Abstract

Given an inner model and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model. The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from V to fail to be as strong as that from W. The results are generalized to, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.