On Computability over Finite Types

Explore a rigorous mathematical definition of computable functions of finite type in this BIMSA Colloquium lecture. Delve into Gödel's concept from his paper "On an extension of finitary mathematics which has not yet been used," where he employed computable functions of finite simple type 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 developed through collaborative research. Compare this new definition with classical definitions of computable functions over natural numbers. Learn from the joint work conducted by researchers from National University of Singapore, Nanyang Technological University, and Amir Kabir University. Gain insights into advanced topics in mathematical logic, recursion theory, and the foundations of mathematics through this detailed exploration of computability theory over finite types.