Mathematical Sciences: The Definable Continuum Hypothesis
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At this time it is known that there are many propositions of            mathematics independent of the ordinary axioms of Zermelo-Frankel               set theory.  Foremost among these is the continuum hypothesis.                  This has led investigators to pose stronger axioms, known as                    large cardinal axioms to try to settle these problems.  Professor               Foreman plans to investigate the relationship of strong axioms of               set theory to the continuum hypothesis.  In particular he                       conjectures that ordinary large cardinal axioms should imply that               there are no definable or constructable counterexamples to the                  continuum hypothesis.  He has proven substantial partial results                in this direction and plans to continue work on this conjecture.                Such investigations reach to the most basic foundations of                      mathematics, clarifying our understanding of the infinite.
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