Abstract

We prove a theorem (1.7) about partial orders which can be viewed as a version of the Barwise compactness theorem which does not mention logic. The Barwise compactness theorem is easily equivalent to 1.7 + "Every Henkin set has a model". We then make the observation that 1.7 gives us the definability of forcing for quantifier-free sentences in the forcing language and use this to give a direct proof of the truth and definability lemmas of forcing