Indestructible strong unfoldability

  • J. D. Hamkins and T. A. Johnstone, “Indestructible strong unfoldability,” Notre Dame J.~Form.~Log., vol. 51, iss. 3, pp. 291-321, 2010.  
    @ARTICLE{HamkinsJohnstone2010:IndestructibleStrongUnfoldability,
    AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},
    TITLE = {Indestructible strong unfoldability},
    JOURNAL = {Notre Dame J.~Form.~Log.},
    FJOURNAL = {Notre Dame Journal of Formal Logic},
    VOLUME = {51},
    YEAR = {2010},
    NUMBER = {3},
    PAGES = {291--321},
    ISSN = {0029-4527},
    MRCLASS = {03E55 (03E40)},
    MRNUMBER = {2675684 (2011i:03050)},
    MRREVIEWER = {Bernhard A.~K{\"o}nig},
    DOI = {10.1215/00294527-2010-018},
    URL = {http://dx.doi.org/10.1215/00294527-2010-018},
    file = F
    }

Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed + $\kappa^+$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of ${\lt}\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by ${\lt}\kappa$-directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.

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