Mathematical Sciences: Set Theory

9322442 Woodin项目的重点是大基数公理和确定性公理之间的关系。这在某种程度上被广泛地解释，并意味着包括，例如，对一类新模型的研究。这些模型是作为确定性公理成立的模型的强制扩展而产生的。它们之所以有趣，是因为它们提供了第一批证明连续统假设是错误的典型模型的例子。库尔特·哥德尔60年前的结论之一是，永远存在无法解决的数学问题。30年前，当保罗•科恩（Paul Cohen）证明康托尔的连续统假设（Continuum Hypothesis）无法成立时，这个问题首次成为一个现实问题。自从科恩的研究以来，许多问题都被证明是无法解决的。在许多情况下，大的基数等级制度提供了解决独立问题的办法。要做到这一点，不是通过解决问题，而是通过校准问题的不可解决程度。这种公理的层次，每一个都断言存在一个新的无穷大，作为基本的数学现实而出现。研究者打算研究这一层次结构的多种表现形式。＊＊＊

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. ***