Singular cardinals and strong extenders

  • A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Cent. Eur. J. Math., vol. 11, iss. 9, pp. 1628-1634, 2013.  
    @article {ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders,
    AUTHOR = {Apter, Arthur W. and Cummings, James and Hamkins, Joel David},
    TITLE = {Singular cardinals and strong extenders},
    JOURNAL = {Cent. Eur. J. Math.},
    FJOURNAL = {Central European Journal of Mathematics},
    VOLUME = {11},
    YEAR = {2013},
    NUMBER = {9},
    PAGES = {1628--1634},
    ISSN = {1895-1074},
    MRCLASS = {03E55 (03E35 03E45)},
    MRNUMBER = {3071929},
    MRREVIEWER = {Samuel Gomes da Silva},
    DOI = {10.2478/s11533-013-0265-1},
    URL = {http://jdh.hamkins.org/singular-cardinals-strong-extenders/},
    eprint = {1206.3703},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Brent Cody asked the question whether the situation can arise that one has an elementary embedding $j:V\to M$ witnessing the $\theta$-strongness of a cardinal $\kappa$, but where $\theta$ is regular in $M$ and singular in $V$.

In this article, we investigate the various circumstances in which this does and does not happen, the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing “$\kappa$ is $\mu$-strong”, such that $\mu$ is singular in $Ult(V, E)$.

Large cardinals with few measures

  • J. Apter Arthur W.~and Cummings and J. D. Hamkins, “Large cardinals with few measures,” Proc.~Amer.~Math.~Soc., vol. 135, iss. 7, pp. 2291-2300, 2007.  
    @ARTICLE{ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures,
    AUTHOR = {Apter, Arthur W.~and Cummings, James and Hamkins, Joel David},
    TITLE = {Large cardinals with few measures},
    JOURNAL = {Proc.~Amer.~Math.~Soc.},
    FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {135},
    YEAR = {2007},
    NUMBER = {7},
    PAGES = {2291--2300},
    ISSN = {0002-9939},
    CODEN = {PAMYAR},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {2299507 (2008b:03067)},
    MRREVIEWER = {Tetsuya Ishiu},
    DOI = {10.1090/S0002-9939-07-08786-2},
    URL = {http://dx.doi.org/10.1090/S0002-9939-07-08786-2},
    eprint = {math/0603260},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F
    }

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $\kappa^+$ many normal measures on the least measurable cardinal $\kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $\lambda$-strong compactness or $\lambda$-supercompactness measures on $P_\kappa(\lambda)$ can be exactly $\lambda^+$, if $\lambda>\kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.