Canonical models, cardinal invariants and idealized forcing

As one of the greatest achievements of axiomatic set theory, in the 1960s, Paul Cohen was able to show that one of the biggest open problems in mathematics, the so-called continuum hypothesis, is actually unsolvable. In other words, it is neither possible to prove that the continuum hypothesis is true nor that it is false, using the usual mathematical tools. The method that Cohen used was further extended in the years following its discovery and is known as "forcing". In the 60s and 70s, a veritable explosion of results occurred, all of which were able to show that certain previously open questions in mathematics were logically undecidable. Many different forcing techniques were developed to adapt the method to the problem at hand. Today, forcing is the central subject of set theory. The aim of our project is to develop a new forcing technique that can be applied to questions concerning inequalities of so-called "cardinal characteristics". Put simply, cardinal characteristics are quantities that measure how big a set (usually a set of real numbers) must be, for it to be large in some other sense. For example, one can ask what must be the minimal size of a set of numbers for a randomly selected number to fall into it with a non-zero probability. Here, "size" refers to the so-called cardinality of a (possibly infinite) set. The technique we want to develop in this project is based on a new approach that has the potential to overcome some of the drawbacks of the techniques that are currently in use, such as finite support iterations. It is expected that our technique can also be applied to other problems.