A Laver-like indestructibility for hypermeasurable cardinals

Archive for Mathematical Logic 58 (3-4):275-287 (2019)
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Abstract

We show that if \ is \\)-hypermeasurable for some cardinal \ with \ \le \mu \) and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model \ in which the \\)-hypermeasurability of \ is indestructible by the Cohen forcing at \ of any length up to \ is \\)-hypermeasurable in \). The preservation of hypermeasurability is useful for subsequent arguments. The construction of \ is based on the ideas of Woodin and Cummings :1–39, 1992) for preservation of measurability, but suitably generalised and simplified to achieve a more general result. Unlike the Laver preparation :385–388, 1978) for a supercompact cardinal, our preparation non-trivially increases the value of \, which is equal to \ in \ is still true in \ if we start with GCH).

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References found in this work

The lottery preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
The negation of the singular cardinal hypothesis from o(K)=K++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.
Aronszajn trees on ℵ2 and ℵ3.Uri Abraham - 1983 - Annals of Mathematical Logic 24 (3):213-230.

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