TITLE: Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?
QUESTION [11 upvotes]: Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ unary relation symbols and an $\mathcal{L}[\Sigma']$-sentence $\eta$ such that the following are equivalent for all $\Sigma$-structures $\mathfrak{A},\mathfrak{B}$:

$\mathfrak{A}\equiv_\mathcal{L}\mathfrak{B}$.

There is a $\Sigma'$-structure $\mathfrak{S}$ such that $\mathfrak{S}\models \eta$, $A^\mathfrak{S}\upharpoonright\Sigma\cong\mathfrak{A}$, and $B^\mathfrak{S}\upharpoonright\Sigma\cong\mathfrak{B}$.


For example, Fraisse showed that $\mathsf{FOL}$ is SED (this is used crucially in the proof of Lindstrom's theorem - it's a very happy construction). That same argument gives as a corollary that $\mathsf{SOL}$ is also SED, roughly because $(i)$ "$X$ is the powerset of $Y$" is second-order expressible and $(ii)$ $\mathsf{SOL}$-elementary equivalence between two structures amounts to $\mathsf{FOL}$-elementary equivalence between their "power-structures."
The nicest logic whose SED status I don't know is $\mathcal{L}_{\omega_1,\omega}$. This logic isn't powerful enough to perform the same sort of "cheat" as $\mathsf{SOL}$; on the other hand, the direct game-theoretic attack analogizing the situation for $\mathsf{FOL}$ results in a game which is a bit too complicated for $\mathcal{L}_{\omega_1,\omega}$ to handle appropriately (see here, and note that this was also an issue in this earlier question of mine). So my question is:

Is $\mathcal{L}_{\omega_1,\omega}$ SED?

I'm separately interested in the general situation of which infinitary logics are SED. It's not hard to show that for $\kappa$ satisfying appropriate large cardinal properties we have that $\mathcal{L}_{\kappa,\omega}$ is SED, but I don't see how to extract large cardinal strength from SEDness.
Note (per Peter LeFanu Lumsdaine's comment below) that looking at a single sentence $\eta$, as opposed to a theory $E$, is needed to avoid every (regular) logic being trivially SED; I messed this up in the original version of this question.

REPLY [11 votes]: (Working in ZFC.)
No (re $\mathcal{L}_{\omega_1,\omega}$). Suppose it is. Consider the signature $\Sigma$ with just one binary relation symbol $<$. Let $\Sigma',\eta$ witness SED-ness for $\Sigma$.
Let $\pi:M\to V_\eta$ be elementary,with $\eta$ some sufficiently large limit ordinal, $M$ transitive, $M$ of cardinality $\kappa=2^{\aleph_0}$, with $\mathbb{R},2^{\aleph_0}\subseteq M$. So $\mathrm{crit}(\pi)=\kappa^{+M}<\kappa^+$ and $\pi(\kappa^{+M})=\kappa^+$. Let $\mathfrak{A}=(\kappa^{+M},{\in}\upharpoonright\kappa^{+M})$ and $\mathfrak{B}=(\kappa^+,{\in}\upharpoonright\kappa^+)$. Note that $\mathfrak{A}\equiv_{\mathcal{L}}\mathfrak{B}$, since $\pi$ is elementary and all the sentences in $\mathcal{L}$ are in $M$, since $\mathbb{R}\subseteq M$. So by hypothesis, there is some $\Sigma'$-structure $\mathfrak{G}$ such that $\mathfrak{G}\models\eta$, $A^{\mathfrak{G}}\upharpoonright\Sigma\approx\mathfrak{A}$ and $B^{\mathfrak{G}}\upharpoonright\Sigma\approx\mathfrak{B}$.
Now let $G$ be $V$-generic for $\mathrm{Coll}(\omega,\max(\kappa^{+},|\mathfrak{G}|))$.
In $V[G]$, there is an $\mathcal{L}_{\omega_1,\omega}$ sentence $\varphi$ in the signature $\Sigma$ asserting that the model is (isomorphic to) $\kappa^{+M}$. So in $V[G]$, $\mathfrak{A}\models\varphi$ but $\mathfrak{B}\models\neg\varphi$. So $V[G]$ models the statement "there are countable structures $\mathfrak{A}_0,\mathfrak{B}_0$ in the signature $\Sigma$ and an $\mathcal{L}_{\omega_1,\omega}$-sentence $\varphi_0$ such that $\mathfrak{A}_0\models\varphi_0$ and $\mathfrak{B}_0\models\neg\varphi_0$
and there is a countable structure $\mathfrak{G}_0$ in the signature $\Sigma'$ such that $\mathfrak{G}_0\models\eta$ and $A^{\mathfrak{G}}\upharpoonright\Sigma\approx\mathfrak{A}_0$ and $B^{\mathfrak{G}}\upharpoonright\Sigma\approx\mathfrak{B}_0$". But this statement is $\Sigma^1_2$ in a real coding $\eta$. (I don't see that it is $\Sigma^1_1$ in such a real, because to assert that $\varphi_0$ is really a sentence of $\mathcal{L}_{\omega_1,\omega}$ involves saying that it is built along a real ordinal.) Since $\eta\in V$ and by Shoenfield absoluteness, $V$ models the same statement. But this contradicts our assumptions.