Unfoldable cardinals and the GCH

  • [DOI] J. D. Hamkins, “Unfoldable cardinals and the GCH,” Journal of Symbolic Logic, vol. 66, iss. 3, p. 1186–1198, 2001.
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Unfoldable cardinals and the {GCH}},
    JOURNAL = {Journal of Symbolic Logic},
    VOLUME = {66},
    YEAR = {2001},
    NUMBER = {3},
    PAGES = {1186--1198},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E55 (03E35 03E40)},
    MRNUMBER = {1856735 (2002i:03059)},
    MRREVIEWER = {Eva Coplakova},
    DOI = {10.2307/2695100},
    URL = {http://wp.me/p5M0LV-28},
    eprint = {math/9909029},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},

Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, $\kappa$ is $\theta$-unfoldable when for any transitive structure $M$ of size $\kappa$ that contains $\kappa$ as an element, there is an elementary embedding $j:M\to N$ with critical point $\kappa$ for which $j(\kappa)$ is at least $\theta$. Define that $\kappa$ is fully unfoldable, then, when it is $\theta$-unfoldable for every $\theta$. In this paper I show that the embeddings associated with these unfoldable cardinals are amenable to some of the same lifting techniques that apply to weakly compact embeddings, augmented with methods from the strong cardinal context. Using these techniques, I show by set-forcing over any model of ZFC that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$. This result contradicts expectations to the contrary that class forcing would be required.

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