A Note on Power-OTMs

TL;DR

This paper investigates the computational capabilities of Power-OTMs, a type of ordinal Turing machine with a power set operator, and explores their relation to set register machines and implications for the axiom of choice.

Contribution

It establishes the equivalence of Power-OTMs with set register machines under certain conditions and analyzes their differing computational strengths in various set-theoretical contexts.

Findings

Power-OTMs are equivalent to set register machines with parameters and a global well-ordering.

Results on realizability of Power-OTMs are similar to Passmann's but differ in the context of the axiom of choice.

The computational strength of Power-OTMs varies depending on the set-theoretical background.

Abstract

We consider the computational strength of Power-OTMs, i.e., ordinal Turing machines equipped with a power set operator, and study a notion of realizability based on these machines. When parameters are allowed, these machines are, modulo access to a global well-ordering, equivalent to the Set Register Machines defined by Robert Passmann in \cite{Passmann}, and while most of the results on the realizability of Power-OTMs are analogous to results obtained by Passmann, the settings lead to different results concerning the axiom of choice. As we will see, the computational strength of power-OTMs can, depending on the set-theoretical background, also differ from that of Set Register Machines.