Descriptive Inner Model Theory, Large Cardinals, and Combinatorics

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The standard axioms of set theory, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), have been widely accepted as a foundation for mathematics; virtually all branches of mathematics that have been studied and applied to other scientific fields can be interpreted in ZFC. However, it turns out that there are natural and important mathematical theories that cannot be decided within ZFC alone. The Large Cardinal Axioms (LCAs) are extensions of ZFC designed to settle all such theories. Thus LCAs pursue Godel's program in set theory. How can one test for "correctness" of an LCA? The inner model program, a major program in modern set theory, justifies correctness by constructing canonical models for LCAs much like the natural numbers are the canonical model for the Peano Axioms of arithmetic (PA) (and as such PA is a correct theory). This research project contributes to the inner model program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The project focuses on studying the connections between inner models, sets of reals, hybrid structures (such as hereditarily ordinal definable sets (HOD) of determinacy models), forcing, and strong combinatorial principles (such as the Proper Forcing Axiom (PFA), (generalizations of) the tree property, the Unique Branch Hypothesis (UBH)). This research project aims to advance the basic theory of hybrid structures, as well as developing methods for the core model induction beyond its current state. In particular, the project aims to make advancements in answering two fundamental questions in descriptive inner model theory: (1) Is HOD of a determinacy model fine-structural (e.g. do the Generalized Continuum Hypothesis (GCH) and various square principles hold in HOD)? (2) What is the consistency strength of PFA?