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 and descriptive set theory. 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.
In model theory we work in both pure and applied model theory, deepening the classification of first-order theories as well as uncovering connections between model theory and other areas of mathematics, particularly algebra. Another area of our interest is recursive model theory, which studies the models of first-order or infinitary theories which can be presented in a computable way.
Our research in recursion theory emphasizes its set-theoretic aspects, such as infinite-time Turing machine computation and the complexity of computations on sets.
Together with the work of the logicians of the Technical University of Vienna on proof theory and theoretical computer science, the principal themes in logic emanating from Gödel's fundamental work are well-represented in Vienna.
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.