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Explore the foundational concepts of reverse mathematics applied to mathematical analysis in this introductory lecture from a summer school on "Reverse Mathematics: New Paradigms." Begin by reviewing the "big five" subsystems of second-order arithmetic and their fundamental properties, then establish precise definitions of open and closed sets alongside continuous functions within Polish spaces. Delve into the reverse mathematical analysis of basic analytical concepts, examining how different axiom systems relate to fundamental theorems in analysis. Discover how reverse mathematics provides a framework for understanding the logical strength required for various mathematical statements, with potential extensions to more advanced topics in analysis depending on time constraints.
