# Research

The era of modern Mathematical Logic began in the 1930s with the work of Kurt Gödel. His results on the completeness of first-order logic, the incompleteness of formal systems extending basic arithmetic and on constructibility served to provide the foundation for its four principal subfields: model theory, proof theory, recursion theory and set theory.

Research at the KGRC focuses most strongly on set theory, where we work on large cardinals, forcing, descriptive set theory and set-theoretic topology. Large cardinal axioms and forcing together provide powerful tools for studying the consistency of set-theoretic properties which are not resolved by the traditional axioms of set theory. A central theme in descriptive set theory is the study of classifiability versus unclassifiability of natural classes of mathematical structures, a distinction analogous to Gödel's distinction between completeness and incompleteness of theories. Set-theoretic topology has a long tradition at the University of Vienna tracing back to Menger and is today greatly enriched by new forcing methods.

In model theory we work primarily in pure model theory, deepening the classification of first-order theories. Another of central interest is recursive model theory, which studies the models of first-order or infinitary theories which can be presented in a computable way.

In addition to recursive model theory, our research in recursion theory emphasizes both its set-theoretic aspects, such as infinite-time Turing machine computation and the complexity of computations on sets, as well as the study of computational complexity through the application of the methods of proof theory.

Our scientific activities are documented on this website in the following:

- Publications published at the KGRC.
- Research grants of KGRC members.
- Conferences (co)organized by the KGRC or by KGRC members.
- The frequent visitors to the KGRC, among them many leading researchers.
- The KGRC research seminar.
- The excellent evaluations we received.
- Some academic career milestones of (former) KGRC members.
- The international orientation and appeal of the KGRC. For example, in the period 2005-2010 there were 48 researchers working at the KGRC, 7 of them Austrians, the rest from Australia, Bosnia, Canada, Colombia, the Czech Republic, Finland, Germany, India, Iran, Italy, the Netherlands, Poland, Spain, the UK and the USA.