Problems in Combinatorial Set Theory

我提出了一个关于组合集合论一般领域的研究计划，重点是谢拉风格的“奇异基数组合学”。我建议研究的这类问题的一个例子是(由于伍丁的)问题，即Sch在单一基数的失败是否意味着在该基数的继任者处存在Aronszajn树。我对获得ZFC结果以及互补的一致性结果很感兴趣：我计划使用的方法包括PCF理论、内部模型、Radin作用力和一些最近适当强迫的推广。组合集合论可以被看作是一种尝试，将一些非常基本的数学概念(计数、排序、置换)在无限集合的背景下加以理解。典型的问题可能涉及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，这样相邻的两个顶点都不会有相同的颜色。事实证明，这类问题的深度令人惊讶，并在数学的许多领域得到了应用。我的建议的重点是涉及所谓的“奇异基数”的问题，从某种程度上说，这是最难理解的无限集合。

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.