Mathematical Sciences: The Definable Continuum Hypothesis

此时，已知有许多数学命题独立于Zermelo-Frankel集合论的普通公理。其中最重要的是连续体假说。这导致调查人员提出了更强大的公理，即所谓的大基数公理，试图解决这些问题。福尔曼教授计划研究集合论的强公理与连续统假设之间的关系。特别是，他推测，普通的大型基数公理应该暗示，对于连续统假设，没有可定义或可构造的反例。他已经在这个方向上证明了大量的部分结果，并计划继续研究这个猜想。这样的研究触及了数学的最基本基础，澄清了我们对无限的理解。

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.