Applied Set Theory: Ideals, wellordering and combinatorics

Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems at the University of Vienna 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 applied set theory, and will explore the following topics: ideals, definable wellorderings, measures, L-combinatorics and partition cardinals. Concerning ideals, we will discuss the saturation of ideals on a regular cardinal and the stationarity of certain subsets of P. Definable wellorderings will be considered in the context of forcing axioms and absoluteness principles. We will relate the previous two topics to measures defined on sets of reals as well as to measures defined on subsets of a regular cardinal. Our study of L-combinatorics is concerned with Suslin trees, morasses and the solvability in L of combinatorial problems with respect to cardinal-preserving extensions. And we will explore the role of partition cardinals in uniqueness of generic classes and generic saturation.

Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems at the University of Vienna 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 applied set theory, and will explore the following topics: ideals, definable wellorderings, measures, L-combinatorics and partition cardinals. Concerning ideals, we will discuss the saturation of ideals on a regular cardinal and the stationarity of certain subsets of P. Definable wellorderings will be considered in the context of forcing axioms and absoluteness principles. We will relate the previous two topics to measures defined on sets of reals as well as to measures defined on subsets of a regular cardinal. Our study of L-combinatorics is concerned with Suslin trees, morasses and the solvability in L of combinatorial problems with respect to cardinal-preserving extensions. And we will explore the role of partition cardinals in uniqueness of generic classes and generic saturation.