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    Lévy and Solovay showed that large cardinal properties ar... — Carmelics
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    Challenges→A weakly compact inaccessible cardinal is exceedingly large

    Lévy and Solovay showed that large cardinal properties are highly malleable under forcing, suggesting they track structural roles rather than intrinsic size.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 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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    Reasons Against

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    Reason against
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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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    Related

    A cardinal's cofinality and unboundedness in V are intrinsic properties that for...A weakly compact inaccessible cardinal is exceedingly largeForcing can create models where inaccessible cardinals become countable, showing...If large cardinals tracked only structural roles, the hierarchy of consistency s...
    +3 moreShow less
    Large cardinal axioms remain consistent across diverse forcing extensions, sugge...Large cardinals characterize closure properties and definability hierarchies tha...The absolute undefinability of large cardinals in smaller models suggests they m...

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    claim
    Perspectives
    2 (1 for, 1 against)
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    1 edit