Reduction and Elimination

Abstract

In Gentzen’s famous paper cut-elimination is a constructive method for proving the “Hauptsatz” which is used to constitute such important principles as the existence of a mid-sequent and the decidability of propositional intuitionistic logic. The idea of the Hauptsatz is connected to the elimination of ideal objects in mathematical proofs according to Hilbert’s program. In this sense LK (with the standard axiom set) is consistent because the empty sequent is not cut-free derivable; any proof of a contradiction would need ideal (indirect) arguments. By shift of emphasis mathematicians began to focus on the proof transformation by cut-elimination itself. In fact, cut-elimination is an essential tool for making implicit contents of proofs explicit. It also allows the construction of Herbrand disjunctions and interpolants for real mathematical proofs. Furthermore elementary proofs can be obtained from abstract ones; one of the most important examples from literature is the transformation of the Fürstenberg–Weiss proof into the original (van der Waerden’s) proof [40].