THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL. (23rd October 2018)

• Record Type: Journal Article
• Authors: FUCHS, GUNTER; SCHINDLER, RALF
• Is Part Of: Journal of symbolic logic. Volume 83:Number 3(2018)
• Page Start: 920
• Page End: 938
• Publication Date: 2018-10-23
• Subjects: 03E40, -- 03E45, -- 03E55, -- 03E57; core model theory, -- forcing, -- solidity; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2018.45
• Languages: English
• ISSNs: 0022-4812

Abstract

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 .