Explore the historical controversies and foundational issues in mathematics through this 47-minute lecture on computability and set theory. Delve into Cantor's set theory definition, the formalist approach to resolving difficulties, and Gödel's impact on Hilbert's program. Examine the Zermelo-Fraenkel axiomatic approach to sets and investigate Alan Turing's ideas on computability using Turing machines. Analyze the consequences of these concepts, including the countability of computable sequences. Review historical perspectives on infinite sets from Aristotle to Abraham Robinson, and consider the ongoing debates in mathematical foundations. Gain insights into the work of influential mathematicians like Kurt Gödel and Émile Borel, and explore the connections between set theory, computability, and measure theory.