Infinite combinatorics and large continuum

The main goal of this project is to improve our understanding of the properties of the real line. A large number of the combinatorial properties of the real line as well as its measure theoretic and categorical properties are described by the combinatorial cardinal characteristics and the cardinal characteristics corresponding to measure and category. The subject of cardinal characteristics of the real line can be traced back to the works of Georg Cantor and Rene Baire from more than a century ago and the early development of real analysis. Even though its long history, there are still many open questions in the field, that is questions regarding the properties of the real line which are expressible through cardinal characteristics. Many of those problems are directly related to the topological, combinatorial and algebraic properties of the real line. The purpose of this project is to study exactly such questions. Of particular interest is the study of models of set theory in which the real line has large cardinality, that is cardinality strictly larger than the first uncountable cardinal. On one side there are many open problems regarding the structure of the continuum whose solution will involve such models. Examples are the consistency of the splitting number being of countable cardinality, the existence of a model in which there are no P-points and no Q-points, the existence of a model in which the pseudo intersection number is strictly smaller than the tower number. On the other side there are very few techniques, in particular forcing and iteration techniques, leading to such models. In this project we suggest to look at specific problems involving models of set theory with large continuum, with the objective of not only finding solutions to these problems but also applying and further developing some of the most recent forcing techniques in this process. Among the problems which we consider are for example the definability of certain combinatorial objects on the real line in the presence of large continuum; a generalization of a forcing techniques used to show that the minimal size of a maximal almost disjoint family can be of countable cofinality, so that it can be applied in a much broader context; consistency results which were prior known to hold only under the additional assumption of the existence of a measurable cardinal. Among the existing techniques which appear relevant to the suggested problems are: 1. iteration along a template, a technique originating in the work of Saharon Shelah on the consistency of the dominating number being strictly smaller than the almost disjointness number; 2. ultrapowers of forcing notions; 3. matrix iterations of ccc posets, a technique which appeared originally in the work of Andreas Blass and Saharon Shelah on the dominating and ultrafilter numbers, and which was recently further developed in my joint project with Jörg Brendle where we study the almost disjointness, bounding and splitting numbers; 4. the construction of generic posets and applications of generalized mixed support iterations. One may expect that the results and the associated forcing techniques will find direct applications beyond the scope of the suggested problems and so further broaden and enrich our understanding not only of the structure of the real line, but also in a more general context, of the subject of set theory. The Kurt Gödel Research Center for Mathematical Logic at University of Vienna will provide an excellent environment to implement the suggested project. Of particular value will be collaborations with some of the many excellent researchers in the field, currently working there. Among them are Sy-David Friedman, Jakob Kellner, Hiroaki Minami, Miguel Anguel Mota. The research center is often visited by other leading scientists in mathematical logic, which creates an additional possibility for joint discussions and collaborations.

The cardinal characteristics of the real line are usually defined as the minimal cardinality of a set of reals, which is characterized by a certain property. These properties often correspond to the topological, measure theoretic and combinatorial properties of the reals. An important topic in the study of the combinatorial properties of the real line is the study of the regularity of the associated cardinal characteristics. Introducing a new method of generically adding a maximal cofinitary group of desired cardinality, that is a group of permutations of the natural numbers all non-trivial elements of which have only finitely many fixed point which is maximal with respect to this property, we show that consistently the minimal size of such group can be of countable cofinality. The new techniques which we introduce to obtain this consistency result have already been successfully applied to settle at least two other open problems regarding maximal cofinitary groups. Another important theme of the subject is the study of the combinatorial properties of the real line in the presence of projective wellorders of the reals and large continuum. As a major contribution to this area, it is shown that all known relations between the invariants of measure and category are in fact consistent with the existence of a projective wellorder on the reals. The work on this Lise-Meitner Project benefited from many national and international cooperation, which I fully expect will continue to produce interesting and important mathematical results. 1