The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$

We show that, contrary to the commonly held view, there is a natural and optimal compactness theorem for $\mathrm{L}_{\infty\infty}$ which generalizes the usual compactness theorem for first order logic. The key to this result is the switch from Tarski semantics to Boolean valued semantics. On the way to prove it, we also show that the latter is a (the?) natural semantics both for $\mathrm{L}_{\infty\infty}$ and for $\mathrm{L}_{\inftyω}$.