CCC creatures and cardinal characteristics

Mathematics uses many notions of ``smallness for sets of reals numbers. Important examples are Lebesgue null or meager. It is easy to see that the countable union of Lebesgue null sets is again Lebesgue null; and the same holds (by definition) for meager. On the other hand it is clear that there is a family of size continuum of Lebesgue null (and meager) sets whose union is the whole set of real numbers: For example, the family of sets containing exactly one real number. The Continuum Hypothesis (CH) states that every infinite set of real numbers is either countable or has size continuum. CH is not provable or refutable in the usual axioms of mathematics (ZFC). If we assume that CH fails, then the following question is natural: What is the minimal size of a family of Lebesgue null set whose union is not Lebesgue null any more? This size is called the additivity of the null ideal and denoted by add(N). It is an example of a cardinal characteristic. Others include non(N), the minimal size of a set that is not null; cov(N), the minimal size of a family of null sets whose union is the real line; and cof(N), the minimal size of a family of null sets such that every null set is subset of a set in the family. The same definitions work for the meager ideal M instead of Lebesgue null, leading to a total of eight cardinal characteristics, often summed up as part of the so-called Cichon diagram. Between some of these characteristics, there are provable inequalities. For example, add(N) is less or equal to add(M) (and consistently can be strictly less). For each pair (x,y) of entries in Cichons diagram it has either been either proved that x is less or equal than y, or that x can consistently be greater than y. For a sequence of more than two entries this question becomes harder. In the proposed project, we will attempt to construct a model where all entries of the diagram are pairwise different.

During this project we could show that consistently Cichon's Maximum holds, i.e., all ten independent entries in Cichon's diagram could be pairwise different. In mathematics one can precisely define the notion of infinity, and define for two infinite sets what it means that one is bigger than the other. It turns out the the set of natural numbers has the smallest possible infinite size (the same on as the set of rationals), and that the set of reals is bigger. The Continuum Hypothesis states that there are no infinities between the natural numbers and the reals. Gödel and Cohen have shown that the Continuum Hypothesis is neither provable nor refutable. Cichon's diagram contains 12 important definitions of infinite sizes (only 10 of them independent). All these sizes are bigger than the natural numbers, and at most as big as the reals. So under the Continuum Hypothesis all entries are the same. It has been known for quite some time that each two entries of the diagram can consistently be different. The new result shows that it is even possible that all entries are pairwise different. The result was published in the Annals of Mathematics and was reported in various newspaper articles and in popular science magazines such as Scientific American.