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Explore the mathematical foundations of definability theory in this lecture that begins with an overview of how mathematicians study definability across different mathematical contexts. Examine computability within sets of integers, including the fundamental Halting Problem and its associated Turing jump operation, while investigating arithmetic and hyperarithmetic hierarchies. Discover how these concepts relate to topological complexity in real numbers, particularly through continuous functions and Borel sets. Follow detailed proofs of several basic theorems that demonstrate the core methodologies used in this field. Delve into Martin's Conjecture, which provides a precise framework showing how definability analysis is both intrinsic and inevitable in pure theory. Conclude by examining two practical case studies where definability theory illuminates phenomena from outside mathematical logic: normality to integer bases and Hausdorff dimension, demonstrating the broad applicability of these theoretical foundations.
