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

  • G. Fuchs and J. D. Hamkins, “The Bukovský-Dehornoy phenomenon for Boolean ultrapowers.” (manuscript under review)  
    @ARTICLE{FuchsHamkins:TheBukovskyDehornoyPhenomenonForBooleanUltrapowers,
    AUTHOR = {Gunter Fuchs and Joel David Hamkins},
    TITLE = {The {Bukovsk\'y-Dehornoy} phenomenon for {Boolean} ultrapowers},
    JOURNAL = {},
    YEAR = {},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {manuscript under review},
    abstract = {},
    keywords = {},
    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.

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