Set theory: Combinatorics and Large Cardinals

[9970536] Mitchell调查的主要主题是关于核心模型，特别是它的结构和在伍迪纳尔附近的存在。核心模型的结构已知远高于伍定基数，但它的存在取决于期望模型之上是否存在可测量基数。因此，当伍丁基数存在时，没有核心模型，但没有更大的模型。然而，初步结果表明，一些核心模型结构仍然存在。米切尔将尝试更好地理解这种结构。他还将研究涉及封闭无界集和更大基数的核心模型的存在性和结构的相关主题。Larson将研究可数序数和大序数的Ramsey性质，这些序数不是基数的小有限幂。集合论通常被认为是所有数学的基础，所以这类研究在某种意义上是在探索数学可能性的极限。顾名思义，核心模型构成了集合世界的坚实基础，是研究和验证公理的主要工具，这些公理扩展了集合理论的力量，从而扩展了数学。序数构成了集合的骨架，理解它们的组合性质对于理解集合的整体是至关重要的

9970536Mitchell The main topics of Mitchell's investigation concern the core model,particularly its structure and existence in the vicinity of a Woodincardinal. The structure of the core model is known far above a Woodincardinal, but its existence depends on the existence of a measurablecardinal lying above the desired model. Thus there is no core modelin the important case when a Woodin cardinal exists, but nothinglarger. Preliminary results suggest, however, that some of the coremodel structure remains. Mitchell will attempt to understand thisstructure better. He will also study related topics involving closed,unbounded sets and the existence and structure of the core model forlarger cardinals. Larson will study Ramsey properties of countableordinals and of larger ordinals that are not small finite powers ofcardinals. Set theory is usually considered the foundation of all mathematics,so studies of this kind are in a sense exploring the limits ofmathematical possibility. The core model forms, as its name implies,the rigid foundation of the universe of sets and is the major tool forstudying and validating axioms that extend the power of set theory andhence of mathematics. The ordinals form the skeleton of the universeof sets, and understanding their combinatorial properties are vital tounderstanding the universe of sets.***