The Boolean compactness theorem for L ∞ ∞ $\mathrm{L}_{\infty \infty }$

Abstract

We show that, contrary to the commonly held view, there is a natural and optimal compactness theorem for $L_{\infty \infty }$ 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 $L_{\infty \infty }$ and for $L_{\infty \omega }$.