Can I say something about the cadinality of this model?

Question

Let $L$ be some first-order language. Suppose $A$ is existentially closed in $K$, a class of $L$-structures whose age is at most countable, and age($A$) is at most countable set . Can we say anything about $A$'s cardinality?

Answer 1

Not without more information. Here's a trivial example: Let $L$ be the language with no symbols other than equality, and $K$ the class of infinite sets. Every element of $K$ is existentially closed with countable age (consisting of one representative for each finite cardinality).

I'm assuming that by $\text{age}(A)$, you mean the set of finite substructures up to isomorphism. If you mean to take the set of finite substructures of $A$, not identified up to isomorphism, then only a countable structure will have countable age.