Complexity and Definability at Higher Cardinals - Studies in Generalized Descriptive Set Theory

• Applicant: Privatdozent Dr. Philipp Lücke
• Subject Area: Mathematics
• Term: from 2015 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 274718383

Descriptive set theory is the study of definable or topologically simple sets of real numbers and their structural properties. It originates in the work of the French analysts Borel, Baire and Lebesgue, emerged as a distinct discipline with the fundamental work of Alexandrow, Hausdorff, Souslin and others in the 1910's and forms a central part of contemporary set theory. Descriptive set theory shows that simply definable sets of real numbers are well-behaved in the sense that they have many regularity properties, such as Lebesgue measurability and the Baire property. Since many objects of set-theoretic interest, like well-orders of the reals and ultrafilters on the natural numbers, lack such properties, it follows that they cannot have simple definitions. Moreover, the fundamental results of Martin, Steel and Woodin show that in the presence of large cardinals, all definable sets of reals have all these regularity properties and hence well-orders and ultrafilters are undefinable in this case. It is natural to ask whether these concepts can be adapted to objects of higher cardinalities. This research was initiated by Mekler and Väänänen and continued by S. Friedman, Hyttinen, Shelah and others. While small parts of the classical theory carry over to higher cardinalities, the rich combinatorial nature of uncountable cardinals causes their theory of definability to differ significantly. In particular, it is possible that the higher analogues of well-orders and ultrafilters are definable in very simple ways. Moreover, this is possible also in the presence of (very) large cardinals. The proposed project studies the nature of definability at higher cardinals. In one direction, we want to refine hierarchies of complexity to obtain classes with provable regularity properties. In the other direction, we want to consider extensions of the axioms of set theory by various axioms and investigate the structure theory of classes of sets with simple definitions under these additional hypotheses. This analysis is planned to make use of methods and techniques from several branches of set theory like uncountable combinatorics, forcing, large cardinals, inner models and classical descriptive set theory. The applicants have worked and published extensively on topics related to this project. We plan to attract a postdoctoral researcher (Peter Holy) with proven expertise in the above branches of set theory, whose previous research is strongly connected with this project. This will allow us to incorporate important new aspects and methods into the project. Furthermore, since the fall of 2014, a Ph.D. student (Ana Njegomir) in the logic group in Bonn is working on topics related to this project. In 2016, we plan to organize an international workshop on generalized descriptive set theory in Bonn.

• DFG Programme: Research Grants
• Co-Investigators: Professor Dr. Peter Koepke; Dr. Philipp Schlicht