliza¸c˜aoda teoria de integra¸c˜aode Riemann. Apesar do integral de Lebesgue ter muitas vantagens, este n˜aoconsegue medir todos os conjuntos de reais. No entanto, existe uma alternativa quando adoptamos uma vers˜aomais fraca do axioma da escolha, nomeadamente o princ´ıpiodas escolhas dependentes (DC). O Teorema de Solovay afirma que se existe um modelo de ZFC com um cardinal inacess´ıvel, ent˜ao existe um modelo de ZF+DC onde todos os conjuntos de reais s˜aomensur´aveis `aLebesgue. Por outro lado, o Teorema de Shelah esclarece que quando existe um modelo de ZF+DC onde todos os conjuntos de reais s˜aomensur´aveis `aLebesgue, tamb´emexiste um modelo de ZFC com um cardinal inacess´ıvel. Esta tese ´euma exposi¸c˜aodetalhada dos teoremas de Solovay e de Shelah, e das respectivas demons- tra¸c˜oes.Os pr´e-requisitos para as demonstra¸c˜oess˜aoprincipalmente a teoria dos modelos para conjuntos; o m´etodo de forcing, em particular o colapso de L´evye a ´algebraaleat´oria;e a teoria descritiva, com foco na mensurabilidade `aLebesgue. Estes pr´e-requisitoss˜aoapresentados ap´osuma revis˜aobreve da medida de Lebesgue. Depois os dois teoremas principais s˜aodemonstrados com o devido detalhe. Palavras-chave: Medida de Lebesgue, Cardinal Inacess´ıvel, Princ´ıpiodas Escolhas Depen- dentes, Colapso de L´evy, Algebra´ Aleat´oria,Filtro R´apido vii viii Abstract One of the significant driving factors behind major developments in set theory has been the Lebesgue measure. The Lebesgue measure is the cornerstone of Lebesgue’s theory of integration, which is a gen- eralization of Riemann’s theory of integration. Despite the many advantages of the Lebesgue integral, it cannot measure every set of reals. However, an alternative can be found if we adopt a weaker version of the axiom of choice, namely the principle of dependent choices (DC). Solovay’s Theorem establishes that if there is a model of ZFC with an inaccessible cardinal, then there is a model of ZF+DC in which every set of reals is Lebesgue measurable. On the other hand, Shelah’s Theorem clarifies that when there is a model of ZF+DC in which every set of reals is Lebesgue measurable, there is also a model of ZFC with an inaccessible cardinal. This thesis is a detailed exposition of the Solovay and Shelah theorems, and their respective proofs. The prerequisites for the proofs are mainly the model theory of sets; the method of forcing, particularly the L´evycollapse and the random algebra; and descriptive set theory, focused on the aspect of Lebesgue measurability. These prerequisites are presented after a brief overview of the Lebesgue measure. Then the two main theorems are shown with due detail. Keywords: Lebesgue Measure, Inaccessible Cardinal, Principle of Dependent Choices, L´evy Collapse, Random Algebra, Rapid Filter ix x Contents Acknowledgments............................................v Resumo................................................. vii Abstract................................................. ix List of Figures............................................. xiii Nomenclature..............................................1 Glossary.................................................1 1 Introduction 1 2 The Lebesgue Measure5 2.1 The Lebesgue Measure......................................6 3 Preliminaries of Set Theory 11 3.1 Models of Set Theory...................................... 11 3.2 Forcing and the L´evyCollapse................................. 18 3.2.1 The Theory of Forcing.................................. 19 3.2.2 Examples......................................... 30 3.2.3 The L´evyCollapse.................................... 32 3.3 The Lebesgue Measure and Descriptive Set Theory...................... 35 4 The Lebesgue Measure and Large Cardinals 51 4.1 Solovay’s Theorem........................................ 51 4.2 Shelah’s Theorem......................................... 55 4.3 Further Results.......................................... 63 5 Conclusions 67 Bibliography 69 A Proof of Solovay’s Technical Lemma 71 B Proof of Shelah’s Auxiliary Lemma 75 xi xii List of Figures 3.1 Geometric representation of the forcing method........................ 23 xiii xiv Chapter 1 Introduction Set theory is a branch of Mathematics pioneered by Georg Cantor in the late XIX century. He established for the first time that there are different sizes of infinities and that there is no upper bound for them. In the subsequent period, set theory was developed and expanded. However, it was still in a nonaxiomatic stage in the early XX century. There was a naive assumption that the set of all sets existed, but Bertrand Russell found a logical paradox following from this assumption. To avoid this kind of paradoxes, mathematicians had to find an appropriate axiomatization for set theory. The set of axioms that eventually became the standard axiomatization of set theory were the Zermelo-Fraenkel axioms with choice, abbreviated to ZFC. From ZFC almost all of Mathematics can be derived. Because of this, set theory is regarded as the foundation of Mathematics. Set theory became a branch of mathematical research in its own right in great part due to Kurt G¨odel.He proved that the axiom of choice (AC) and the continuum hypothesis (CH) are consistent with ZF, provided that ZF is consistent. G¨odelaccomplished this by defining the class L of constructible sets and proving that L is a model of ZFC+CH. This put to rest early concerns about the axiom of choice. G¨odelalso established his famous incompleteness theorems in 1931. These theorems entail that any consistent system of axioms strong enough to interpret first-order arithmetic is necessarily incomplete. These groundbreaking results eliminated the possibility of deciding all statements from a single set of strong enough axioms, such as the axioms of set theory. Among the undecidable statements in ZFC are the large cardinal hypotheses. G¨odelconjectured that if we assumed the existence of a large cardinal, in addition to the ZFC axioms, then we could prove new results that were previously undecidable in ZFC. G¨odelalso conjectured that this process could be repeated by adding even larger cardinals to improve the strength of the previously strengthened axiom system. Since there is no maximum cardinal, the process of adding larger cardinal hypotheses would eventually decide all propositions by resorting to strengthened enough systems of axioms. This was G¨odel’sprogram for set theory. The next great innovation in set theory came from Paul Cohen, in 1963. He proved the independence of AC and CH from ZF using the method of forcing, a mathematical technique that he created. Forcing is a method of adding a convenient object to a proposed model of set theory to force the resulting extended model to satisfy desired properties. This method, in its multiple forms, proved to be incredibly powerful. 1 It has been used to establish that there are propositions that are consistent relative to ZFC, or even independent of ZFC. It has also been used to measure the strength of certain propositions by comparing them with large cardinal hypotheses. Among the most common of these propositions are the axiom of choice, the continuum hypothesis, and the axiom of constructibility (V = L). A (partial) hierarchy of the size of large cardinals was established. As G¨odelconjectured, large cardinals hypotheses allowed to prove new results and to shed light on the other areas of set theory. The study of large cardinals is now inseparably interwoven with other areas of set theory. Despite its many successes, the known large cardinal hypotheses do not decide CH. Set theory is a branch of mathematics that is more isolated from the other branches than the norm. Its direct consequences to mathematics as a whole have been comparatively limited — for now. But the study of large cardinals, for example, may have an enormous impact through the whole of mathematics.