Core models in the presence of Woodin cardinals

In this paper we show that the core model might exist even if there are Woodin cardinals in V . This observation is not new. Woodin [8], in his proof that ADR implies the ADR hypothesis, constructed models of ZFC in which there are fully iterable extender models with Woodin cardinals which satisfy (among other things) a weak covering property. Steel [7], in his proof that Mn satisfies V = HOD, gave an argument which appears to be a special case of what we shall do in this paper. However, the general method for constructing the core model in the theory ZFC + “there is a measurable cardinal above n Woodin cardinals” + “M n+1 does not exist” which we shall present here does seem to be new. This method might in turn admit generalizations, but we do not know how to do it. We shall indicate that our method might have applications; we shall prove that if M is an ultrapower of V by an extender with countably closed support then K is an iterate of K (although K might not be fully iterable).