Core Models and Combinatorial Set Theory

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

This is a mathematical research project on core models and their fine structure, which is a subfield of set theory that was pioneered by Jensen. Core models are generalizations of Goedel's constructible universe, L, that arose in the the study of large cardinal axioms. The field today builds on the fundamental discoveries of Dodd, Jensen, Martin, Mitchell, Steel and Woodin. The project is on pure core model theory as well as its interactions with other areas, such as descriptive set theory and infinitary combinatorics. Several of the research problems are on combinatorial principles such as Diamond and Square, which were isolated by Jensen in his work on L. They are among the most commonly used tools in set theoretic constructions and have applications in neighboring fields such as model theory and topology. Other research problems have their origins outside of core model theory. These include open questions about descriptive set theory, forcing, singular cardinal combinatorics and cardinal invariants. The axioms of set theory are the foundation of mathematics. Core model theory addresses some of the most fundamental questions in set theory in direct and highly sophisticated ways. Ideas are constantly exchanged between core model theory and other parts of set theory. The research problems to be addressed in this project are central in these respects. Some ask for further development of the basic tools of core model theory, while others are about applications to neighboring fields.