On Gentzen’s first consistency proof for arithmetic

sequent of d) forms an inference tp(d) in cutfree ω-arithmetic with repetition rule Rep. Obviously, if d is a derivation of falsum ⊥, i.e. if End(d) = ⊥, then tp(d) can only be an instance of Rep, so that d[0] is again a derivation of ⊥. In a second step, to each d an ordinal o(d) < ε0 is assigned such that o(d[n]) < o(d) for all n ∈ |tp(d)|. Then the consistency of Z follows by (quantifierfree) transfinite induction up to ε0. Actually Gentzen’s terminology is somewhat different. First (in §13 of [Ge36]) Gentzen defines reduction steps on sequents. Such a reduction step I may involve a certain ‘option’ (Wahlfreiheit), so that the result of applying I to a sequent Π actually is a family of sequents ( I(Π, n) ) n∈|I|. Then (in §14 of [Ge36]) for each Z-derivation d (whose endsequent is not an axiom) a reduction step on derivations, d (d[n])n∈|I|, is defined such that ∀n ∈ |I| ( End(d[n]) = I(End(d), n) ) , where I is a reduction step on sequents, uniquely determined by d. Here, in contrast to Gentzen, we also regard Rep as a reduction step on sequents — with |Rep| = {0} and Rep(Π, 0) = Π. The outline of the paper is as follows. In §1 and §2 we repeat relevant parts of [Ge36] using to a great extent Gentzen’s own words (in the translation by M. E. Szabo [Sz69]). Thereby we do not hesitate to deviate from the original text (in content or form) whenever we think it is appropriate or facilitates understanding. The main point where we deviate from [Ge36] (besides omitting conjunction &) is the following: In the reduction steps on sequents concerning an antecedent formula ∀xF or ¬A (13.51, 13.53) we always require that this formula is retained in the reduced sequent while Gentzen allows to omit it. As a consequence we also have to modify the reduction steps on atomic Z-derivations (which will be deferred till §5). In §3 we present the main definitions and proofs of §2 in a more condensed style (and with some further modifications). This facilitates the work in §4 where we assign to each Z-derivation d an ordinal o(d) < ε0 and prove that each reduction step on a derivation d lowers its ordinal, i.e. we prove that o(d[n]) < o(d) for all n ∈ |tp(d)|. Our ordinal assignment is essentially that of [KB81] which on first sight looks very different from Gentzen’s original assignment in [Ge36], where certain finite decimal fractions were used as notations for ordinals < ε0. But in the appendix we will show that actually both ordinal assignments are rather closely related. In §6 we give an interpretation of Z in an infinitary system Z∞. This way we obtain a semantic explanation for Gentzen’s reduction steps on Z-derivations and for the ordinal assignment of §4. Finally, in §7 we indicate how the approach of §§3,4 can easily be adapted to calculi with multisuccedent sequents.