Large-cardinal principles expressible through arbitrary objects

Determine which large-cardinal principles can be expressed in terms of arbitrary objects and which large-cardinal properties correspond to basic principles governing arbitrary objects.

Background

The paper proves a specific connection between arbitrary objects and measurable cardinals: under a hierarchy-recovery condition, an ultrafilter realizes an unbounded arbitrary object such as the identity function as an ordinary set in a larger well-founded elementary extension. The recovery condition characterizes measurability in the construction considered.

The authors propose investigating whether this connection extends beyond measurable cardinals. The unresolved direction has two parts: identifying large-cardinal principles definable through arbitrary-object constructions, and determining which large-cardinal properties are reflected by basic tenets of arbitrary-object theory.

References

On the one hand we could ask which other large cardinals we can express in terms of arbitrary objects, while on the other hand we could ask to which large cardinals properties correspond basic tenets of arbitrary objects.

— Models of Arbitrary Sets  (2610.23219 - Jockwich et al., 16 Oct 2026) in Section 4, subsection “Three further questions,” item (3); see also Section 3, subsection “Arbitrary sets and measurable cardinals”