Problems in Combinatorial Set Theory

组合集合论中的问题

• 批准号: 0400982
• 负责人: James Cummings
• 金额: $ 10.01万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400982&HistoricalAwards=false
• 关键词: Problems; Combinatorial; Set; Theory

I propose a program of research in the general area of combinatorial set theory, with an emphasis on "singular cardinalcombinatorics" in the style of Shelah. An example of the sort of questions that I propose to work on is the question (due to Woodin)whether failure of SCH at a singular cardinal implies the existence ofan Aronszajn tree at the successor of that cardinal. I am interested inobtaining ZFC results as well as complementary consistency results: the methods that I plan to use include PCF theory, inner models, Radin forcingand some recent generalisations of proper forcing. Combinatorial set theory can be seen as an attempt to take some veryelementary mathematical ideas (counting, ordering, permuting) andmake sense of them in the context of infinite sets. Typical problems mightinvolve finding an infinite path through an infinite tree, or colouring the vertices in an infinite graph so that no two adjacent vertices have the same colour. Problems of this kind turn out to be surprisinglydeep and have found applications in many areas of mathematics. The focusof my proposal is problems involving so-called "singular cardinals",which (by some measure) are the hardest infinite sets to understand.

我提出了一个计划的研究在一般领域的组合集理论，重点是“奇异cardinalcombinatorics”的风格的希拉。 一个例子的排序的问题，我建议工作的问题（由于Woodin）是否失败的SCH在一个单一的基数意味着存在aanAronszajn树的继任者，基数。 我感兴趣的是获得ZFC结果以及互补的一致性结果：我计划使用的方法包括PCF理论，内部模型，Radin forcing和最近的一些适当的强迫概括。 组合集合论可以被看作是试图把一些非常基本的数学思想（计数、排序、置换）放在无限集合的上下文中来解释。典型的问题可能涉及到找到一条通过无限树的无限路径，或者给无限图中的顶点着色，使得相邻的顶点没有相同的颜色。这类问题被证明是非常深奥的，并且在数学的许多领域都有应用。我的建议的重点是问题涉及所谓的“奇异基数”，这（在某种程度上）是最难理解的无限集。