Combinatorics of Filters, Large Cardinal and Prikry-Type Forcing

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Our understanding of the mathematical universe, or more precisely, the behavior of the infinite, is quite limited. The main reason for this limitation is the incapability of the Zermelo-Fraenkel axiomatic system of set theory (ZFC), which is the global standard for the formal foundations of mathematics, to determine basic questions about the infinite. This channeled the research in modern set theory to mainly two directions. The first direction is the search for new restraints imposed by ZFC on the behavior of the infinite. This direction had great success in the realm of singular cardinal arithmetic where new surprising principles were discovered by Silver and later by Shelah. This project aims to study some of these principles. The second direction is the subtle interaction between extensions of ZFC to stronger axiomatic systems and statements which are unsettled by ZFC. Perhaps the most prominent axiomatic systems are extensions of ZFC by the so-called large cardinal axioms. This project contributes to the development of new constructions with large cardinals, implementing them to analyze the interaction between those large cardinal axioms and unsettled statements. The PI will lean of his extensive experiences of mentoring underprivileged students in his teaching activities. This project deals with several central areas in set theory: (i) forcing theory and more particularly Prikry-type forcing; (ii) cardinal arithmetic; and (iii) infinitary combinatorics. Forcing with a Prikry-type forcing notion is perhaps the most important technique to generate models with non-trivial patterns of singular cardinal arithmetic. This project contributes to the investigation and sophistication of these techniques through several aspects: the development of combinatorics of ultrafilters, the discovery of new connections of this theory with other areas of set theory such as inner model theory and infinite combinatorics, the characterization of intermediate models of several Prikry-type models such as the Magidor-Radin model and the tree Prikry forcing, and obtaining stationary reflection at the successor of the first singulars of uncountable cofinalities. One particularly interesting combinatorial property of ultrafilters which is investigated in this project is the Galvin property which recently gained renewed interest due to the surprising connections of this property to the structure of ultrafilters in canonical inner models, the Tukey order, Prikry forcing, partition relations and more. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.