ZFC proof of Rubin elementary extensions

Determine whether ZFC alone proves that every countable structure in a countable first-order language has an elementary extension of cardinality ℵ1 that is a Rubin model—namely, a model in which every definable directed poset without a maximum has a cofinal ω-chain and any maximal filter on a definable poset with a cofinal ω-chain is coded in the model.

Background

The paper proves Theorem 5.15 under the additional combinatorial principle ♦(ω1), asserting that every countable structure in a countable language has an elementary extension of cardinality ℵ1 that is a Rubin model. Rubin models ensure strong definability properties for chains and filters on definable posets.

In contrast, the paper establishes in Theorem A.1 (and Theorem 5.17) that ZFC suffices to obtain weakly Rubin models as elementary extensions. The author explicitly notes uncertainty about whether the full Rubin result can be achieved in ZFC without ♦(ω1), posing it as an unresolved issue.