Set theory: Combinatorics and Large Cardinals

9970536Mitchell Mitchell 调查的主要主题涉及核心模型，特别是其结构和在 Woodincardinal 附近的存在。 核心模型的结构远高于伍丁基数，但其存在取决于位于所需模型之上的可测量基数的存在。 因此，在伍丁红衣主教存在的重要情况下，不存在核心模型，但也没有更大的模型。 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个结构。 他还将研究涉及闭无界集合以及较大基数核心模型的存在和结构的相关主题。 拉森将研究可数序数和较大序数的拉姆齐性质，这些序数不是基数的有限幂。 集合论通常被认为是所有数学的基础，因此此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，核心模型形成了集合论的刚性基础，并且是研究和验证公理的主要工具，这些公理扩展了集合论以及数学的力量。 序数构成集合宇宙的骨架，理解它们的组合属性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.***