Inner Models, Combinatorics, and Determinacy

The Large Cardinal Axioms (LCAs) are extensions of the standard axioms of set theory, Zermelo-Fraenkel set theory with the axiom of choice (ZFC). LCAs are designed to settle all natural theories that are independent of ZFC. This is Gödel'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). The canonicity of the models justifies the correctness of the LCAs much like the canonicity of the natural numbers justifies the correctness of PA. This research project contributes to the inner model program by advancing the state of the current knowledge regarding canonical models for LCAs and their relationship with other foundational frameworks of set theory. The project provides research opportunities for graduate students.

This project builds on and expands the PI’s previous work on computing consistency lower bounds for the Proper Forcing Axiom (PFA), on studying canonical models of AD^+ up to the minimal model of the Largest Suslin Axiom (LSA), and on the Sealing phenomenon concerning universally Baire sets. These are important theories in our set-theoretic landscape. The PI plans to explore further various aspects of Sealing (particularly a weak form of Tower Sealing and its implications) and develop further techniques of the core model induction with the eye towards determining the consistency strength of various fragments of Martin’s Maximum. The PI plans to study the theory of short-tree strategy mice for the least-branch hierarchy; this not only completes the general structure theory for least-branch hod mice but also will have potential applications, particularly in core model induction contexts. The PI also plans to continue the ongoing joint work with W. Chan and S. Jackson on the general analysis of combinatorial structures of determinacy; this allows the PI and his co-authors to understand deeper how sets are related to one another in this context. This, in particular, leads to the discovery and resolution of the ABCD Conjecture, solutions to several classical descriptive set theoretic conjectures, and the study of general cardinalities and cofinalities of sets in choiceless contexts.

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.

Award Number: 2449780
Principal Investigator: Nam Trang
Funds Obligated: $100,000
State: TX