# THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

@article{Fuchs2018THESA, title={THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL}, author={Gunter Fuchs and Ralf Schindler}, journal={The Journal of Symbolic Logic}, year={2018}, volume={83}, pages={920 - 938} }

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…

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The main result here is that if there is an inner model with a Woodin cardinal, then the solid core of a model of set theory is a fine-structural extender model.

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