Abstract
For f: On → On let supp(f): = ξ: 0, and let S := {f : On → On : supp(f) finite}. For f,g ϵ S define
f ≤ g : ↔ (∃h : On → On)[h one‐to‐one ⁁ (∀ξ)f(ξ) ≤ g(h(ξ))].
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 Cl~ψ~(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 = ω^n^]}. Let φ be the induced Schütte‐Klammerterm‐function (see [11]) which generates the Schütte‐Veblen‐hierarchy of ordinals. Then φ is monotonic increasing. We show: If ψ: S → On is monotonic increasing, then otyp(Cl~φ~(0)) ≤ min {ξ : ξ = Φ, where 1~ξ~(α) = 1 if α = ξ and 1~ξ~(α) = 0 if α ↔ = 0 if α ≠ ξ. For τ an ordinal let S~τ~ := {f ϵ S : supp(f) ⊆ τ}. A function ψ : S~τ~ → On is called τ‐monotonic increasing if __f__ξ (f) and if f ≤ g implies ψ (f) ≤ (g). For a function ψ S~τ~ → On let Cl~ψ~(0) ↔ τ be the least set T of ordinals, which contains 0 as an element, such that if f ϵ S~τ~ and range (f) ϵ T, then ψ (f) ϵ T. We show: if ≥2 and if ψ S~τ~ → On is τ‐monotonic increasing, then otyp(Cl~ψ~(0) ↔ τ) ≤(1~ψ~). We also prove a generalization of this theorem in terms of well‐partial orderings. MSC: 03F15, 03E10, 06A06.