"Local" compactness properties beyond $\mathcal{L}_{\omega_1,\omega}$?

Question

Below, all languages are finite for simplicity.

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This question is about generalizations of Barwise compactness for logics more complicated than $\mathcal{L}_{\omega_1,\omega}$: properties of the form "For every 'small' $\mathcal{L}$-theory $T$, if every 'relatively simple' $D\subseteq T$ is satisfiable then $T$ is satisfiable" (which I'll call "local compactness properties").

Both classical and Barwise compactnesses fit this paradigm: in classical compactness "small" means arbitrary and "relatively simple" means finite, and in Barwise compactness "small" means countable and $D$ is "simple relative to $T$" iff $D\in \mathbb{A}$ for every admissible structure $\mathbb{A}$ over which $T$ is $\Sigma_1$-definable (or r.e.).

Now classical compactness generalizes easily to appropriate higher logics, but Barwise compactness is more finicky - in particular, the "definition" of local compactness properties above is horribly vague and I see no natural precisiation of it. I'm interested in pinning down exactly what "local compactness property" ought to mean in whatever its appropriate general context is.

Towards that end, I'd like to put the cart ahead of the horse and ask:

• Are there (reasonably natural) examples of logics strictly stronger than $\mathcal{L}_{\omega_1,\omega}$ which satisfy some local compactness property?

Of course this is necessarily vague, but I hope the notion of local compactness property above is sufficiently clear that this will admit a non-silly answer. I'm especially interested in logics which contain $\mathcal{L}_{\omega_2,\omega}$ and in local compactness properties which are "computability-flavored," but I'll be happy with anything.

Answer 1

Interestingly, the logic $\mathcal{L}_{\infty,\omega_1}$ - despite characterizing well-foundedness and so being "wild" in most senses - does have such a local compactness property! This was proved by S.-D. Friedman, in Model theory for $\mathcal{L}_{\infty,\omega_1}$. Unsurprisingly, it is significantly more technical than Barwise compactness (the following is essentially verbatim Theorem 11 in Friedman's paper):

If $\alpha$ is $\omega$-admissible and has cardinality $\aleph_1$ and $T\subseteq\mathcal{L}_{\alpha,\omega_1}$ is an effectively scattered $\Sigma_1\langle L_\alpha, \lambda x.x^\omega\rangle$-theory such that each $\alpha$-finite subtheory of $T$ has an $\omega$-closed model, then $T$ has an $\omega$-closed model $\mathfrak{M}$ with $0_\omega(\mathfrak{M})\le\alpha$.

All definitions can be found in Friedman's paper, and putting them all here would be inconvenient. I will however give the definition of $\omega$-admissible and $0_\omega$:

• An admissible set $A$ is $\omega$-admissible iff $A$ is closed under and admissible with respect to the function $\lambda x. x^\omega$.

• $0_\omega(\mathfrak{M})=\mathsf{HYP}(\mathfrak{M}^{<\omega_1})\cap Ord$.

The really new ingredient here is effective scatteredness, which is essentially a complicated "few types" condition.