Forcing Methods: Creatures, Products and Iterations

The focus of this project is research in set theory, a subfield of mathematical logic. In set theory we consider cardinalities, which measure the size of infinite sets. We are in particular interested in the set of all real numbers, and in subsets of this set. Many questions about properties of these sets (e.g. in connection with the Lebesgue measure) cannot be answered with the usual set-theoretic axioms. The method of forcing allows us to construct set-theoretic universes in which these answers can be depending on what you want yes or no. In this project we try to further develop the method of forcing. On the one hand, to construct new set- theoretic universes, and on the other hand, in order to better understand the basic properties of the set of real numbers and its subsets.

In this project we have investigated several infinities (or or "infinite cardinal numbers"), which appear as cardinalities of sets of real numbers. Set Theory, a subfield of Mathematical Logic deals with the investigation of infinite sets (whereas in everyday life we mainly deal with finite objects); examples for such sets are the set of all real numbers, or its subsets such as the set of natural numbers, or the set of numbers between 0 and 1. Not only finite sets but also infinite sets can be compared in terms of their "size" (we use the technical term "cardinality", to point out that this is different from the notion of size of a finite set), and it has long been known that there are infinitely many cardinalities of infinite sets; for example the set of natural numbers has the same cardinality as the set of all rational numbers, but not the same cardinality as the set of all real numbers. What exactly the difference between these two cardinalities is has long been a central question in set theory; Georg Cantor's Continuum Hypothesis, the first of Hilbert's 23 problems), was the conjecture that no cardinality lies strictly between those two. In the project we have investigated cardinalities of special (often pathological) sets which appear in other areas of mathematics. For example: What is the least cardinality of a subset of 3-dimensional space to which one cannot assign a "volume" in a meaningful way. (Such sets appear in the Banach-Tarski paradox.) When we consider, for example, subsets of 3-dimensional space, we distinguish "null sets" (i.e., sets to which we assign the volume 0), for example finite subsets or 2-dimensional subsets such as planes) and "positive" sets (=all other sets). We are interested, for example, in the smallest cardinality of a positive set, in the number of null sets needed whose join will the whole space. The two cardinalities just described, together with 8 more, are collected in Cichon's diagram, which also describes the relations between them. They all are uncountable (that is, larger than the cardinality of the set of all natural numbers). But one can construct a set theoretic universe in which all these cardinalities are equal, and further universes where some of them have distinct values. By combining known methods with new ideas, we could (for the first time) describe a set theoretic universe in which all these cardinalities have different values. Our new method has also found other applications. There is a long list of further cardinalities, whose relation to the cardinalities considered so far is still open.