Computability and Definability - Theory and Application - Lecture 4

Explore the mathematical study of definability through this advanced lecture that examines both theoretical foundations and practical applications. Begin with an overview of definability in the context of sets of integers, covering computability theory, the Halting Problem, and the Turing jump operation, before progressing to 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 this field of study. Examine Martin's Conjecture, which provides insight into the intrinsic and inevitable nature of definability analysis within pure mathematical theory. Conclude by exploring two comprehensive case studies that demonstrate how definability theory applies to phenomena originating outside mathematical logic: normality to integer bases and Hausdorff dimension, illustrating the broader relevance of these theoretical concepts to mathematical analysis.