Explore the evolution of computation in mathematical foundations through this 40-minute lecture by Reinhard Kahle. Delve into the historical context, beginning with Hilbert's Programme, and examine how computational concepts became integral to mathematical foundations. Gain insights into the crucial distinction between syntax and semantics. Investigate key topics such as Tarski's Theorem, the foundational question in mathematics, formalization, Gödel's contributions, and the recursive turn. Discover the implications of the uncared corollary and the complete "axiomatization" of Arithmetic. Analyze the relationship between syntax and semantics in first-order theories, and understand the role of computations in shaping modern mathematical foundations.