Computability Theory and Logic with Scheme

Have you ever wondered what makes a problem solvable by a computer, or how we can mathematically prove that a program behaves exactly as intended? This course bridges the gap between abstract mathematical logic and practical programming by using the elegant Scheme language to explore the fundamental limits of computation. You will transition from writing basic expressions to understanding the deep theoretical boundaries of software, algorithms, and logical systems. By reading through clear explanations and structured code analysis, you will master the mechanics of computation from first principles. The course starts with essential terminology, establishing how evaluation works as algebraic manipulation and term-rewriting. You will then progress to complex theoretical concepts, including self-application paradoxes, formal semantics, and the famous Halting Problem. What you'll learn: - Understand the foundations of computability theory using Scheme as a model of computation - Analyze evaluation as a form of algebraic manipulation and term-rewriting - Explore the mechanics of self-application, recursion, and fixed-point combinators - Prove the undecidability of the Halting Problem and examine its implications - Study recursively enumerable sets and their connection to incompleteness theorems - Apply formal logic principles to program specification and verification This text-based curriculum is designed to guide you step-by-step through dense theoretical concepts. We begin with foundational definitions and simple substitution models before building up to advanced proofs, incompleteness, and program verification logic. Each concept is paired with readable Scheme code snippets to make abstract mathematics concrete. This course is designed for curious programmers, computer science students, and self-directed learners who want to understand the mathematical soul of computation. No prior background in advanced logic or computability theory is required, though a basic familiarity with programming concepts is helpful. Begin reading today to unlock the deepest principles of computer science and formal logic.