Computability and Definability - Theory and Application - Lecture 3

Explore the mathematical study of definability through this advanced lecture that examines computability theory and its applications across different mathematical contexts. Begin with an overview of definability in sets of integers, covering computability, the Halting Problem, the Turing jump operation, and arithmetic and hyperarithmetic hierarchies. Transition to studying definability in real numbers, where objects relate directly to topological complexity including continuous functions and Borel sets. Examine proofs of fundamental theorems that demonstrate the core methodologies of definability theory. Delve into Martin's Conjecture, which provides precise understanding of how definability analysis is intrinsic and inevitable in pure theory. Investigate two detailed case studies showing how definability theory applies to phenomena originating outside mathematical logic: normality to integer bases and Hausdorff dimension, illustrating the practical relevance of these theoretical concepts.