Superdestructibility: a dual to Laver's indestructibility

  • [DOI] J. D. Hamkins and S. Shelah, “Superdestructibility: a dual to Laver’s indestructibility,” Journal of Symbolic Logic, vol. 63, iss. 2, p. 549–554, 1998.
    [Bibtex]
    @article {HamkinsShelah98:Dual,
    AUTHOR = {Hamkins, Joel David and Shelah, Saharon},
    TITLE = {Superdestructibility: a dual to {L}aver's indestructibility},
    JOURNAL = {Journal of Symbolic Logic},
    VOLUME = {63},
    YEAR = {1998},
    NUMBER = {2},
    PAGES = {549--554},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E55 (03E40)},
    MRNUMBER = {1625927 (99m:03106)},
    MRREVIEWER = {Douglas R.~Burke},
    DOI = {10.2307/2586848},
    URL = {http://jdh.hamkins.org/dual/},
    note = {[HmSh:618]},
    eprint = {math/9612227},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

After small forcing, any $<\kappa$-closed forcing will destroy the supercompactness, even the strong compactness, of $\kappa$.

Destruction or preservation as you like it

  • [DOI] J. D. Hamkins, “Destruction or preservation as you like it,” Annals of Pure and Applied Logic, vol. 91, iss. 2-3, p. 191–229, 1998.
    [Bibtex]
    @article {Hamkins98:AsYouLikeIt,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Destruction or preservation as you like it},
    JOURNAL = {Annals of Pure and Applied Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {91},
    YEAR = {1998},
    NUMBER = {2-3},
    PAGES = {191--229},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E55 (03E35)},
    MRNUMBER = {1604770 (99f:03071)},
    MRREVIEWER = {Joan Bagaria},
    DOI = {10.1016/S0168-0072(97)00044-4},
    URL = {http://jdh.hamkins.org/asyoulikeit/},
    eprint = {1607.00683},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of ${<}\kappa$-directed closed posets which preserve a supercompact cardinal $\kappa$ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible.

Canonical seeds and Prikry trees

  • [DOI] J. D. Hamkins, “Canonical seeds and Prikry trees,” Journal of Symbolic Logic, vol. 62, iss. 2, p. 373–396, 1997.
    [Bibtex]
    @article {Hamkins97:Seeds,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Canonical seeds and {P}rikry trees},
    JOURNAL = {Journal of Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {62},
    YEAR = {1997},
    NUMBER = {2},
    PAGES = {373--396},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E40 (03E05 03E55)},
    MRNUMBER = {1464105 (98i:03070)},
    MRREVIEWER = {Douglas R.~Burke},
    DOI = {10.2307/2275538},
    URL = {http://jdh.hamkins.org/seeds},
    }

Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.

A class of strong diamond principles

  • J. D. Hamkins, “A class of strong diamond principles,” ArXiv e-prints, 2002.
    [Bibtex]
    @ARTICLE{Hamkins:LaverDiamond,
    author = {Joel David Hamkins},
    title = {A class of strong diamond principles},
    journal = {ArXiv e-prints},
    year = {2002},
    eprint = {math/0211419},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://wp.me/p5M0LV-C},
    }

In the context of large cardinals, the classical diamond principle $\Diamond_\kappa$ is easily strengthened in natural ways. When $\kappa$ is a measurable cardinal, for example, one might ask that a $\Diamond_\kappa$ sequence anticipate every subset of $\kappa$ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function $\ell:\kappa\to V_\kappa$ such that for any $A\in H(\kappa^+)$ there is an embedding $j:V\to M$ having critical point $\kappa$ with $j(\ell)(\kappa)=A$. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.