The cut-elimination theorem
Pietro Mancosu, Sergio Galvan, Richard Zach · 2021
Abstract All the rules of the sequent calculus have the property that all the formulas that are present in the premises also occur in the conclusion. There is only one exception, the cut rule. In this chapter, it is shown using double induction that every theorem provable in Gentzen’s sequent calculi using the cut rule can also be proved without. One first proves that the mix rule is equivalent in proof-theoretic strength to the cut rule. Then one proves by induction on two complexity measures, the degree and the rank of a mix, that every application of mix can be eliminated. Several results follow from the cut-elimination theorem, including the so-called “mid-sequent” theorem and Herbrand’s theorem. In addition, one can prove the consistency of minimal, intuitionistic, and classical logic.