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Archive for Mathematical Logic

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

A Laver-like indestructibility for hypermeasurable cardinals

  • Radek HonzikEmail author
Article
  • 38 Downloads

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 The construction of \(V^*\) is based on the ideas of Woodin (unpublished) and Cummings (Trans Am Math Soc 329(1):1–39, 1992) for preservation of measurability, but suitably generalised and simplified to achieve a more general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

Keywords

Laver indestructibility Hypermeasurable cardinals Strong cardinals 

Mathematics Subject Classification

03E35 03E55 

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Notes

Acknowledgements

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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Copyright information

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

Authors and Affiliations

  1. 1.Department of LogicCharles UniversityPraha 1Czech Republic

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