Boolean valued semantics for infinitary logics
It is well known that the completeness theorem for $\mathrm{L}_{ω_1ω}$ 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 $\mathrm{L}_{\inftyω}$). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for $\mathrm{L}_{\inftyω}$ 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 $\mathrm{L}_{\inftyω}$ and the forcing method in set theory.