Inner Model Theory and Descriptive Set Theory

斯蒂尔致力于大型基数假设的规范内部模型理论和描述性集合论。在内模理论中，他专注于与基本迭代性问题相关的问题，以及关于如何在各种情况下构造内模以获得一致性强度下界的问题。他特别感兴趣的是从适当的强迫公理，从詹森的平方原理在奇异基数上的失败，从独特的分支假设的失败中获得大的基数力量。在描述集合论中，斯蒂尔正在研究与满足强确定性形式的模型的结构有关的各种问题。强大的无穷大公理，或者更常被称为大型基数假设，至少出于两个原因，在三四十年来一直是集合论和数学基础的工作重点。第一，大的基本假设可以用来以自然的方式决定许多问题，这些问题不能根据普遍接受的数学公理体系来决定，第二，大的基本假设提供了一种组织和考察这一普遍接受的系统的所有可能的自然延伸的方法。研究大型基数假设的一个重要方法是构造规范的极小内模型，在该模型中这些假设为真。这类模型允许对其内部结构进行系统、详细的分析，这使它们在两种大型基本假设的应用中都成为一种宝贵的技术工具。目前，我们已经有了一个很好的正则内模型理论来满足“有一个Woodin基数”，甚至更强的大型基数假设。一个基本的公开问题，也是集合论中最重要的公开问题之一，是将这个理论推广到满足‘有一个超紧基数’的模型。钢铁公司正朝着这个方向努力。

Steel is working in the theory of canonical inner models for large cardinal hypotheses, and in descriptive set theory. In inner model theory, he has focussed on questions related to the fundamental iterability problem, and on questions concerning how to construct inner models in various situations so as to obtain consistency-strength lower bounds. He is particularly interested in obtaining large cardinal strength from the Proper Forcing Axiom, from the failure of Jensen's square principle at a singular cardinal, and from the failure of the Unique Branches Hypothesis. In descriptive set theory, Steel is working on various questions related to the structure of models satisfying strong forms of determinacy. Strong axioms of infinity, or as they are more often called, large cardinal hypotheses, have been a focal point of work in set theory and the foundations of mathematics for thirty or forty years, for at least two reasons. First, large cardinal hypotheses can be used to decide in a natural way many questions which cannot be decided on the basis of the commonly accepted system of axioms for mathematics, and second, large cardinal hypotheses provide a way of organizing and surveying all possible natural extensions of this commonly accepted system. One important way to study large cardinal hypotheses is to construct canonical minimal inner models in which these hypotheses are true. Such models admit a systematic, detailed analysis of their internal structure which makes them an invaluable technical tool in both sorts of application of large cardinal hypotheses. At the moment, we have a good theory of canonical inner models satisfying ``There is a Woodin cardinal", and even slightly stronger large cardinal hypotheses. A fundamental open problem, one of the most important open problems in set theory, is to extend this theory to models satisfying ``There is a supercompact cardinal". Steel is working in this direction.