Set theory: Combinatorics and Large Cardinals

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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.***