Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

  • A. W.~Apter and J. D. Hamkins, “Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata,” Math.~Logic Q., vol. 47, iss. 4, pp. 563-571, 2001.  
    @ARTICLE{ApterHamkins2001:IndestructibleWC,
    AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
    TITLE = {Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata},
    JOURNAL = {Math.~Logic Q.},
    FJOURNAL = {Mathematical Logic Quarterly},
    VOLUME = {47},
    YEAR = {2001},
    NUMBER = {4},
    PAGES = {563--571},
    ISSN = {0942-5616},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1865776 (2003h:03078)},
    DOI = {10.1002/1521-3870(200111)47:4%3C563::AID-MALQ563%3E3.0.CO;2-%23},
    URL = {http://jdh.hamkins.org/indestructiblewc/},
    eprint = {math/9907046},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

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