In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of size at most $\aleph_1$. (only countable signatures are considered)
It is then asked whether in fact the stronger result holds, that every structure has an elementary substructure of size at most $\aleph_1$.
In a footnote on p.222 it is mentioned that this question depends on set theory: "If V = L, then there is a structure of size $\omega_2$ with no L(aa)-elementary submodel of power $\omega_1$"
I'm looking for a reference where this argument is worked out in more detail or if it is not too long maybe someone here can give me a more detailed argument.
Edit: L(aa) arises from first order logic in the following way: We add new variables $X_1,X_2,\dots$ for unary countable relations, which lead to new atomic formulas $Xt$ for a first order term t. Now the formation rules for formulas are that of first order logic together with the new quantifier (aa) which binds one relation variable, i.e if $\varphi$ is a formula, so is $(aa)X_i\varphi$ the meaning of which is defined by $$\mathfrak{A}\models(aa)X_i\varphi \Leftrightarrow \{R\in P_{\omega_1}(A)|\mathfrak{A}\models \varphi[R]\}\in D(A)$$ where $D(A)$ is the club filter on $P_{\omega_1}(A)$. So aa says for almost all relations.