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arXiv · 1607.00684

Small Forcing Makes Any Cardinal Superdestructible

Joel David Hamkins
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Abstract

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $κ$ becomes superdestructible---any further ${<}κ$-closed forcing which adds a subset to $κ$ will destroy the measurability, even the weak compactness, of $κ$. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.

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Joel David Hamkins. 2016-07-03. Small Forcing Makes Any Cardinal Superdestructible. https://doi.org/10.2307/2586586

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