Set theory: Combinatorics and Large Cardinals

小行星9970536 米切尔研究的主要课题是核心模型，特别是它的结构和存在附近的一个Woodincardinal。 核心模型的结构远高于Woodincardinal，但它的存在取决于所需模型之上的可测量cardinal的存在。 因此，在存在Woodin基数的重要情况下，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。 Mitchell将尝试更好地理解这个结构。 他还将研究相关主题，包括封闭，无界集和存在和结构的核心模型forlarger基数。 拉尔森将研究拉姆齐属性countableordinals和较大的序数是不小的有限权力的基数。 集合论通常被认为是所有数学的基础，因此这类研究在某种意义上是在探索数学可能性的极限。 核心模型形式，正如它的名字所暗示的那样，是集合论的刚性基础，是研究和验证公理的主要工具，这些公理扩展了集合论的力量，因此也扩展了数学的力量。 序数构成了集合论域的骨架，理解它们的组合性质对于理解集合论域至关重要。

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.***