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.