Lower Bounds on $\beta (\alpha )$

Abstract

For an ordinal $\alpha$, an $\alpha$-ITRM is a machine model of transfinite computability that operates on finitely many registers, each of which can contain an ordinal $\rho <\alpha$; they were introduced by Koepke in [11]. In [4], it was shown that the $\alpha$-ITRM-computable subsets of $\alpha$ are exactly those in a level $L_{\beta (\alpha )}$ of the constructible hierarchy. It was conjectured in [4] that $\beta (\alpha )$ is the first limit of admissible ordinals above $\alpha$. Here, we show that this is false; in particular, even the computational strength of $\omega ^{\omega }$-ITRMs goes far beyond $\omega _{\omega }^{\text {CK}}$. To this end, we prove lower bounds on this computational strength, using a strategy for iterating $\alpha$-ITRM-computable operators for $\eta$ many steps on $\alpha ^{\eta }$-ITRMs.