Herbrand's Theorem and Proof-Theory

Abstract

Herbrand's “Théorème Fondamental,” one of the milestones of mathematical logic, is one of the few basic results of proof theory, only challenged by Gentzen's Hauptsatz. The chapter discusses the importance of Herbrand's theorem for the proof theorist of 1981, 50 years after Herbrand's death. Direct uses or extensions of Herbrand's result are discussed, but also general ideas in proof theory that are still alive and can be—at least partly—ascribed to Herbrand, The chapter analyzes the main ideas in Herbrand's theorem—as they appear 50 years later—and presents their posterity in recent work, not connected with Herbrand's theorem. One of the major defects of Herbrand's theorem is the bad behavior with respect to implication. A regular notion with respect to this problem is obtained by Gödel's functional interpretation, which can be thought as an unwinding of the no-counter example interpretation by means of functionals of finite type.