Gentzen’s consistency proof without heightlines

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This paper gives a Gentzen-style proof of the consistency of Heyting arithmetic in an intuitionistic sequent calculus with explicit rules of weakening, contraction and cut. The reductions of the proof, which transform derivations of a contradiction into less complex derivations, are based on a method for direct cut-elimination without the use of multicut. This method treats contractions by tracing up from contracted cut formulas to the places in the derivation where each occurrence was first introduced. Thereby, Gentzen’s heightline argument, which introduces additional cuts on contracted compound cut formulas, is avoided. To show termination of the reduction procedure an ordinal assignment based on techniques of Howard for Godel’s T is used.