Set theory of the real line and its subsets

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 PI'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.


The PI plans to work on a related collection of projects involving the complexity of sets of real numbers. Two of these projects aim to extend the work published in two of his recent books. The first of these is another book exposing part of W. Hugh Woodin's work on his axiom AD^+. The first part of this book is to be on methods for producing maximal models of AD^+ via strengthenings of Woodin's derived model methods. The second half of the book concerns reversals, methods for showing that a given model of AD^+ arises as a derived model. This approach resolves many basic questions about models of AD^+. The second project involves applications of the methods of Geometric Set Theory, which is the topic of the PI's book with Jindrich Zapletal. There are many open questions about the relationships between fragments of the Axiom of Choice which appear to be amenable to the techniques developed in this book. The PI plans to use this approach to study discontinuous homomorphisms of the additive group of the real line, probability measures on the integers, and Blackwell determinacy. Another project involves the universally measurable sets of real numbers, those which are measured by every Borel measure on the real line. Despite their fundamental nature, many basic questions about these sets remain open, among them the question of whether it is consistent for every universally measurable set to have the property of Baire. Another project involves using techniques from inner model theory to study canonical inner models derived from strengthenings of first-order logic, such as logics allowing quantifiers for cofinality or stationarity. These models are typically analyzed by understanding which mice they contain, and which mice construct then. Finally, the PI plans to work with Saharon Shelah on an approach to 0-1 laws. Results in this area show that for many natural properties of finite structures, the probability that a finite structure has the property in question approaches either 0 or 1 as the size of the structure approaches infinity. Results in this project show that under natural assumptions 0-1 laws are preserved while adding additional random predicates to the structures in question.

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: 2452139
Principal Investigator: Paul Larson
Funds Obligated: $80,000
State: OH