The downward directed grounds hypothesis and very large cardinals

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    5 Citations (Scopus)

    Abstract

    A transitive model (Formula presented.) of ZFC is called a ground if the universe (Formula presented.) is a set forcing extension of (Formula presented.). We show that the grounds of(Formula presented.)(Formula presented.)(Formula presented.) are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) (Formula presented.) has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.

    Original languageEnglish
    JournalJournal of Mathematical Logic
    DOIs
    Publication statusAccepted/In press - 2017

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    Large Cardinals
    Forcing
    Geology

    Keywords

    • downward directed grounds hypothesis
    • Forcing method
    • generic multiverse
    • large cardinal
    • set-theoretic geology

    ASJC Scopus subject areas

    • Logic

    Cite this

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    abstract = "A transitive model (Formula presented.) of ZFC is called a ground if the universe (Formula presented.) is a set forcing extension of (Formula presented.). We show that the grounds of(Formula presented.)(Formula presented.)(Formula presented.) are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) (Formula presented.) has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.",
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    author = "Toshimichi Usuba",
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