Set theory is focused on the mathematical study of infinity. Fundamental questions are considered and analysed, such as "How many real numbers exist?", or "Are all simply definable (for example, projective) sets of real numbers Lebesgue measurable?". The standard axioms of set theory (ZFC, the Zermelo-Fraenkel axioms with the Axiom of Choice) give us only partial information here. There are at least ℵ1 many real numbers (more than ℵ0, the least infinity), and all continuous images of Borel sets are Lebesgue measurable. But they leave many questions unanswered. Set theorists have developed various natural extensions of ZFC, which resolve some of these questions.
Our work at Münster is focused on various topics in set theory, notably large cardinals, inner models, and forcing, and some combinatorics and descriptive set theory.
Large cardinal axioms posit the existence of strongly transcendental kinds of infinite sets (for example, inaccessible cardinals cannot be reached from below by standard set operations, such as power sets). These axioms do not resolve the first question above, but mid-strength instances do resolve the second one, positively.
Inner model theory, a sub-field of set theory, is focused on the construction and study of canonical models of set theory with large cardinals. (They are sub universes of the universe of all sets.) This provides some of our best understanding of large cardinal axioms. Using inner models, we can also often determine bounds on the consistency strength of combinatorial statements in terms of large cardinal hypotheses. Such techniques were used by Gödel in his proof of the relative consistency of the Generalized Continuum Hypothesis.
Forcing is a method of extending a universe of sets, analogous to forming a field extension of a field, for example. It is also a fundamental technique for consistency proofs in set theory. Forcing axioms assert that the universe is rich, in that it looks roughly like there has already been a lot of forcing done. Standard forcing axioms do actually resolve the first question above, but in a surprising manner: they imply that there are exactly ℵ2 many real numbers (contradicting the Continuum Hypothesis).
Descriptive set theory seeks to understand subsets of the reals, through an analysis of their (relative) complexity. Determinacy Axioms assert that certain two-player games are determined, in that one of the players has a winning strategy. These axioms lead to a rich theory of the sets of reals - but one which contradicts the Axiom of Choice.
These seemingly disparate axiom systems and principles, and many others, are highly connected and interrelated, and set theory is focused on their understanding.