Set Theory at the University of Munich
People
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:
- The theory of ZFC
- Large Cardinals
- Inner Models and Fine Structure
- Descriptive Set Theory
- Forcing
- 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).