Set Theory: Forcing, projective sets and morasses

Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems in Vienna in the 1930`s. This project, to be based at my institute (the Institute of Discrete Mathematics and Geometry of the Technical University of Vienna) in cooperation with Sy Friedman`s group at the Kurt Gödel Research Center for Mathematical Logic of the University of Vienna, focuses on set theory, the area of logic that most interested Gödel in his later years. The proposed topics are: oracle-cc forcing, projective regularity, morasses and Axiom A forcing.

Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems in Vienna in the 1930`s. This project, based at my institute (the Institute of Discrete Mathematics and Geometry of the Technical University of Vienna) in cooperation with Sy Friedman`s group at the Kurt Gödel Research Center for Mathematical Logic of the University of Vienna, focused on set theory, the area of logic that most interested Gödel in his later years. We were specifically interested in analysing subsets of the real line. Set theory started with Cantor`s discovery that there there are many different kinds of infinity; the smallest possible infinity is the infinity of countable sets, and Cantor showed that the set of real numbers is larger than that. The real line (the set of all real numbers, rational and irrational) has long been an important tool in mathematics, but also a subject of foundational investigations. Only with the advent of set theory did it turn out that several basic properties of the real line (such as: are all definable set measurable?) cannot be decided on the basis of the usual axioms of mathematics. One main result of our project concerned the notions of strong measure zero sets and strongly meager sets; both notions pick a special family of very small subsets of the real line. It was an open question if all those small sets could be countable. In a joint paper with Kellner, Shelah and Wohofsky we constructed a set-theoretic universe in which all these sets are indeed countable.