Filters, Ultrafilters and Connections with Forcing

The project focuses on research in Set Theory & Foundations of Mathematics, in particular on Combinatorial Set Theory and Forcing. It proposes to investigate new techniques for constructing ultrafilters, which are complex combinatorial objects appearing in many different areas of mathematics. Roughly speaking an ultrafilter is a comprehensive measure of largeness. Finding these objects with additional special properties often requires complicated techniques and involved arguments; there are many important unresolved questions and the understanding of these objects is far from satisfactory. In the project we will try to answer some of these questions applying methods originating from Forcing Theory, Topology and Combinatorics. It is known that current methods cannot provide full answers so we will try novel approaches. We will try to find new so-called preservation theorems for forcing extensions and new "diamond like" constructions.

Short version: As is typical for set theory, we discovered "gaps" in the mathematics of infinite sets, or more precisely proved that certain mathematical statements cannot be neither proved nor refuted. In more detail: Set Theory, a subfield of Mathematical Logic, deals with the investigation of infinite sets. Filters (and in particular a special kind of filters, the "ultrafilters") are useful tools in these investigations and at the same time object to be investigated. For finite sets S there are several variants of the pair "many/few elements of S have a certain property". such the minimal variant "majority/minority" = more than / less than 50 percent, the stronger "more than 2/3 vs less than 1/3" or the extreme variant "all/none". For example, taking S as the set of all members of a committee, and the property as "agree with Opinion X" or "vote for candidate Y". This example leads also to other variants of "majority", e.g. by assigning different weights (or voting power) to different elements of the set, or accepting only unanimous decisions. In set theory we consider such notions of "majority/minority", "many/few" or "large/small"also for infinite sets, and it turns out that there are such notions with the additional property that the union of two small sets (or the union of two minorities) is again small, which is not the case in the finite scenario. The mathematical notion of "filter" formalizes this notion. In the current project we investigate properties of such "filters". Kurt Gödel's famous incompleteness theorem tells us that the usual axioms of mathematics or of set theory are not sufficient to answer all questions in mathematics. It turns out that there are very simple questions about filters (such as "do p-points exist?" or a more detailed version "does the existence of p-points follow from the statement that there are many different cardinalities of infinite sets of real numbers?") which cannot be decided by the axioms, that is: one can construct one mathematical universe where the answer is "yes", and another one where the answer is "no".