Diamond (on the regulars) can fail at any strongly unfoldable cardinal

  • M. D{u{z}}amonja and J. D. Hamkins, “Diamond (on the regulars) can fail at any strongly unfoldable cardinal,” Ann.~Pure Appl.~Logic, vol. 144, iss. 1-3, pp. 83-95, 2006. (Conference in honor of sixtieth birthday of James E.~Baumgartner)  
    @ARTICLE{DzamonjaHamkins2006:DiamondCanFail,
    AUTHOR = {D{\u{z}}amonja, Mirna and Hamkins, Joel David},
    TITLE = {Diamond (on the regulars) can fail at any strongly unfoldable cardinal},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {144},
    YEAR = {2006},
    NUMBER = {1-3},
    PAGES = {83--95},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E05 (03E35 03E55)},
    MRNUMBER = {2279655 (2007m:03091)},
    MRREVIEWER = {Andrzej Ros{\l}anowski},
    DOI = {10.1016/j.apal.2006.05.001},
    URL = {http://dx.doi.org/10.1016/j.apal.2006.05.001},
    month = {December},
    note = {Conference in honor of sixtieth birthday of James E.~Baumgartner},
    eprint = {math/0409304},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

If $\kappa$ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which $\Diamond_\kappa(\text{REG})$ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

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