Abstract#R##N##R##N#For f: On On let supp(f): = ξ: 0, and let S := {f : On On : supp(f) finite}. For f,g ϵ S define#R##N##R##N##R##N##R##N#f ≤ g : (∃h : On On)[h one-to-one ⁁ (∀ξ)f(ξ) ≤ g(h(ξ))].#R##N##R##N##R##N##R##N#A function ψ : S On is called monotonic increasing, if f(ξ)≤ψ(f) and if f ≤ g implies ψ (f) ≤ ψ(g). For a mapping ψ : S On let Clψ(0) be the least set T of ordinals which contains 0 as an element and which is closed under the following rule: If f ϵ S, range(f) ⊆ T and supp(f) ⊆ T, then ψ(f) ϵ T. Let φ be the enumeration function of the class {ξ: (∃n)[ξ = ωn]}. Let φ be the induced Schutte-Klammerterm-function (see [11]) which generates the Schutte-Veblen-hierarchy of ordinals. Then φ is monotonic increasing. We show: If ψ: S On is monotonic increasing, then otyp(Clφ(0)) ≤ min {ξ : ξ = Φ, where 1ξ(α) = 1 if α = ξ and 1ξ(α) = 0 if α = 0 if α ≠ ξ. For τ an ordinal let Sτ := {f ϵ S : supp(f) ⊆ τ}. A function ψ : Sτ On is called τ-monotonic increasing if fξ (f) and if f ≤ g implies ψ (f) ≤ (g). For a function ψ Sτ On let Clψ(0) τ be the least set T of ordinals, which contains 0 as an element, such that if f ϵ Sτ and range (f) ϵ T, then ψ (f) ϵ T. We show: if ≥2 and if ψ Sτ On is τ-monotonic increasing, then otyp(Clψ(0) τ) ≤(1ψ). We also prove a generalization of this theorem in terms of well-partial orderings. MSC: 03F15, 03E10, 06A06.
['Andreas Weiermann']
An order-theoretic characterization of the Schütte-Veblen-Hierarchy