Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

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The area of research of this project 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 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. The project has three parts: first, to continue the study of a class of very large cardinals; second, to continue the study of the relation of these cardinals to problems in finite combinatorics; and third, to study some unrelated topics. While the details of proofs in this area are mainly meant for experts, it is a remarkably intriguing discovery that some very basic mathematical propositions not only have not yet been proved or disproved, but can be proved to be unprovable (or disprovable).