Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 275–287 | Cite as

A Laver-like indestructibility for hypermeasurable cardinals

  • Radek HonzikEmail author


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 The construction of \(V^*\) is based on the ideas of Woodin (unpublished) and Cummings (Trans Am Math Soc 329(1):1–39, 1992) for preservation of measurability, but suitably generalised and simplified to achieve a more general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).


Laver indestructibility Hypermeasurable cardinals Strong cardinals 

Mathematics Subject Classification

03E35 03E55 


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The author wishes to thank to Sy Friedman and Š. Stejskalová for helpful discussions regarding this paper. In particular, the proof of Lemma 2.3(ii) was suggested by Sy Friedman.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of LogicCharles UniversityPraha 1Czech Republic

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