A Laver-like indestructibility for hypermeasurable cardinals

Abstract

We show that if $\kappa$ is $H(\mu )$-hypermeasurable for some cardinal $\mu$ with $\kappa < \mathrm {cf}(\mu ) \le \mu$ and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model $V^*$ in which the $H(\mu )$-hypermeasurability of $\kappa$ is indestructible by the Cohen forcing at $\kappa$ of any length up to $\mu$ (in particular $\kappa$ is $H(\mu )$-hypermeasurable in $V^*$). The preservation of hypermeasurability (in contrast to preservation of mere measurability) is useful for subsequent arguments (such as the definition of Radin forcing). The construction of $V^*$ is based on the ideas of Woodin (unpublished) and Cummings (Trans Am Math Soc 329(1):1–39, 1992) for preservation of measurability, but suitably generalised and simplified to achieve a more general result. Unlike the Laver preparation (Isr J Math 29(4):385–388, 1978) for a supercompact cardinal, our preparation non-trivially increases the value of $2^{\kappa ^+}$, which is equal to $\mu$ in $V^*$ (but $2^\kappa =\kappa ^+$ is still true in $V^*$ if we start with GCH).