SFB 878 A05 - Deskriptive Mengenlehre und Modelltheorie

Grunddaten zu diesem Projekt

Beschreibung

Descriptive set theory is one of the oldest areas of modern logic and derives directly from questions arising from the topology of the real line. It is concerned with classifying subsets of the reals with respect to the complexity needed to describe a set as a union or intersection of open and closed subsets, projections and complements. Lately, descriptive set theory has had influence on ergodic theory, model theory as well as other areas of set theory. In particular, ergodic group actions and the orbit equivalence relations arising from them have been studied using descriptive set theory. Similarly, the automorphism group of a countable homogeneous first-order structure naturally carries a polish topology, i.e. it is a completely metrizable separable topological group. This topology has been used for proving simplicity or the Bergmann property for these groups. The methods and results obtained on the reals in this way have been transfered to other ordered structures. By extracting the notion of ominimality from the reals, it has been possible to obtain far-reaching generalizations from known facts about semi-algebraic structures to a much wider setting. While traditionally it were mostly the Borel structures that have attracted attention, it will also be useful to extend this study to more complicated subsets of the reals, such as sets and equivalence relations which are ∞-Borel in models of determinacy. Much less is known about this wider class so that we plan to develop the abstract classification theory for this setting. The aim of this project is to apply and study different aspects of Borel and more generally ∞-Borel sets and equivalence relations naturally occuring in these different areas of logic and mathematics. We here focus on problems which need the interaction of set theory, model theory, group theory and geometry. We plan to focus on the following points which are closely connected to each other via the techniques from model theory, set theory, and descriptive set theory. Automorphism groups of first-order structures Classification of Borel and ∞-Borel equivalence relations Automorphism towers Asymptotic cones Tame ordered structures