Inner Model Theory and Descriptive Set Theory

钢将工作在集理论，特别是对理论的典型内部模型的大型基数假设。他的重点将是扩展这一理论的基础，以更强大的大基数假设。 这里的中心问题是V的每个可数基本子模型是否都是适当的可迭代的。斯蒂尔将研究这个问题，它的碎片，以及相关的问题。这一努力与纯粹描述性集合论密切相关，描述性集合论是对真实的数字集合的可定义性的一般研究。Steel也计划在这一领域开展工作，特别是与无限博弈的确定性相关的问题。一个非常好的问题是，是否所有封闭的（在可数初始段拓扑）游戏的长度aleph-one与投影定义的回报是确定的。 ZFC是公认的数学公理系统，许多数学问题都没有得到解决。扩展ZFC以决定此类问题的最自然、最成功的方法是在ZFC中添加“大基数假设”。这样的假设在数学基础中起着核心作用，因此在过去的30到40年里一直被深入研究。理解和使用大基数假设的一种方法是通过构建规范的最小宇宙，或“内部模型”，其中这些假设成立。内模型理论始于20世纪30年代库尔特·哥德尔的工作，从20世纪60年代末到现在，该理论经历了或多或少的不断发展。集合理论家已经为许多大基数假设构造了规范内部模型并进行了详细的分析，这些模型已经被证明是非常有用的。然而，有一些重要的大基数假设是目前的内模理论所无法达到的。 斯蒂尔将试图扩展这一理论，使其适用于这些假设。

Steel will work in set theory, and in particular on the theory of canonical inner models for large cardinal hypotheses. His focus will be on extending the basics of this theory to stronger large cardinal hypotheses. The central open problem here is whether every countable elementary submodel of V is appropriately iterable. Steel will work on this question, its fragments, and related issues. This effort is closely connected to pure descriptive set theory, the general study of definability for sets of real numbers. Steel plans to work in this area too, and in particular on questions related to the determinacy of infinite games. One very nice question here is whether all closed (in the countable-initial-segment topology) games of length aleph-one with projectively definable payoff are determined. Many questions of mathematical interest are left undecided by ZFC, the commonly accepted system of axioms for mathematics. The most natural and successful way to extend ZFC so as to decide such questions is to add "large cardinal hypotheses" to ZFC. Such hypotheses play a central role in the foundations of mathematics, and because of this have been studied intensively over the last 30 to 40 years. One way to understand and use large cardinal hypotheses is by constructing canonical minimal universes, or "inner models", in which these hypotheses hold true. Inner model theory began with work of Kurt Godel in the 1930's, and from the late 1960's to the present the theory has undergone more or less continuous development. Set theorists have constructed and analyzed in detail canonical inner models for many large cardinal hypotheses, and these models have proved quite useful. Nevertheless, there are important large cardinal hypotheses which are beyond the reach of current inner model theory. Steel will attempt to extend the theory so that it applies to these hypotheses.