Combinatorial Set Theory of the Real Line

We propose to investigate combinatorial properties in the set theory of the real line, a subfield of mathematical logic. The proposed research has applications to open problems in topology, algebra and combinatorics of Aleph_1. Specific applications concern: the number of near-coherence classes of ultrafilters, the existence of subgroups of the Baer-Specker group that are bounded in one dimension but unbounded in a higher dimension, and the connections between "guessing principles`` and the existence of Souslin trees. In the proposed field, independence of ZFC is very likely. Therefore the main part of the proposed work is to develop forcing techniques. I also want to emphasize combinatorial methods in the analysis of existing notions of forcing with respect to new properties. In order to determine whether a given forcing extension has a property, the forcing notion might need to be further specified, because the forcing notion creates a whole class of extension models. This class is given axiomatically and hence Gödel`s incompleteness theorem applies and the question whether the forcing notion forces a statement can be undecided. The art is to combine forcing technology with the right tools from combinatorial set theory. In the proposed work, cardinal characteristics of the continuum describe important combinatorial features of the ZFC models studied. A cardinal characteristic of the continuum locates the smallest size of a set having a property that is typically not exhibited by any countable set, but is exhibited by at least one set of size of the cardinality of the continuum. Usually the value of a cardinal characteristic is not determined by ZFC.

The project belongs to mathematical logic, and therein to the subfield of set theory. We worked on combinatorial questions on the real numbers and on the structure of the subsets of the first uncountable cardinal and answered some of the questions mentioned in the proposal. In the area the commonly accepted axioms of mathematics, i.e., the axiom system ZFC given by Zermelo and Fraenkel, often is too weak to decide a question of the kind ``Does this and this hold?`` Rather, there are models of ZFC in which the answer is positive and other models of ZFC in which the answer is negative. In this case the work is to construct such models of ZFC. These are infinite structures with the binary relation ``is an element of`` that one can imagine as being similar to algebraic structures like fields or groups. Often new ZFC models are constructed by an extension technique or by some shrinking technique from given ZFC models. The most common extension method is forcing, a method of construction by approximations via custom-tailored partial orders. In the project work we developed new forcing orders and investigated combinatorial properties of forcing extensions. In particular we achieved some progress on Axiom A forcings, definable forcings, preservation properties and the use of combinatorics with almost disjoint sets in connection with long finite support iterations of forcings. We constructed new models of ZFC in which the subsets of the first uncountable cardinal can be predicted along an enumeration and in which nevertheless some cardinals describing the richness of the real numbers are large. We answered some open questions on cardinal invariants coming from ultrafilters and from the group of permutations on the natural numbers. We developed some preservation theorems for definable proper forcings in connection with trees on aleph-1.