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Explore groundbreaking research in reverse mathematics focusing on real analysis through this comprehensive lecture that examines the hierarchical structure of mathematical principles. Delve into recent collaborative work with Dag Normann from the University of Oslo, conducted within Kohlenbach's higher-order framework, which reveals fascinating connections between second-order and third-order mathematical systems. Discover how basic theorems about discontinuous functions in real analysis relate to the classical "Big Five" systems of reverse mathematics, and learn about newly identified mathematical principles that extend beyond these traditional boundaries. Examine four emerging "Big systems" including the uncountability of the reals, Jordan decomposition theorem, Baire category theorem, and Tao's pigeonhole principle for Lebesgue measure, while gaining insights into ongoing research on even stronger principles such as Feferman's projection principle and coding principles for open sets of real numbers.
