The computational content of classical arithmetic

Abstract

Almost from the inception of Hilbert's program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the G\"odel-Gentzen double-negation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory.