Comparing "local compactness numbers" of two strong logics Given a logic $\mathcal{L}$ (together with a fixed encoding of $\mathcal{L}$-sentences by sets), let $\beta_\mathcal{L}$ be the smallest limit ordinal $\beta>\omega$ such that every $\mathcal{L}$-theory $T\subseteq L_\beta$ which is $\Sigma_1(L_\beta)$ and has the property that every subtheory which is an element of $L_\beta$ is satisfiable is itself satisfiable. For example, $\beta_{\mathsf{FOL}}=\omega\cdot 2$ by ordinary compactness and $\beta_{\mathcal{L}_{\infty,\omega}}=\omega_1^{CK}$ by Barwise-Kreisel compactness. The ordinal $\beta_\mathcal{L}$ always exists, regardless of how semantically nasty $\mathcal{L}$ is.
I'm curious about comparing the $\beta$-values of the two most natural (to my mind) logics besides first-order and basic infinitary logic - namely, "big" infinitary logic and second-order logic:

How do $\beta_{\mathcal{L}_{\infty,\omega_1}}$ and $\beta_\mathsf{SOL}$ compare?

Intuitively it seems clear that $\beta_{\mathsf{SOL}}$ should be gargantuan compared to $\beta_{\mathcal{L}_{\infty,\omega_1}}$. However, I can't actually show that at the moment. I suspect I'm just missing something obvious.
 A: Assume ZFC. Then $\beta_{\mathrm{SOL}}>\beta_{\mathcal{L}_{\infty,\omega_1}}$.
Lemma: For each limit ordinal $\beta$ there is a $\Sigma_1$ surjection $F_\beta:[\beta]^{<\omega}\to L_\beta$,  uniformly $\Sigma_1^{L_\beta}$ without parameters.
In fact, we can restrict the domain of $F_\beta$ to be sets of limit ordinals and integers.
(This is a standard fact.)
Claim 1: $\beta_{\mathrm{SOL}}\geq\beta_{\mathcal{L}_{\infty,\omega_1}}$.
Suppose otherwise. Then there is a finite set of limit ordinals $S\subseteq(\beta_{\mathrm{SOL}}+1)$ with $\beta_{\mathrm{SOL}}\in S$ and such that for each $\beta\in S$ there is a $\Sigma_1$ formula $\varphi$ such that $\varphi^{L_\beta}(S\cap\beta,\cdot)$ defines a theory
witnessing that $\beta<\beta_{\mathcal{L}_{\infty,\omega_1}}$. (Let $\beta_0=\beta_{\mathrm{SOL}}$. Let $S_0\subseteq\beta_0$ be a finite set of limit ordinals such that there is a $\Sigma_1$ formula $\varphi_0$
such that $\varphi_0(S_0,\cdot)$ works over $L_{\beta_0}$. Let $\beta_1=\max(S_0)$. Let $S_1\subseteq\beta_1$ be a finite set of limits and $\varphi_1$ be such that $\varphi((S_0\cup S_1)\cap\beta_1,\cdot)$ works over $L_{\beta_1}$. Let $\beta_2=\max((S_0\cup S_1)\cap\beta_1)$. Etc. Note that $\beta_0>\beta_1>\beta_2>\ldots$, so the process eventually stops with $S_n=\emptyset$. Set $S=S_0\cup S_1\cup\ldots\cup S_{n-1}$.)
Let $S=\{\beta_0>\beta_1>\ldots>\beta_n\}$ and let $\varphi_k$ be such that
$\varphi_k(S\cap\beta_k,\cdot)$ works over $L_{\beta_k}$ for each $k\leq n$.
Using the formulas $\varphi_0,\ldots\varphi_n$, we can write down a second-order statement whose unique model $M$ is $V_{\beta_0+\omega}$. For easily there is statement whose models are exactly those of the form $V_{\beta+\omega}$ with $\beta$ infinite (up to isomorphism). So assert this, and also that $M$ thinks that there is a sequence $\gamma_0>\ldots>\gamma_n$ of limit ordinals such that for each $k\leq n$, letting $T_k=\{\psi\bigm|L_{\gamma_k}\models\varphi_k(\{\gamma_{k+1},\ldots,\gamma_n\},\psi)\}$, then $T_k$ is not satisfiable, but for each $x\in L_{\gamma_k}$, $T\cap x$ is satisfiable.
Note that $V_{\beta+\omega}$ (with $\beta$ infinite) is always correct about satisfiability for $\mathcal{L}_{\infty,\omega_1}$, as a Lowenheim-Skolem argument shows. (The basic point is to consider hulls closed under $\omega$-sequences, and coding these to get them in $V_{\beta+\omega}$. Having enough of these uses some Choice, though.). So let $M=V_{\beta+\omega}$ be a model of these things. Then $\gamma_n=\beta_n$. For let $T^{\gamma_n}_n$ and $T_n^{\beta_n}$ be the interpretations of $\varphi_n(\emptyset,\cdot)$ over $L_{\gamma_n}$ and $L_{\beta_n}$ respectively. Then because $\varphi_n$ is $\Sigma_1$ and no parameter is used,
if $\gamma_n\leq\beta_n$ then $T_n^{\gamma_n}\subseteq T_n^{\beta_n}$,
and if $\beta_n\leq\gamma_n$ it is vice versa. And if $\gamma_n<\beta_n$ then $T_n^{\gamma_n}\in L_{\beta_n}$, and if $\beta_n<\gamma_n$ it is vice versa. But then the satisfiability/non-satisfiability of the various fragments (with correctness of $M$ regarding this) implies $\beta_n=\gamma_n$. Proceeding by inuction, at stage $k$ we know $\gamma_{k+1}=\beta_{k+1},\ldots,\gamma_n=\beta_n$, so $T_k^{\beta_k}$ and $T_k^{\gamma_k}$ are defined from the same parameter, and so we can argue otherwise like when $k=n$. So in particular we get $\gamma_0=\beta_0=\beta_{\mathrm{SOL}}$.
Now let $\psi$ be the (second order) statement used above to describe $V_{\beta_0+\omega}$. We now define a second order theory $T$ over $L_{\beta_0}$, with constant symbols $\alpha$ for each $\alpha<\beta_0$, plus two further constant symbols $\dot{\xi},\dot{\beta}$: $T$ asserts $\psi$ + "$\dot{\xi},\dot{\beta}$ are  ordinals" + "$\alpha$ is an ordinal" for each $\alpha<\beta_0$ + "$\alpha_0<\alpha_1<\dot{\xi}<\dot{\beta}$" for each $\alpha_0<\alpha_1<\beta_0$ + "$\dot{\beta}$ is the ordinal $\gamma_0$ specified by $\psi$". If $x\in L_{\beta_0}$ then $T\cap x$ is satisfiable, as $V_{\beta_0+\omega}$ models it (the point being that we interpret $\dot{\beta}$ as $\beta_0$ and $\dot{\xi}$ as some ordinal $<\beta_0$), but $T$ is not satisfiable (because there is no longer space left to interpret $\dot{\xi}$). It is also $\Sigma_1^{L_{\beta_0}}$.
This contradicts that $\beta_0=\beta_{\mathrm{SOL}}$.
Claim 2: $\beta_{\mathrm{SOL}}\neq\beta_{\mathcal{L}_{\infty,\omega_1}}$.
Since $\mathcal{L}_{\infty,\omega_1}$-satisfiability is absolute to $V_{\xi+\omega}$, note that there is a SOL-statement $\psi$ such that $V_{\beta+\omega}$ is the unique model of $\psi$, where $\beta=\beta_{\mathcal{L}_{\infty,\omega_1}}$.
But then we can use the same trick building a theory $\psi$ + "slightly too many ordinals $<\beta$" as at the end of the proof of Claim 1, for a contradiction.
It would be interesting to know whether one can get rid of the appeals to AC.
