Pure Set Theory: Inner Models, Forcing and Absoluteness

Mathematical logic entered the modern era through the work of Kurt Gödel. This project is part of the ongoing revival of the Gödel tradition at the University of Vienna, where Gödel established his famous Completeness and Incompleteness theorems in the 1930`s. Our topic is set theory, the area of logic that most interested Gödel in his mature years. Set theory today exhibits two interconnected aspects, the pure and the applied. The former is concerned with the analysis of infinity, leading to a picture of the universe of sets as a whole, whereas the latter refers to the many successes of pure set theory either in solving mathematical problems or in showing that they are unsolvable using the traditional axioms of set theory. This project is concerned with pure set theory, and will explore the following topics: constructibility, iterated forcing, class forcing, inner model theory and absoluteness principles. In constructibility, we will discuss some new combinatorial principles that hold in Gödel`s model and further develop the hyperfine structure theory. In iterated forcing, we will consider iterations indexed by morasses and develop a generalization of properness that applies not only to set-forcing. In class forcing we will look for a valid version of Solovay`s dichotomy, a forcing which provably has a unique generic and further investigate generic saturation. In inner model theory, we will consider new extender model constructions, the new coding method in the presence of Woodin cardinals and connections with the theory of projective sets. And in the theory of absoluteness we will study the bounded Martin Maximum and develop an absoluteness principle appropriate for the theory of class forcing.

Mathematical logic became a profound area of research beginning with the ground-breaking work of Kurt Gödel at the University of Vienna in the 1930`s. One of the most active current areas of mathematical logic is set theory, which in addition to being a rich mathematical theory also provides a satisfying foundation for mathematics. Gödel`s Incompleteness Theorem has the striking consequence that set theory, and therefore mathematics as a whole, has multiple interpretations. The aim of this project, in pure set theory, was to analyse these different interpretations of set theory through the study of large cardinal axioms and their inner models, forcing methods and absoluteness principles. Some of the most important results arising out of this project were the following: 1. A gap 1 morass in L can be constructed using the Friedman-Koepke hyperfine structure theory. (Friedman - Koepke - Piwinger) 2. BPFA is consistent with a delta-1-3 wellordering of the reals. (Caicedo - Friedman) 3. It is consistent that the bounding number is greater than aleph-1 but still less than the groupwise density number. (Mildenberger - Shelah) 4. Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. (Dobrinen - Friedman) 5. Infinite sets whose power sets are Dedekind-finite can only carry aleph-0 categorical structures. (Walczak- Typke) 6. Assuming the existence of 0-sharp, there is an inner model where GCH fails at all regular cardinals. (Friedman - Ondrejovic) 7. There are class-generic reals which are not set-generic but preserve Woodin cardinals. (Friedman) 8. L-like principles can be forced, preserving very strong large cardinal properties. (Friedman)