On Computability over Finite Types

Explore a rigorous mathematical approach to defining computable functions of finite type in this colloquium lecture. Delve into Gödel's concept of computable functions of finite simple type from his work on extending finitary mathematics, where he used these functions to prove the consistency of classical number theory. Examine the challenge posed by Gödel's use of the phrase "well-defined mathematical procedure" without further explanation, and discover a possible rigorous definition for computable functions of finite type. Compare this new definition with the classical definition of computable functions over natural numbers, gaining insights into the foundations of computability theory and mathematical logic. Learn from collaborative research conducted with experts from National University of Singapore, Nanyang Technological University, and Amir Kabir University, as presented by a distinguished mathematician with extensive experience in recursion theory and reverse mathematics.