Philipp Lücke

"Frameworks for large cardinals"

Large cardinal axioms play a central role in modern set theory, not only because they are strong candidates for new axioms of mathematics, but also because they allow us to analyse other candidates for such principles through equiconsistency results. Despite their prominence, these principles are still surrounded by many open fundamental questions. In particular, even though set theorists have an intuitive understanding of these axioms, there is no widely accepted definition of what a large cardinal precisely is, and, without such a definition, it seems impossible to develop a general theory of large cardinals that enables proofs of their observed properties.
In my talk, I will present work addressing these issues through the development of uniform frameworks for large cardinal axioms that both incorporate various central notions from all parts of the large cardinal hierarchy and provide concrete tools for investigating the relationships between the notions involved. I will then show how one of these frameworks, based on model-theoretic reflection, enables the formulation of new set-theoretic principles that turn out to be deeply connected to central open problems in contemporary set theory.