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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that Lévy and Solovay showed that large cardinal properties are highly malleable under forcing, suggesting they track structural roles rather than intrinsic size.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
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    • 1.A cardinal's cofinality and unboundedness in V are intrinsic properties that forcing cannot genuinely alter—only our access to them changes.
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    • 2.The absolute undefinability of large cardinals in smaller models suggests they mark genuine structural boundaries, not merely relative positions.
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    • 3.If large cardinals tracked only structural roles, the hierarchy of consistency strength would collapse; instead it's remarkably rigid.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Forcing can create models where inaccessible cardinals become countable, showing size properties aren't intrinsic but context-dependent.
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    • 2.Large cardinal axioms remain consistent across diverse forcing extensions, suggesting they describe structural relationships, not absolute magnitudes.
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    • 3.Large cardinals characterize closure properties and definability hierarchies that persist across models, indicating they identify roles rather than sizes.
      ?

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