PROJECT PROPOSAL

My research project regards the interconnection between two burgeoning areas of mathematical logic: generalized descriptive set theory and set theory of the κ-reals. 1 State of the art and preliminary work State of the art. Set theory is a branch of mathematical logic, which has been extensively developed during the 20th century as a foundation of mathematics. Two important subbranches are the set theory of the real numbers and the descriptive set theory, which concern a carefully investi-gation of definable subsets of the real line. In particular, one of the main purposes was to analyze how much regularity one could obtain. Obviously, the notion of regularity is not uniquely determined; the most common ones are certainly the Lebesgue measurability and the Baire property, which are not only of interest from the set-theoretical viewpoint, and indeed they came from measure theory and topology, respectively. A very old result coming from the beginning of 20th century, due to Vitali, showed that the axio