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    A weakly compact inaccessible cardinal is exceedingly large — Carmelics
    Home/Modality & Possibility
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    A weakly compact inaccessible cardinal is exceedingly large

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 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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    Modality & Possibility

    Related

    A property that can be destroyed or created by set-theoretic forcing does not ro...Cardinality-based magnitude claims require a prior account of 'largeness' that i...Having κ many inaccessibles below κ places κ far beyond any finitely indexed pos...If κ is inaccessible and weakly compact, then there is a set of κ many inaccessi...
    +3 moreShow less
    Lévy and Solovay showed that large cardinal properties are highly malleable unde...The supporting argument establishes relative position within the inaccessible hi...Without a fixed external standard of magnitude, 'exceedingly large' reduces to a...

    Similar

    A weakly compact inaccessible cardinal cannot be the first, second, or...88%A measurable cardinal κ cannot be the least strongly inaccessible card...85%If κ is inaccessible and weakly compact, then there is a set of κ many...84%If κ is inaccessible and weakly compact, then there exists a set of κ ...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-infinitary
    View source passageHide passage
    (3.3) Suppose κ is inaccessible. Then κ is weakly compact  ⇔  L(κ,ω) is weakly κ-compact. Also, Also κ is weakly compact ⇒ there is a set of κ inaccessibles before κ. Thus a weakly compact inaccessible cardinal is exceedingly large; in particular it cannot be the first, second, …, nth, … inaccessible.
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit