Problems in Combinatorial Set Theory

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

I propose a program of research in the general area of combinatorial set theory, with an emphasis on "singular cardinal combinatorics" in the style of Shelah. An example of the sort of questions that I propose to work on is the question (due to Woodin) whether failure of SCH at a singular cardinal implies the existence of an Aronszajn tree at the successor of that cardinal. I am interested in obtaining ZFC results as well as complementary consistency results: the methods that I plan to use include PCF theory, inner models, Radin forcing and some recent generalisations of proper forcing. Combinatorial set theory can be seen as an attempt to take some very elementary mathematical ideas (counting, ordering, permuting) and make sense of them in the context of infinite sets. Typical problems might involve finding an infinite path through an infinite tree, or colouring the vertices in an infinite graph so that no two adjacent vertices have the same colour. Problems of this kind turn out to be surprisingly deep and have found applications in many areas of mathematics. The focus of my proposal is problems involving so-called "singular cardinals", which (by some measure) are the hardest infinite sets to understand.