Computability and Definability - Theory and Application - Lecture 5

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 real numbers, including continuous functions and Borel sets. Examine several basic theorems with complete proofs that demonstrate fundamental methods in the field. Delve into Martin's Conjecture, which provides insight into the intrinsic and inevitable nature of definability analysis in pure theory. Conclude by studying two detailed case studies that showcase how definability theory applies to phenomena outside mathematical logic: normality to integer bases and Hausdorff dimension, illustrating the broader relevance of these theoretical concepts.