Complexity of Effective Reductions with Ordinal Turing Machines
Merlin Carl
TL;DR
This paper refines the notion of transfinite reductions by introducing f-bounded OTM-reducibility, distinguishing the number of calls to an effectivizer in ordinal computations. It develops a framework linking various function types (PowerCard, Pot, DecCard, OrdCard) and truth/separation predicates, and analyzes concrete reductions such as PowerCard from Pot and NextCard to DecCard. The results reveal independence phenomena and model-dependent behaviours (e.g., in $L$ vs. models with $0^{\#}$), and establish bounds and forcing arguments that delineate what can and cannot be achieved with restricted call-usage. Collectively, the work opens a pathway to quantify and compare the complexity of transfinite mathematical principles under different computation models and paves the way for exploring other transfinite computational paradigms.
Abstract
In arXiv:1811.11630, we introduced a notion of effective reducibility between set-theoretical $Π_{2}$-statements; in arXiv:2411.19386, this was extended to statements of arbitrary (potentially even infinite) quantifier complexity. We also considered a corresponding notion of Weihrauch reducibility, which allows only one call to the effectivizer of $ψ$ in a reduction of $φ$ to $ψ$. In this paper, we refine this notion considerably by asking how many calls to an effectivizer for $ψ$ are required for effectivizing $φ$. This allows us make formally precise questions such as ``how many ordinals does one need to check for being cardinals in order to compute the cardinality of a given ordinal?'' and (partially) answer many of them. Many of these anwers turn out to be independent of ZFC.