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    Carmelics

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

    It is not the case that A weakly compact inaccessible cardinal is exceedingly large

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Cardinality-based magnitude claims require a prior account of 'largeness' that is independent of the cardinal hierarchy being described.
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    • 2.The supporting argument establishes relative position within the inaccessible hierarchy, not absolute largeness in any theory-neutral sense.
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    • 3.Without a fixed external standard of magnitude, 'exceedingly large' reduces to a relational claim dressed in absolutist language.
      ?

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    Reason for 2 of 2
    ?
    • 1.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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    • 2.A property that can be destroyed or created by set-theoretic forcing does not robustly ground claims about absolute cardinal magnitude.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If κ is inaccessible and weakly compact, then there is a set of κ many inaccessible cardinals strictly below κ
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    • 2.Having κ many inaccessibles below κ places κ far beyond any finitely indexed position in the inaccessible hierarchy
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