Let $K=\{A_i:i\in\omega\}$ be a countable chain of infinite (not necessarily countable) N−substructures, where N is a binary relation and let A be the limit (union) of K.
Let Ax be a $\Pi_2$ sentence containing, in addition to N, also a unary predicate V (occurring both positively and negatively) and such that every $A_i$ can be made into its model (by interpreting V appropriately). When can also A be made into a model of Ax? Put differently, under what conditions is also A a model of the $\Sigma^1_1$ sentence: $\exists V Ax$ ?
One "obvious" case is when K can be made into a chain of (N,V)-substructures so that all are models of Ax (since $\Pi_2$ sentences are preserved in unions of chains). Are there any other sufficient conditions (on Ax or K, N, or all)?