Extensions with the approximation and cover properties have no new large cardinals

  • J. D. Hamkins, “Extensions with the approximation and cover properties have no new large cardinals,” Fund.~Math., vol. 180, iss. 3, pp. 257-277, 2003.  
    @article{Hamkins2003:ExtensionsWithApproximationAndCoverProperties,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Extensions with the approximation and cover properties have no new large cardinals},
    JOURNAL = {Fund.~Math.},
    FJOURNAL = {Fundamenta Mathematicae},
    VOLUME = {180},
    YEAR = {2003},
    NUMBER = {3},
    PAGES = {257--277},
    ISSN = {0016-2736},
    MRCLASS = {03E55 (03E40)},
    MRNUMBER = {2063629 (2005m:03100)},
    DOI = {10.4064/fm180-3-4},
    URL = {http://dx.doi.org/10.4064/fm180-3-4},
    eprint = {math/0307229},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F,
    }

If an extension $\bar V$ of $V$ satisfies the $\delta$-approximation and cover properties for classes and $V$ is a class in $\bar V$, then every suitably closed embedding $j:\bar V\to \bar N$ in $\bar V$ with critical point above $\delta$ restricts to an embedding $j\upharpoonright V:V\to N$ amenable to the ground model $V$. In such extensions, therefore, there are no new large cardinals above delta. This result extends work in my article on gap forcing.

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  1. Pingback: Approximation and cover properties transfer upward | Joel David Hamkins

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