CAREER: Current and Future Developments of the Core Model Induction

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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 Gödel'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). This project contributes to the Inner Model Program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The educational component includes annual conferences, curriculum development, and outreach and broadening participation activities. The project fits into the general framework of 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 problem of building a canonical inner model for an LCA is referred to as the Inner Model Problem for that LCA. One of the project's main goals is to identify certain obstructions that may prevent the current approaches to resolving the Inner Model Problem for various LCAs from going further as well as proposing ways of overcoming them. Specifically, the project will study obstacles to the current techniques of the Core Model Induction (one such obstacle is the Sealing of the Universally Baire Sets) and develop new approaches to the Core Model Induction that overcome these obstacles. One main application of the above analyses is in constructing canonical inner models of large cardinals from various strong extensions of ZFC, like the Proper Forcing Axiom.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.