Reduction Complexities in Set Theory

Abstract

In \cite{Ca2016} and \cite{Ca2018}, we introduced a notion of effective reducibility between set-theoretical $\Pi_{2}$-statements; in \cite{Ca2025}, 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 Stammes \cite{StammesMaster}, a considerably refined analysis through interpolating between these two notions was proposed, where one asks how many calls to an effectivizer for $\psi$ are required for effectivizing $\phi$. This allows us to 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.