I know that Cohen showed the existence of a set between N and R using forcing, which required the assumption of a large cardinal axiom such as the existence of a Ramsey cardinal, and thereby showed that under this model, the universe is not constructable. However, is it possible to go in the other direction and, from the assumption that the universe is not constructable, but without assuming such a large cardinal axiom (although not, obviously, assuming its negation), that the Continuum Hypothesis is false? In fact, is it true? That is, is a model possible whereby V=L is only violated after P(N)?