Another Reduction of Classical Id Ν to Constructive Id

One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction, resp.). Only quite recently Avigad and Towsner [AT09] succeeded in giving a reduction of classical iterated ID theories to constructive ones by the method of functional interpretation. For a thorough exposition and discussion of all this cf. [Fef]. In the present paper we give yet another reduction of classical IDν to IDν(W) based on cut-elimination arguments. W is a particularly simple accessibility ID; its corresponding operator form W(P,Q, y, x) (cf. [BFPS]) has the shape A(x, y)∧∀z(Q̃(t(x), z)→ Pq(x, z)) with primitive recursive A, t, q, and Q̃(u, z) :≡ u ≥ 1 ∧ (u ≥ 2→ Q(u−· 2, z)). There are two reasons which, as we hope, justify a publication of this additional proof. First, it is considerably more direct then all the existing ones. Second, the method used here stems to a great extent from [Ge36] and therefore may be interesting for historical reasons too. Actually I have used a variant of this method under the label “notations for infinitary derivations” already in several papers (e.g. [Bu91], [Bu97], [Bu01]) without mentioning its close relation to [Ge36]. When writing [Bu91] I was definitely not aware of this connection; but cf. [Bu95]. The method from [Ge36] can be roughly described as follows: To each derivation d in a certain suitably designed proof system Z of first order arithmetic a family (d[i])i∈Id of Z-derivations is assigned such that . . .Γ(d[i]) . . . (i ∈ Id) Γ(d) forms an inference in cutfree ω-arithmetic (with