Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

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The area of research is set theory, in particular, large cardinals and forcing. Set theory may be viewed as containing mathematics, that is, every theorem of classical mathematics may be formulated as a statement about sets, and then formally derived from the standard collection ZFC of axioms for set theory. Beginning with the work of Godel (1936) and Cohen (1963), set theorists have shown that certain problems, on which mathematicians have worked, are independent, i.e. neither provable nor disprovable from ZFC. The first and most well known example of this is the Godel-Cohen result that Cantor's continuum hypothesis is independent. Some statements, independent of ZFC, are nevertheless known to be provable when ZFC is augmented by axioms asserting the existence of large infinite cardinal numbers. This project has two parts: firstly, to continue the study of a class of very large cardinals, and secondly, to study the consistency and independence of von Neumann's conjecture and a collection of related problems. Such studies deepen our understanding of the limits of mathematical proof.