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THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL
Published online by Cambridge University Press: 23 October 2018
Abstract
It is shown that 
$K|{\omega _1}$ 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 
$K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if 
${0^{\rm{\P}}}$ does not exist and 
$\kappa \ge {\omega _2}$ is a cardinal, then 
$K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of 
$\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.
It is also shown, assuming 
${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves 
${\omega _1}$, forces that every real has a sharp, and increases 
$\delta _2^1$, then 
${\omega _1}$ is measurable in K.
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References
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