Department Mathematik



Set Theory at the University of Munich


Research Interests

  • Inner Models and Fine Structure
  • Infinite Combinatorics
  • Equiconsistency Results
  • Large Cardinals

Set Theory was invented by Georg Cantor (1845 - 1918), and revolutionized mathematics. It's main theme is infinity. The research areas of modern set theory are:

  1. The theory of ZFC
  2. Large Cardinals
  3. Inner Models and Fine Structure
  4. Descriptive Set Theory
  5. Forcing
  6. Infinite Combinatorics

Set Theory provides an universal framework in which all of mathematics can be interpreted. There is no competing theory in that respect. A well-known formulation of the basic set theoretic principles is given by the axiomatic system ZFC of Ernst Zermelo and Abraham Fraenkel, formalized in first order logic (the C denotes the axiom of choice).
ZFC, however, does not decide the size of the reals: Cantor's Continuum Hypothesis (CH) is independent of ZFC (Goedel 1938, Cohen 1963). This result marks the beginning of modern Set Theory.
Not only (CH) but many other propositions were shown to be independent. The methods used to prove independence are inner models - natural realizations of the axioms, the most prominent being Goedel's constructible universe L - and forcing - Cohen's method to enlarge given models of set theory keeping control of the new sets.
It should be mentioned here, that on the other hand ZFC is rich enough to prove deep theorems in cardinal arithmetic. This was shown by Shelah's pcf-theory in the 90s.
Questions of cardinal arithmetic ever drove much of the evolution of the field, and (CH) may still be looked upon as an open problem. Hugh Woodin has recently found a canonical model in which (CH) is false.
One might hope to solve (CH) by adding natural axioms to ZFC. And there is indeed a canonical extension of ZFC by so-called large cardinal axioms, which stipulate that the universe of set theory is very rich. The interest in large cardinal axioms is substantiated by the empirical fact that these principles form a well-ordered hierarchy calibrating independence: for every natural proposition B, there is a large cardinal axiom A equiconsistent with B (relative to ZFC), i.e. ZFC + A is consistent iff ZFC + B is consistent. Many natural principles B come from infinitary combinatorics. The two methods used to prove equiconsistency are extensions of the methods used by Goedel and Cohen: core model theory and iterated forcing. Core model theory tries to realize large cardinal axioms in canonical, fine-structured inner models, and "proves" in this way the consistency of these axioms. The current state of the art are core models up to the level of Woodin cardinals. (Fine Structure is due to Ronald Jensen, and was first developed to analyse Goedel's L. Core model theory is due to Tony Dodd, Ronald Jensen, Bill Mitchell and John Steel. Many developments of iterated forcing are due to Saharon Shelah.)
Large cardinal axioms do not settle (CH), but they do settle many questions in descriptive set theory that ZFC cannot answer. The best result here is that infinitely-many Woodin cardinals imply the axiom (PD) of Projective Determinacy This was shown by Donald Martin and John Steel in the 80s. A consequence concerning (CH) is that all definable subsets of the reals are countable or of the size of the continuum in the presence of sufficiently strong large cardinal axioms.
(A list of the ZFC axioms and examples of combinatorial principles and large cardinal axioms are at Oliver Deiser's homepage).