Borel ideals and filters

The field of the proposed project is set theory and its general aim is to deepen our understanding of Borel ideals, their combinatorics, cardinal invariants, and their behaviour under forcing extensions. A Borel ideal is a definable notion of ``smallness`` on the natural numbers. The very first example of such a notion is the family of all finite sets, in other words, this notion declares a set of natural numbers ``small`` if it is finite. In general, an ideal is a family of sets of natural numbers (these sets are considered be ``small`` or ``negligible``) satisfying the following properties: (i) all finite sets belong to the family, (ii) it is closed for taking subsets of its elements, (iii) it is closed for taking unions of two of its elements, and of course (iv) the set of all natural numbers does not belong to the family. There are many examples of ideals motivated by classical results of combinatorics, analysis, measure theory, number theory etc, and most of them have nice, easily accessible definitions - this is what the qualifier Borel refers to. The study of these ideals has become an extremely active and widely discussed research topic in set theory in the past 20 years. Besides that ideals are interesting own their on, it turned out that they are in strong interaction with many classical areas of combinatorial and descriptive set theory. This project is devoted to answer some of the most important questions concerning these interactions. More precisely, the project has the following two main subtasks: (1) Understanding how Borel ideals behave under forcing extensions. Forcing is the most powerful method of proving relative consistency results, in other words, of proving that certain purely mathematical statements are independent of the axioms of set theory. Characterization of forcing (in)destructibility of ideals and its variants such as dominating ideals and certain covering properties are essential to work on cardinal invariants of these ideals. (2) Comparing cardinal invariants of Borel ideals to classical cardinal invariants of the real line. Roughly speaking, these infinite numbers associated to ideals can be seen as coefficients that code certain properties of these objects, akin to the area and perimeter of a convex region on the plane. Studying the still open (in)equalities between these cardinals and classical cardinal invariants provide us new perspectives on and characterisations of these cardinal characteristics.

The project was devoted to studying certain notions of "smallness", that is, ideals on the natural numbers, their combinatorics and cardinal invariants. For instance, we can consider a set A of natural numbers small if it is finite, or if A is of density zero (in limit), or if the sum of a sequence of positive reals (fixed in advance) over A is finite. The project's main achievements are the following: 1) Discovering new interactions between ideals and Banach spaces, one of the main structures studied in functional analysis (co-author: Piotr Borodulin-Nadzieja, University of Wroclaw): We pointed out numerous symmetries between the theories of nice ideals and of Banach spaces. Also, we investigated a special case, the interactions between combinatorics of families of finite sets, topological properties of the Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. We present (a) new examples of both Banach spaces and nice ideals, (b) a new characterization of the widely studied notion of precompactness, and (c) applications to enhance classical results of infinite combinatorics. 2) We generalized the co-author's and S. Shlelah's consistency results on the classical almost-disjointness number to its ideal version for two main classes of ideals. 3) New combinatorial characterization results regarding various notions of destroying ideals in classical forcing extensions (co-author: Lyubomyr Zdomskyy, University of Vienna): Among other results, we gave a simple combinatorial characterization when a real-forcing can destroy a Borel ideal in a strong sense, and when the two most canonical forcing notions associated to the ideal do so.