Complexity of Effective Reductions with Ordinal Turing Machines

Abstract: In arXiv:1811.11630, we introduced a notion of effective reducibility between set-theoretical $\Pi_{2}$-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 $\psi$ in a reduction of $\phi$ to $\psi$. In this paper, we refine this notion considerably by asking how many calls to an effectivizer for $\psi$ are required for effectivizing $\phi$. 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.