Master's Dissertation
DOI
10.11606/D.45.2018.tde-27062018-130056
Document
Author
Full name
Michel Fernandes Gaspar
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Fajardo, Rogerio Augusto dos Santos (President)
Aurichi, Leandro Fiorini
Boero, Ana Carolina
Title in Portuguese
Forcing e regularidade na reta real
Keywords in Portuguese
Forcing
Hierarquia projetiva
Abstract in Portuguese
Title in English
Forcing and regularity in the real line
Keywords in English
Forcing
Idealized forcing
Projective hierarchy
Regularity in the real line
Abstract in English
The study of the regularity properties in the real line is as old as the beginning of set theory at the end of the 19th century. These properties indicate well behavior for subsets of the real line, being the Lebesgue measurability, Baire measurability and perfect set properties the most prominent examples. In this work other regularity properties are explored, such as the Ramsey property, the doughnut property, the Marczewski measurability, the Miller measurability, the Laver measurability, among others. The relationship between regularity properties and forcing is known since the 70's with the work of Robert Solovay, who, for example, constructed a model of set theory in which every subset of the real line is Lebesgue measurable, Baire measurable, and has the perfect set property. All of theses regularity properties are captured by a general definition making use of the powerful technique of \textit{idealized forcing}, introduced by Jindrich Zapletal in 2008. The main systematic study of regularity properties via idealized forcing was done by Yurii Khomskii in 2012 in his Ph.D dissertation. The result of Solovay mentioned above is proved in this general framework. Characterization results for regularity properties of the sets in the second level of the projective hierarchy via forcing over L are also explored. Some historical notes are provided for most of the addressed subjects.