Dependence of degrees of arbitrariness on filtration choices

Determine how the degrees of arbitrariness in filtered subdomains of the Boolean-valued direct-power model [?] depend on the choice of threshold, the notion of which values count as low, and additional closure conditions on the filtration, including rank cuts and size cuts based on the cardinality of transitive closures.

Background

The paper constructs filtered subdomains of the direct-power model of arbitrary sets by retaining all constant functions while restricting nonconstant functions to values above a chosen rank threshold. These subdomains continue to satisfy ZFC and can suppress crisp intermediate cardinals below the threshold while preserving such mixing above it.

The authors explicitly ask how sensitive the resulting degrees of arbitrariness are to the threshold and to alternative criteria for what counts as low. They mention replacing the rank cut with a size cut based on the cardinality of transitive closures and imposing stronger closure conditions, with the aim of understanding how comparison profiles, crisp comparability, and intermediate objects vary across filtrations.

References

This opens a wider family of questions: how sensitive are the resulting degrees of arbitrariness'' to the choice of threshold and, more importantly, to the choice of what counts aslow''? For instance, one may replace the rank cut $\mathcal{M}_{\kappa+1}$ with a size cut (e.g.\ requiring non-constant objects never to take values whose transitive closure has cardinality $< \kappa$ in $\mathcal{M}$), or one may demand stronger closure conditions on the filtered domain. A systematic study of such filtrations would clarify how profiles, crisp comparability, and the availability of intermediate objects depend on these structural choices.

— Models of Arbitrary Sets  (2610.23219 - Jockwich et al., 16 Oct 2026) in Section 4, subsection “Three further questions,” item (1); see also Section 3, subsection “A filtered ZFC submodel realizing the cut”