From Stenius' Consistency Proof to SCHüTTE's Cut Elimination for ω-Arithmetic

The book Das Interpretationsproblem der Formalisierten Zahlentheorie und ihre Formale Widerspruchsfreiheit by Erik Stenius published in 1952 contains a consistency proof for infinite ω -arithmetic based on a semantical interpretation. Despite the proof’s reference to semantics the truth definition is in fact equivalent to a syntactical derivability or reduction condition. Based on this reduction condition Stenius proves that the complexity of formulas in a derivation can be limited by the complexity of the conclusion. This independent result can also be proved by cut elimination for ω -arithmetic which was done by Schutte in 1951. In this paper we interpret the syntactic reduction in Stenius’ work as a method for cut elimination based on invertibility of the logical rules. Through this interpretation the constructivity of Stenius’ proof becomes apparent. This improvement was explicitly requested from Stenius by Paul Bernays in private correspondence (In a letter from Bernays begun on the 19th of September 1952 (Stenius & Bernays, 1951–75)). Bernays, who took a deep interest in Stenius’ manuscript, applied the described method in a proof Herbrand’s theorem. In this paper we prove Herbrand’s theorem, as an application of Stenius’ work, based on lecture notes of Bernays (Bernays, 1961 ). The main result completely resolves Bernays’ suggestions for improvement by eliminating references to Stenius’ semantics and by showing the constructive nature of the proof. A comparison with Schutte’s cut elimination proof shows how Stenius’ simplification of the reduction of universal cut formulas, which in Schutte’s proof requires duplication and repositioning of the cuts, shifts the problematic case of reduction to implications.