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Explore the mathematical study of definability through this lecture that begins with an overview of definability theory and examines its applications in both pure mathematics and external phenomena. Delve into computability theory in the context of sets of integers, covering the Halting Problem, the Turing jump operation, and arithmetic and hyperarithmetic hierarchies. Investigate how these concepts relate to topological complexity in the context of real numbers, including continuous functions and Borel sets. Follow detailed proofs of fundamental theorems that demonstrate the core methodologies used in definability theory. Examine Martin's Conjecture, which provides insight into why the analysis of definability is both intrinsic and inevitable in pure definability theory. Discover two compelling case studies that showcase how definability theory can be applied to study phenomena originating outside mathematical logic: normality to integer bases and Hausdorff dimension, illustrating the broad applicability of these theoretical frameworks.
