Mathematical Sciences: Combinatorial Set Theory, Forcing andPriority Arguments

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Baumgartner will work on problems in the theory of forcing and on combinatorial set theory, in particular on combinatorial problems that seem to raise fundamental questions about forcing. Forcing is a powerful method for examining the extremes of set theoretic behavior permitted by given axioms of set theory. One uses it to construct models of set theory obeying the axioms, the object being to show that some additional property does not follow from the axioms. If one can show that neither a property nor its opposite follows from the axioms, one has found an independence proof for that property. These are highly prized as illuminating the foundations of set theory and thereby of all mathematics.