THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

It is shown that K|ω₁ need not be solid in the sense previously introduced by the authors: 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, previously introduced by the authors. 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 2 1 , then ω₁ is measurable in K.