Combinatorial characterization of pretameness for class symmetric systems

Identify a combinatorial property on class symmetric systems that is equivalent to pretameness; namely, determine a precise combinatorial criterion that holds exactly for those class symmetric systems that admit forcing relations and preserve the Separation and Collection axioms of Gödel–Bernays set theory without the axiom of choice.

Background

The paper develops a general theory of class-sized symmetric systems, introducing analogues of pretameness and tameness from class forcing to control when forcing relations exist and when axioms of Gödel–Bernays set theory are preserved. Pretameness for class symmetric systems is defined semantically as admitting forcing relations and preserving Separation and Collection.

While the authors provide sufficient conditions—combinatorial pretameness together with stratification—to guarantee pretameness, they note that this combination is not necessary. Unlike classical class forcing, for which pretameness has a known combinatorial characterization on class partial orders, an exact combinatorial characterization for class symmetric systems is currently unknown.