Mathematical Sciences: Set Theory and Large Cardinals

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This project will continue to study several topics in set theory, primarily concerning the core model and its applications. The most important aim of this research will be to extend core model theory so that one can prove the existence of a core model in the presence of a Woodin cardinal. The probable structure of this core model is already known; the main problem seems to be to develop techniques which can be used to prove an extender is a member of the core model. Other areas of study concern applications of the core model under the assumption that there is no model with a Woodin cardinal. These potential applications include questions related to absoluteness, to the singular cardinal hypothesis, and to Jonson and Ramsey cardinals. The ultimate aim of this research is to chararcterize the structure of well-founded models of set theory. As their name suggests, the 'core' of a universe of set theory, if it exists, provides a skeleton with a relatively well-defined structure and thus gives a good deal of information about the form of the full model. Since set theory lies at the very foundation of mathematics, this enterprise has important consequences for the soundness of many accepted mathematical practices, although it is customary for most working mathematicians to take this on faith.