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Mathematical logic entered the modern era through the work of Kurt Gödel, who established his famous Completeness and Incompleteness Theorems at the University of Vienna in the 1930's. In his later work, Gödel focused on set theory, the topic of this proposal. Set theory today has two closely related aspects, the pure and the applied. The former is concerned with an analysis of infinity, leading to a picture of the universe of sets as a whole, accompanied by a detailed study of definability within set theory. This proposal focuses on several aspects of pure set theory: Borel quotients,
forcing axioms, the definable SCH and strong absoluteness.




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