Boolean ultrapowers, the Bukovský-Dehornoy phenomenon, and iterated ultrapowers

  • G. Fuchs and J. D. Hamkins, “The Bukovský-Dehornoy phenomenon for Boolean ultrapowers,” , 2017. (manuscript under review)  
    @ARTICLE{FuchsHamkins:TheBukovskyDehornoyPhenomenonForBooleanUltrapowers,
    AUTHOR = {Gunter Fuchs and Joel David Hamkins},
    TITLE = {The {Bukovsk\'y-Dehornoy} phenomenon for {Boolean} ultrapowers},
    JOURNAL = {},
    YEAR = {2017},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {manuscript under review},
    abstract = {},
    keywords = {under-review},
    source = {},
    eprint = {1707.06702},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://jdh.hamkins.org/boolean-ultrapowers-bukovsky-dehornoy-phenomenon},
    }

Abstract. We show that while the length $\omega$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Příkrý forcing, it is consistent that no iteration of length greater than $\omega$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Příkrý forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct limit model by the critical sequence as the Bukovský-Dehornoy phenomenon, and we develop a criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers.

Canonical seeds and Prikry trees

  • J. D. Hamkins, “Canonical seeds and Prikry trees,” J.~Symbolic Logic, vol. 62, iss. 2, pp. 373-396, 1997.  
    @article {Hamkins97:Seeds,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Canonical seeds and {P}rikry trees},
    JOURNAL = {J.~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 = {},
    }

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.