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: set theory, proof theory, recursion theory, and model theory.

The current research interests at the KGRC are mainly in set theory and model theory.

Research at the KGRC has traditionally covered most major branches of set theory, including large cardinal axioms, forcing, inner models, descriptive set theory, infinitary combinatorics, and set-theoretic topology. Large cardinal axioms and forcing together provide powerful tools for studying the consistency of mathematical statements not resolved by the traditional axioms of set theory. A central theme in descriptive set theory is the division between classifiability versus unclassifiability of classes of mathematical structures, analogous to Gödel's distinction between completeness and incompleteness of theories; this field traditionally has strong connections to analysis, ergodic theory, and topology. Set-theoretic topology has a long tradition at the University of Vienna tracing back to Karl Menger and is today greatly enriched by new forcing methods.

As the logical study of mathematical structures, model theory has found numerous applications in algebra, number theory, and analysis. This part of logic has seen a strong geometric orientation in the last few decades. Our main interests are in applications of model theoretic tools in other fields of mathematics. Among other things we are currently engaged in the development of asymptotic differential algebra, a subject on the interface between algebra, analysis, and logic, with roots in work of Hans Hahn in Vienna, and in the study of o-minimal structures, which provide a framework explaining and generalizing the finiteness phenomena typically encountered in real algebraic geometry.