Does Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$?

Question

Does descendant or ascendant Löwenheim-Skolem fail for $\mathcal{L}_{\omega_1\omega}$ -logic?

Answer 1

The downward Löwenheim-Skolem theorem continues to hold. The upward one does not; for example the infinitary formula $\bigvee_{n \in \omega} (x = n)$ has no uncountable models. This implies that the compactness theorem also fails, because it implies the upward Löwenheim-Skolem theorem.

This is all described in the usual references. You might start with the article on Infinitary logic at the Stanford Encyclopedia.

Answer 2

I think, if ultrafilters everywhere in the proof are replaced by $\omega_1$-ultrafilters (those which are closed under intersection of $\omega$ many sets), then it is still valid in that form.