Local Computability for Ordinals

Abstract

We examine the extent to which well orders satisfy the properties of local computability, which measure how effectively the finite suborders of the ordinal can be presented. Known results prove that all computable ordinals are perfectly locally computable, whereas $\omega_1^\mathrm{CK}$ and larger countable ordinals are not. We show that perfect local computability also fails for uncountable ordinals, and that ordinals $\alpha\geq \omega_1^\mathrm{CK}$ are θ-extensionally locally computable for all $\theta<\omega_1^\mathrm{CK}$, but not when $\theta>\omega_1^\mathrm{CK}$, nor when $\theta=\omega_1^\mathrm{CK}\leq\alpha<\omega_1^\mathrm{CK}\cdot\omega$.