Forcing categoricity at regular limit and singular cofinality ω cardinals

Establish whether there exists a forcing notion that forces, for a given cardinal κ, that every finite complete second‑order theory with a model of cardinality κ is categorical in the two remaining cases: (i) κ is a regular non‑measurable limit cardinal; or (ii) κ is a singular cardinal of cofinality ω.

Background

The paper proves that categoricity of all finitely axiomatizable complete second‑order theories can be forced at successor cardinals of regulars (Theorem 18) and at certain singular strong limits of uncountable cofinality (Theorem 19). These results rely on forcing definable well‑orders and related structural features.

However, the techniques do not extend to regular non‑measurable limit cardinals or to singular cardinals of cofinality ω, leaving the existence of suitable forcing to achieve categoricity in these cases as an open problem.