Forcing, inner models, and large cardinals.

This project seeks to contribute to our understanding of structural properties of the universe of mathematics. Our knowledge of mathematics comes through deduction from axioms. This knowledge is inherently incomplete, in the sense that there is a wide range of questions that can never be answered, either positively or negatively, from the standard axioms. Set theorists study these questions, to see how they relate to each other, and how they relate to a backbone of additional axioms, called large cardinal axioms, that assert strong reflection properties for very large sets. This project explores some of these questions through three interrelated aspects. The first involves a property of cardinal numbers that can be viewed as a remnant of a large cardinal axiom. This property has been studied since the 1970s, and work on it has been driving substantial developments in the area of consistency proofs. The second involves the interactions between combinatorial properties of the line of real numbers, and the size of the set of real numbers. In both these aspects, proofs that the relevant principles can hold use large cardinal axioms. The third aspect is to study the large cardinal axioms themselves, to develop specific models for these axioms, and to further strengthen the known connection between some of these axioms and properties of the line of real numbers. The overall purpose in all cases is to further our understanding of what is, and what is not, possible in the universe of mathematics. This project will support the development and training of graduate students in mathematical logic at UCLA. In addition this project builds on previous work of the PI at the undergraduate level. The PI plans to help talented UCLA undergraduates acquire graduate level knowledge in mathematical logic, and continue to graduate studies. This project deals with three interrelated areas in set theory: (i) the tree property; (ii) consistency proofs with large continuum; and (iii) large cardinals and inner models, in the region of Woodin cardinals and of supercompact cardinals. In connection with (i), the project is particularly concerned with forcing the tree property at regular cardinals, above the first uncountable cardinal, in increasingly large intervals. The ultimate goal is to see whether the tree property on some of these cardinals can prevent the property from holding on other cardinals, or whether it is consistent that the tree property holds at all these cardinals. We are very far from an answer to this question, but there has been some impressive progress in recent years and this project seeks to build on this progress in order to push the boundaries further. In connection with (ii), the project is particularly concerned with some of the central structural consequences of the proper forcing axiom (PFA), for example Todorcevic's open coloring axiom and p-ideal dichotomy. This project seeks to determine whether these principles are consistent with large continuum, meaning larger than its size under PFA. The ultimate goal is to develop a robust framework of consistency results with large continuum, or determine actual mathematical obstacles for such frameworks. In connection with (iii), the project is particularly concerned with descriptive set theoretic applications of inner models theory at the level of Woodin cardinals, and with pushing the theory of inner models to the level of supercompact cardinals. The former should lead to solutions for some of the more intractable, still open, questions in descriptive set theory under determinacy. The latter should help with our understanding of a key level of the large cardinal hierarchy. 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.