Taming Koepke's zoo. II: Register machines (Q2067510)

This paper extends some previous results of its author [the author et al., Lect. Notes Comput. Sci. 10936, 126--135 (2018; Zbl 1436.03236)] on \(\alpha\)-ITRMs, register machines in which each register may hold an ordinal below the ordinal \(\alpha\). For exponentially closed \(\alpha\), the author examines the collection of \(\alpha\)-ITRM-computable subsets of \(\alpha\). This collection is shown to consist of the subsets of \(\alpha\) in \(L_{\alpha+1}\) iff \(\alpha\) models ZF\({}^-\); otherwise, the collection consists of the subsets of \(\alpha\) in \(L_\beta\), where \(\beta\) is the sup of the \(\alpha\)-ITRM-computable ordinals.