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

• Final Report Year: 2017

Final Report Abstract

Descriptive set theory is the study of definable sets of real numbers and their structural properties. Its results show that simply definable sets of real numbers are well-behaved in the sense that they possess many regularity properties, such as Lebesgue measurability and the Baire property. Moreover, fundamental results of Martin, Steel and Woodin show that in the presence of large cardinals, these conclusions can be extended to all reasonably definable sets of reals. Since a great variety of mathematical objects can be identified with definable sets of reals, the concepts and results of descriptive set theory have fruitful applications in diverse fields of mathematics. As these applications are limited to the study of objects of continuum size, it is natural to ask whether these concepts can be adapted to study objects of higher cardinalities. Among set theorists, this question led to an interest to study the generalized Baire spaces of uncountable regular cardinals, consisting of all functions from the cardinal to itself, and the definable subsets of these spaces. This research was initiated by Mekler and Vāānānen and has developed in its own right, with an active research community and internally motivated open questions. Initial results showed that the rich combinatorial nature of uncountable cardinals causes their theory of definability to differ significantly from the countable setting and therefore many classical results cannot be transferred to higher cardinals. Furthermore, many fundamental statements about simply definable sets turn out to be independent of the standard axioms of set theory together with large cardinal assumptions. The goal of this project was the development of new concepts for studying definable sets of large cardinalities, and for measuring their complexity, as well as the construction of specific models of set theory that provide a strong structure theory for classes of such definable sets. On the one hand, we investigated more restricted classes of simply definable subsets of generalized Baire spaces that still contain many interesting mathematical objects, but also have the property that large cardinal assumptions provide a strong structure theory for them. The results obtained in the course of this project show that the collection of all subsets of the generalized Baire space of the first uncountable cardinal ω1 which are definable by a Σ1 -formula only using ω1 and real numbers as parameters is an example of such a class. This class contains many well-studied set-theoretic objects, like the non-stationary ideal on ω1, and our results show that large cardinal assumptions as well as forcing axioms answer many fundamental questions about the members of this class, which are left open by the standard axioms of set theory. For example, we show that these assumptions imply that this class does not contain a well-ordering of the reals. Additional results extend the above to formulas containing universally Baire sets of reals as parameters, and derive similar conclusions for the generalized Baire spaces of large cardinals. Moreover, we studied the influence of large cardinals on the global existence of very simply definable well-orders and other restricted classes whose definitions generalize concepts from computability theory. A second focal part of our project was the investigation of specific models of set theory that provide a strong structure theory for definable subsets of their generalized Baire spaces. Our results produce models in which all definable sets possess strong regularity properties that generalize the perfect set property, the determinacy of the Banach-Mazur game and the Hurewicz dichotomy to the uncountable context, and have several strong consequences, like the non-existence of definable well-orderings with these properties. We also consider extensions of the axioms of set theory by maximality principles and resurrection axioms, showing that, in contrast to large cardinal axioms, these principles provide a strong structure theory for the class of all Σ1-subsets of the generalized Baire spaces. Motivated by the aim to construct models with an even stronger structure theory for definable sets, we also contributed to the study of large cardinals and forcing, focusing on alternative large cardinal characterizations, on strong chain conditions and on their productivity. Finally, starting from the topic of generalized definability, we were led to investigate certain aspects of class forcing. In the course of the project, we were able to answer some of the most fundamental questions in this important research area, and to arouse strong international interest in this part of our project, culminating in a fruitful collaboration with the New York logic group.

Publications

• Class forcing, the forcing theorem and Boolean completions. The Journal of Symbolic Logic, 81(4):1500–1530, 2016
  Peter Holy, Regula Krapf, Philipp Lücke, Ana Njegomir, and Philipp Schlicht
  
• The Hurewicz dichotomy for generalized Baire spaces. Israel Journal of Mathematics, 216(2):973–1022, 2016
  Philipp Lücke, Luca Motto Ros, and Philipp Schlicht
  
• Ascending paths and forcings that specialize higher Aronszajn trees. Fundamenta Mathematicae, 239(1):51–84, 2017
  Philipp Lücke
  
• Σ1(κ)-definable subsets of H(κ+). The Journal of Symbolic Logic, 82(3):1106–1131, 2017
  Philipp Lücke, Ralf Schindler, and Philipp Schlicht