Boolean valued semantics for infinitary logics

TL;DR

This paper improves boolean valued semantics for infinitary logic $ ext{L}_{}$, establishing stronger completeness, interpolation, and omitting types results, and explores connections with set-theoretic forcing.

Contribution

It introduces a stronger form of boolean completeness for $ ext{L}_{}$ using forcing and derives related model-theoretic properties, linking infinitary logic with set theory.

Findings

Established boolean completeness for $ ext{L}_{}$ using forcing.

Proved Craig interpolation and omitting types theorems under boolean semantics.

Connected infinitary logic with forcing methods in set theory.

Abstract

It is well known that the completeness theorem for $\mathrm{L}_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces Tarski semantics with boolean valued semantics. We use forcing to improve his result in order to obtain a stronger form of boolean completeness (but only for $\mathrm{L}_{\infty\omega}$). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for $\mathrm{L}_{\infty\omega}$ with respect to boolean valued semantics. We also show that a weak version of these results holds for $\mathrm{L}_{\infty\infty}$ (if one leverages instead on Mansfield's completeness theorem). Furthermore we bring to light (or in some cases just revive) several connections between the infinitary logic $\mathrm{L}_{\infty\omega}$ and the forcing method in set theory.