Theory of Computation: Countability & Halting Problem

This lecture covers the fundamental questions in the theory of computation: what can be solved with an algorithm and what can be solved efficiently. It explores countability, the halting problem, decision problems, and the concept of denumerability. Through examples, it illustrates how to count infinite sets, the denumerability of pairs of positive integers, and the undecidability of the halting problem using self-reference. The lecture also discusses the countability of programs and the undecidability of certain decision problems, emphasizing the limitations of algorithmic solutions.