Large Cardinals, Small Sets and Absoluteness

Miami University, Oxford OH

Investigators

Abstract

Techniques from mathematical logic can be used to measure the complexity of mathematical concepts. Areas of mathematics with physical applications tend to appear in the low levels of the corresponding complexity hierarchy. Passing to higher levels of complexity enables mathematicians to make connections between different areas of mathematics, and to develop productive general theories. There is a corresponding division of the universe of mathematics into an absolute part, where natural questions tend to be resolved by the standard axioms, and a more abstract part where extensions of the standard axiom system are needed to resolve many fundamental questions. The main focus of Larson's research is the relationship between these parts. The technical machinery involved in this project includes Cohen's forcing technique, axioms asserting the existence of winning strategies in infinite games, and axioms asserting the existence of infinite objects whose existence cannot be proved from the standard axioms for mathematics. These techniques originate in set theory, which serves as the most commonly accepted foundations for mathematics. Larson's work applies them to other areas, including model theory, topology and analysis. Larson hopes to make progress with this approach on several longstanding well-known open problems. One aspect of Larson's work concerns forcing over models of determinacy to produce canonical models. Woodin's Pmax forcing, when applied to a model of determinacy, produces a model which is maximal for subsets of the set of countable ordinals. One project, initiated by Woodin, and continued by Larson in collaboration with other researchers, is the extent to which this maximality can be made to hold for larger cardinals. Another product, largely in collaboration with Jindrich Zapletal, converts a number of classical ZFC constructions into partial orders, which when applied to determinacy models produce models of fragments of the Axiom of Choice. This approach has resolved a number of classical questions about the relationship between forms of the Axiom of Choice, but may also be able to shine light on newer problems in the theory of Borel equivalence relations. Other aspects of the project include a new approach to Vaught's Conjecture, one of the oldest problems in model theory, and the study of the notion of universally measurability, a fundamental concept from analysis which is still not well understood. Finally, Larson is working on a book on unpublished work of W.H. Woodin, on extensions of the Axiom of Determinacy. 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.