Large cardinals, set-theoretic definability and singular cardinal combinatorics

• Applicant: Privatdozent Dr. Philipp Lücke
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567184449

Singular cardinals, set-theoretic definability and large cardinals are central themes of modern set theory, with crucial open questions that determine the direction of research in this discipline positioned at the intersection of these topics. A primary example of such a question is Hugh Woodin’s HOD Conjecture, which asks whether strong axioms of infinity imply that the set-theoretic universe is closely approximated by definable sets. This conjecture is one of the most prominent open problems in set theory due to its far-reaching implications for central endeavors in this field. In his ongoing Heisenberg project, the applicant explores new approaches to problems located at the intersection of the above three topics. The results achieved in this project so far demonstrate the strong potential of these approaches, as key milestones of the Heisenberg project have already been reached. Most notably, a collaboration with researchers from Vienna and Barcelona has identified the first examples of natural large cardinal axioms that cause the mathematical universe to be so complex that non-ordinal-definable sets exist and these newly discovered “large cardinals beyond HOD” provide a clear path to refute the HOD Conjecture through strong axioms of infinity. Additionally, a collaboration initiated a program to study the Ramsey-theoretic properties of singular cardinals through the structural properties of simply definable sets and established a new approach to tackle the notoriously open question of whether the first singular cardinal can be a Jónsson cardinal. The present project now seeks to substantially broaden the scope of these ongoing research activities and significantly increase their ambitions. In one direction, it will explore the newly uncovered landscape of large cardinals beyond HOD, focusing on their interactions with other strong large cardinal notions and the effects of their combinatorics on set-theoretic definability. These plans aim to gain new fundamental insights into the nature of the large cardinal hierarchy and ultimately refute the HOD Conjecture. In another direction, this project will expand on a new approach to analyze very strong large cardinal axioms through generalizations of concepts from the study of definable sets of real numbers in descriptive set theory. Finally, the project will advance the investigation of simply definable sets at uncountable cardinals with strong combinatorial properties, aiming to further restrict the Ramsey-theoretic properties that the first singular cardinal can possess.

• DFG Programme: Research Grants