Boolean ultrapowers and other new forcing techniques

Every not too chaotic subset of the plane can be assigned an area, the Lebesgue measure. This can be a finite real number, or infinite. Sets with measure 0 are called nullsets. This is a useful notion of vanishingly small: Nullsets can be safely ignored in many mathematical arguments and applications. The number of elements of the plane is called continuum. Already Cantor has shown that this cardinality (which is the same as the size of the real numbers) is bigger than the size of the natural numbers (which in turn is the same as the size of rational numbers). Sets that are as big as the natural numbers, or finite, are called countable. Countable subsets of the plane are always null (and in particular measurable). There are measurable, continuum-sized subsets of the plane with infinite measure (e.g., the plane itself), with positive finite measure (e.g., a square with side length 1), or measure zero (e.g., a line). But there are also non- measurable continuum-sized sets. A natural question is: What is the minimal size of a non-null set? This size is called non(Null), and as we have seen it is bigger than countable and at most continuum. non(Null) is a so-called cardinal characteristic of the continuum. Another on, add(Null), is: What is the smallest size of a family of nullsets whose union is non-null? A variant, cov(Null) asks the same for the union being the whole plane. It is easy to see that both non(Null) and cov(Null) are at least as big as add(Null). These and some additional characteristics, and some inequalities between them, make up the so-called Cichon diagram. This diagram has 10 independent entries. Set theory, more specifically: the axiom system ZFC, provides a foundation of the whole of mathematics: Every mathematical sentence that is considered to be proven can be formally proven in ZFC; and every formal ZFC proof is accepted by the mathematical community. But already Gödel has shown that no reasonable, sufficiently strong axiom system can truly be complete: There are always sentences that are neither provable not refutable (incompleteness theorem). The most famous example of such a sentence for ZFC is the Continuum Hypothesis (CH): Every subset of the real numbers is countable or has size continuum. If one assumes CH, then all entries in Cichons diagram are equal. On the other hand it has been known since the 1980s that any pair of entries can be different. 2019 it was shown that actually all ten independent entries can be simultaneously different. As usual in mathematics, the result opened up new questions and problems; and the project will develop new methods to tackle some of these questions.