## Notre Dame Journal of Formal Logic

### A Lifting Argument for the Generalized Grigorieff Forcing

#### Abstract

In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal $\kappa$ from the optimal hypothesis, while adding new unbounded subsets to $\kappa$. In some ways these forcings are closer to the Cohen-type forcings—we show that they are not minimal—but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 2 (2016), 221-231.

Dates
Accepted: 26 November 2013
First available in Project Euclid: 7 January 2016

https://projecteuclid.org/euclid.ndjfl/1452175099

Digital Object Identifier
doi:10.1215/00294527-3459833

Mathematical Reviews number (MathSciNet)
MR3482744

Zentralblatt MATH identifier
1350.03036

#### Citation

Honzík, Radek; Verner, Jonathan. A Lifting Argument for the Generalized Grigorieff Forcing. Notre Dame J. Formal Logic 57 (2016), no. 2, 221--231. doi:10.1215/00294527-3459833. https://projecteuclid.org/euclid.ndjfl/1452175099

#### 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.
• [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.
• [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.
• [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.
• [5] Friedman, S.-D., and R. Honzík, “Supercompactness and failures of GCH,” Fundamenta Mathematicae, vol. 219 (2012), pp. 15–36.
• [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.
• [7] Friedman, S.-D., and M. Magidor, “The number of normal measures,” Journal of Symbolic Logic, vol. 74 (2009), pp. 1069–80.
• [8] Friedman, S.-D., and K. Thompson, “Perfect trees and elementary embeddings,” Journal of Symbolic Logic, vol. 73 (2008), pp. 906–18.
• [9] Friedman, S.-D., and L. Zdomskyy, “Measurable cardinals and the cofinality of the symmetric group,” Fundamenta Mathematicae, vol. 207 (2010), pp. 101–22.
• [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.
• [11] Grigorieff, S., “Combinatorics on ideals and forcing,” Annals of Mathematical Logic, vol. 3 (1971), pp. 363–94.
• [12] Kanamori, A., “Perfect-set forcing for uncountable cardinals,” Annals of Mathematical Logic, vol. 19 (1980), pp. 97–114.
• [13] Repický, M., “Collapsing of cardinals in generalized Cohen’s forcing,” Acta Universitatis Carolinae. Mathematica et Physica, vol. 29 (1988), pp. 67–74.