Herbrand's Theorem and Gentzen's Notion of a Direct Proof

Abstract

Publisher Summary This chapter presents the Herbrand's theorem and Gentzen's notion of a direct proof. To explain the results and methods characteristic of the theory of proofs, the chapter reconsiders a special case of a corollary to the Completeness Theorem. The result is known in the literature as Herbrand's Theorem. The existence of this method, and of a partial recursive one independent of the derivation, follows from the soundness and completeness of the rules. The non-existence of a recursive method depending only on the formula follows from the recursive undecidability of validity. The principal aim of the theory of proofs is to make differences among proofs, previously judged only by aesthetic criteria of elegance or convenience, objects of study.