Given a logic $\mathcal{L}$ and a signature $\Sigma$, let the $\Sigma$-system of $\mathcal{L}$ be the pair $Sys_\Sigma(\mathcal{L})=(Struc(\Sigma),\preccurlyeq_\mathcal{L})$ of structures partially ordered by the $\mathcal{L}$-elementary substructure relation. This is sometimes, but not always, an abstract elementary class; the first four AEC axioms always hold, but the Tarski-Vaught or Lowenheim-Skolem axioms could easily fail. Say that a logic yields AECs iff its $\Sigma$-system is an AEC for every signature $\Sigma$. Examples of logics which yield AECs include first-order logic and the "small" infinitary logic $\mathcal{L}_{\omega_1,\omega}$, while the most natural examples of logics not yielding AECs in my opinion are second-order logic $\mathsf{SOL}$ and the "big" infinitary logic $\mathcal{L}_{\omega_1,\omega_1}$.

Quick caveat: note that the Lowenheim-Skolem axiom for AECs is stronger than what one might expect coming from non-AEC land. Also, the Tarski-Vaught axiom has nothing to do with the Tarski-Vaught test.

Now suppose we have a logic which doesn't yield AECs, but we want to find a reasonably large and canonically definable fragment of it which does. There's a trick which seems promising, at least as far as the Tarski-Vaught axiom goes. For a logic $\mathcal{J}$, let $\hat{\mathcal{J}}$ be the set of $\mathcal{J}$-formulas stable under elementary chains, that is, the set of $\varphi\in\mathcal{J}$ such that whenever $(\mathfrak{A}_i)_{i<\eta}$ are structures with $\mathfrak{A}_i\preccurlyeq_\mathcal{L}\mathfrak{A}_j$ for all $i<j<\eta$ we have $\varphi^{\mathfrak{A}_0}=\varphi^{\bigcup_{i<\eta}\mathfrak{A}_i}\cap\mathfrak{A}_0^{arity(\varphi)}$. Now given a logic $\mathcal{L}$ we can iterate this process through the ordinals, starting with $\mathcal{L}$ itself and taking intersections at limit stages as usual; we eventually hit a fixed point $\mathcal{L}^*$, and it's easy to show that $(i)$ for each $\Sigma$, if $Sys_\Sigma(\mathcal{L}^*)$ satisfies the Lowenheim-Skolem axiom then it is an AEC and $(ii)$ every sublogic of $\mathcal{L}$ which yields AECs is a sublogic of $\mathcal{L}^*$.

I'd like to know what happens to the two most natural logics, in my mind, which fail to yield AECs:

Does $\mathsf{SOL}^*$ yield AECs? Does $\mathcal{L}_{\omega_1,\omega_1}^*$ yield AECs?

I strongly suspect the answer to the first question is negative: failures of the Lowenheim-Skolem axiom for $Sys_\emptyset(\mathsf{SOL})$ are easy to come by, e.g. there is a second-order sentence true in exactly those structures of infinite limit cardinality. However, while there are lots of such failures, each one I can think of goes away when we pass to $\mathsf{SOL}^*$. So I don't immediately see an obstacle to $\mathsf{SOL}^*$ yielding AECs.

I'm much more ambivalent about $\mathcal{L}_{\omega_1,\omega_1}^*$. Tentatively I'll suspect that it does, but I really don't know.



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