Existence of ω-measurable cardinals

Determine whether any ω‑measurable cardinal exists; equivalently, ascertain whether the class L1 of ω‑measurable cardinals is empty. Here, a cardinal κ is ω‑measurable if there is a nonprincipal ultrafilter on κ that is ω‑closed (i.e., closed under countable intersections), and L1 denotes the class of all such cardinals.

Background

The paper introduces classes of large cardinals: L1 (ω‑measurable cardinals), L2 (measurable cardinals), and L3 (strongly inaccessible cardinals), and notes the inclusions L2 ⊆ L1 and L2 ⊆ L3. An ω‑measurable cardinal is defined via the existence of a nonprincipal ultrafilter that is closed under countable intersections.

This question is significant for the paper because several results restrict cardinalities by excluding ω‑measurability, and the existence or nonexistence of ω‑measurable cardinals influences the scope of these restrictions within ZFC.