##
**A lifting argument for the generalized Grigorieff forcing.**
*(English)*
Zbl 1350.03036

In this paper, the authors study a generalized version of Grigorieff forcing at inaccessible cardinals and use it to present a new proof of Woodin’s celebrated result for forcing the failure of GCH at a measurable cardinal from optimal hypotheses.

The forcing, like the case of S.-D. Friedman and K. Thompson [J. Symb. Log. 73, No. 3, 906–918 (2008; Zbl 1160.03035)], is more uniform than Woodin’s original proof (which was based on Cohen forcing), in the sense that the required guiding generic is obtained directly without going to some further extension of the universe; but it is different from that of Friedman-Thompson, as it does not have a treelike structure. Also, unlike Sacks forcing at an inaccessible which is minimal, the resulting generalized Grigorieff forcing is not minimal.

The authors think that the method might be useful for obtaining new results concerning cardinal invariants at uncountable regular cardinals (see the open questions at the end of the paper).

The forcing, like the case of S.-D. Friedman and K. Thompson [J. Symb. Log. 73, No. 3, 906–918 (2008; Zbl 1160.03035)], is more uniform than Woodin’s original proof (which was based on Cohen forcing), in the sense that the required guiding generic is obtained directly without going to some further extension of the universe; but it is different from that of Friedman-Thompson, as it does not have a treelike structure. Also, unlike Sacks forcing at an inaccessible which is minimal, the resulting generalized Grigorieff forcing is not minimal.

The authors think that the method might be useful for obtaining new results concerning cardinal invariants at uncountable regular cardinals (see the open questions at the end of the paper).

Reviewer: Mohammad Golshani (Tehran)

### Citations:

Zbl 1160.03035
PDFBibTeX
XMLCite

\textit{R. Honzík} and \textit{J. Verner}, Notre Dame J. Formal Logic 57, No. 2, 221--231 (2016; Zbl 1350.03036)

### References:

[1] | Adersen, B. M., and M. J. Groszek, “Grigorieff forcing on uncountable cardinals does not add a generic of minimal degree,” Notre Dame J. Formal Logic , vol. 50 (2009), pp. 195-200. · Zbl 1188.03033 |

[2] | Brown, E. T., and M. J. Groszek, “Uncountable superperfect forcing and minimality,” Annals of Pure and Applied Logic , vol. 144 (2006), pp. 73-82. · Zbl 1110.03041 |

[3] | Cummings, J., “Iterated forcing and elementary embeddings,” pp. 775-883 in vol. 2 of Handbook of Set Theory , edited by M. Foreman and A. Kanamori, Springer, Dordrecht, 2010. · Zbl 1198.03060 |

[4] | Friedman, S.-D., and R. Honzík, “A definable failure of the Singular Cardinal Hypothesis,” Israel Journal of Mathematics , vol. 192 (2012), pp. 719-62. · Zbl 1300.03028 |

[5] | Friedman, S.-D., and R. Honzík, “Supercompactness and failures of GCH,” Fundamenta Mathematicae , vol. 219 (2012), pp. 15-36. · Zbl 1284.03234 |

[6] | Friedman, S. D., R. Honzík, and L. Zdomskyy, “Fusion and large cardinal preservation,” Annals of Pure and Applied Logic vol. 164 (2013), no. 12, pp. 1247-73. · Zbl 1320.03078 |

[7] | Friedman, S.-D., and M. Magidor, “The number of normal measures,” Journal of Symbolic Logic , vol. 74 (2009), pp. 1069-80. · Zbl 1183.03042 |

[8] | Friedman, S.-D., and K. Thompson, “Perfect trees and elementary embeddings,” Journal of Symbolic Logic , vol. 73 (2008), pp. 906-18. · Zbl 1160.03035 |

[9] | Friedman, S.-D., and L. Zdomskyy, “Measurable cardinals and the cofinality of the symmetric group,” Fundamenta Mathematicae , vol. 207 (2010), pp. 101-22. · Zbl 1196.03063 |

[10] | Gitik, M., “The negation of singular cardinal hypothesis from \(o(\kappa)=\kappa^{++}\),” Annals of Pure and Applied Logic , vol. 43 (1989), pp. 209-34. · Zbl 0673.03043 |

[11] | Grigorieff, S., “Combinatorics on ideals and forcing,” Annals of Mathematical Logic , vol. 3 (1971), pp. 363-94. · Zbl 0328.02041 |

[12] | Kanamori, A., “Perfect-set forcing for uncountable cardinals,” Annals of Mathematical Logic , vol. 19 (1980), pp. 97-114. · Zbl 0453.03056 |

[13] | Repický, M., “Collapsing of cardinals in generalized Cohen’s forcing,” Acta Universitatis Carolinae. Mathematica et Physica , vol. 29 (1988), pp. 67-74. · Zbl 0673.03041 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.