Descriptive Inner Model Theory and Its Applications

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The Zermelo-Fraenkel axioms plus 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 which cannot be decided by ZFC alone. The Large Cardinal Axioms (LCAs) are extensions of ZFC designed to settle all such theories. This is the Godel's program in Set Theory. If an LCA is "correct", then the theories it decides are also correct. 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). The proposed 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)). The PI proposes to work on advancing the basic theory of hybrid structures, focusing on strategy mice and hod mice as well as developing methods for the core model induction beyond its current state. In particular, the project aims to make advancements in answering the following fundamental questions in descriptive inner model theory: (1) Is HOD of a determinacy model fine-structural (e.g. do the Generalized Continuum Hypothesis (GCH), various square principles hold in HOD)? (2) What is the consistency strength of PFA? (3) Does PFA (or any other strong combinatorial theory) imply models in various partially backgrounded constructions iterable?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.