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

TL;DR

This paper establishes a natural and optimal compactness theorem for the logic $ ext{L}_{ ext{∞} ext{∞}}$, generalizing the classical first-order compactness theorem by using Boolean valued semantics.

Contribution

It introduces a new compactness theorem for $ ext{L}_{ ext{∞} ext{∞}}$ based on Boolean valued semantics, challenging the traditional view and showing its naturalness for this logic.

Findings

Boolean valued semantics is natural for $ ext{L}_{ ext{∞} ext{∞}}$ and $ ext{L}_{ ext{∞} ext{ω}}$

A generalization of the compactness theorem for $ ext{L}_{ ext{∞} ext{∞}}$

Switching semantics enables the new compactness result

Abstract

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 $\mathrm{L}_{\infty\omega}$.