Suppose $\kappa$ is a supercompact cardinal. Is it possible to find a forcing which collapses $\kappa^{+\omega+1}$ to $\kappa^{+\omega}$ (all those $\kappa^{+n}$'s are preserved) while the supercompactness of $\kappa$ is preserved?
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1$\begingroup$ Suppose that there is a Woodin cardinal above $\kappa$, can you do some tricks to ensure that the stationary tower will preserve the supercompactness? $\endgroup$– Asaf Karagila ♦Commented Sep 2, 2020 at 15:07
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$\begingroup$ @AsafKaragila, I doubt it. Isn't it the case that such a forcing, let's say $P_\lambda$ always singularize some inaccessible $\theta$ which lives in between $\kappa$ and $\lambda$ to be of confinality $\omega$? $\endgroup$– Jiachen YuanCommented Sep 3, 2020 at 2:12
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1$\begingroup$ @JiachenYuan Yes that should be true. But it’s possible that you need to add the full proper-class sized generic to restore supercompactness. $\endgroup$– Monroe EskewCommented Sep 3, 2020 at 9:53
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2$\begingroup$ The point is that $\kappa$ is supercompact in the generic ultrapower by elementarity, but with the proper-class tower, the ultrapower is the generic extension. $\endgroup$– Monroe EskewCommented Sep 3, 2020 at 9:57
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1$\begingroup$ @AsafKaragila Paul Larson’s book. $\endgroup$– Monroe EskewCommented Sep 3, 2020 at 16:03
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