Mathematical Sciences: Set Theory

小行星9322442 该项目的重点是大基数公理和确定性公理之间的关系。 这是广义上的解释，例如，包括对一类新模型的研究。 这些模型是作为强制扩展的模型，其中公理的确定性举行。 它们很有趣，因为它们提供了第一个典范模型的例子，在这些模型中，连续统假设是错误的。 一个后果库尔特哥德尔的结果60年前是，将永远是数学问题无法解决。 30年前，当保罗·科恩证明康托的连续统假说不能被解决时，这第一次成为一个实际的问题。 自科恩的工作以来，许多问题已被证明是无法解决的。 在许多情况下，大的基数等级制为独立性问题提供了解决方案。 这不是通过解决问题，而是通过校准它无法解决的程度来实现的。 这一系列公理，每一个都断言一个新的无穷大的存在，作为基本的数学现实出现。 研究人员打算研究这种等级制度的许多表现形式。 ***

9322442 Woodin The focus of the project is the relationship between large cardinal axioms and determinacy axioms. This is interpreted somewhat broadly and is meant to include, for example, the study of a new class of models. These models are generated as forcing extensions of models in which the axiom of determinacy holds. They are interesting because they provide the first examples of canonical models in which the Continuum Hypothesis is false. One consequence of Kurt Godel's results of 60 years ago is that there will always be mathematical problems which cannot be solved. This first became a practical concern 30 years ago when Paul Cohen showed that Cantor's Continuum Hypothesis cannot be settled. Since Cohen's work, numerous problems have been shown to be unsolvable. The large cardinal hierarchy provides in many cases a resolution to the problem of independence. This is done, not by resolving the question, but rather by calibrating the degree to which it is unsolvable. This hierarchy of axioms, each of which asserts the existence of a new infinity, emerges as the fundamental mathematical reality. The investigator intends to study this hierarchy in its many manifestations. ***