The solidity and nonsolidity of initial segments of the core model

It is shown that K|ω1 need not be solid in the sense of [FS]: It is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that K|ω1 ∈ W [x] \ W . However, if 0¶ does not exist and κ ≥ ω2 is a cardinal, then K|κ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of ¬0¶, the core model is contained in the solid core, introduced in [FS]. It is also shown, assuming 0¶ does not exist, that if there is a forcing that preserves ω1, forces that every real has a sharp, and increases δ 1 2, then ω1 is measurable in K.