Mathematical Sciences: Large Cardinals and Forcing

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8901730 Foreman The principal investigator intends to work on several well-known problems in Set Theory and the foundations of mathematics. The first involves the relationship between the "generic" large cardinal axioms and conventional large cardinal axioms. Closely related to this is the problem of whether aleph-omega is Jonsson. A successful resolution of these problems would unify the various proposed strengthenings of the ordinary assumptions of mathematics. If the results conjectured by the P.I. are true it would also shed considerable light on the continuum hypothesis. The second type of problem examines the combinatorial characteristics of definable partial orderings and graphs. Frequently, adding a definability hypothesis to a class of partial orders implies that they have very nice properties, devoid of the "pathologies" induced by arbitrary choice. The third kind of problem involves investigating applications of set theory in functional analysis.