Sums of Computable Ordinals

Abstract

We prove that for any computable ordinal α there exists a notation a ∈ O and a partial computable function possessing the following property: if b and c are given notations in the set {t ∈ O | t <O a} for ordinals β and γ respectively and β +γ < α, then this function finds a notation for β+γ in the same set. We construct examples demonstrating that not all notations for ordinals α ≥ ω2 possess this property. Bibliography: 2 titles.