Boolean valued semantics for infinitary logics

It is well known that the completeness theorem for L_{omega_1,omega} fails with respect to Tarski semantics. Mansfield showed that it holds for L_{infinity,infinity} 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 L_{infinity,omega}). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for L_{infinity,omega} with respect to Boolean valued semantics. We also show that a weak version of these results holds for L infinity infinity (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 L infinity omega and the forcing method in set theory.(c) 2023 Elsevier B.V. All rights reserved.