Set theory: Combinatorics and Large Cardinals

9970536米切尔调查的主要主题与核心模型有关，特别是它的结构和在红衣主教附近的存在。核心模型的结构是已知的，远远高于WoodinCardina，但它的存在取决于所需模型上方是否存在可测量的Cardinal。因此，在伍丁基数存在的重要情况下，没有核心模型，而是没有更大的核心模型。然而，初步结果表明，仍保留了一些岩心重建结构。米切尔将试图更好地理解这种结构。他还将学习与闭集、无界集以及较大基数的核心模型的存在和结构相关的主题。拉森将研究可数序数和不是基数的有限小幂的较大序数的Ramsey性质。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宇宙的刚性基础，是研究和验证扩展集合论和数学力量的公理的主要工具。序数构成了集合宇宙的骨架，了解它们的组合性质对于理解集合宇宙至关重要。

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.***