Abstract
It is well known that the completeness theorem for Lω1ω fails with respect to Tarski semantics. Mansfield showed that it holds for L∞∞ 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∞ω). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for L∞ω with respect to Boolean valued semantics. We also show that a weak version of these results holds for L∞∞ (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∞ω and the forcing method in set theory.
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