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    A weakly compact inaccessible cardinal cannot be the firs... — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    A weakly compact inaccessible cardinal cannot be the first, second, or any finitely indexed inaccessible cardinal

    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 exists a set of κ inaccessible cardinals below κ
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    • 2.If there are κ inaccessible cardinals below κ, then κ is exceedingly large and surpasses any finite position in the inaccessible hierarchy
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The claim conflates ordinal indexing with cardinal hierarchy position, since 'finitely indexed' presupposes a well-ordered enumeration of inaccessibles that may not be definable within standard ZFC.
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    • 2.Lévy and Vaught showed that inaccessible cardinals can be indexed only relative to a model; across models, the same cardinal may occupy different finite positions in the hierarchy.
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    • 3.If κ's position in the inaccessible hierarchy is model-relative, then 'κ cannot be the nth inaccessible' is not an absolute claim but a schema relativized to a background universe.
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    Reason against 2 of 2
    ?
    • 1.P1 of the supporting argument assumes κ-many inaccessibles below κ, but this requires assuming consistency strength beyond what the definition of weak compactness alone entails.
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    • 2.Kanamori's 'The Higher Infinite' establishes that weak compactness is a Π¹₁-indescribability property, and indescribability does not by itself generate κ-many inaccessible predecessors without additional large cardinal axioms.
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    • 3.An argument whose conclusion about cardinality position depends on premises requiring stronger axioms than those cited commits a suppressed-premise fallacy that undermines the claim's generality.
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    Topics

    Modality & Possibility

    Connections

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    Truth & Knowledge1 linked

    Related

    An argument whose conclusion about cardinality position depends on premises requ...If there are κ inaccessible cardinals below κ, then κ is exceedingly large and s...If κ is inaccessible and weakly compact, then there exists a set of κ inaccessib...If κ's position in the inaccessible hierarchy is model-relative, then 'κ cannot ...
    +4 moreShow less
    Kanamori's 'The Higher Infinite' establishes that weak compactness is a Π¹₁-inde...Lévy and Vaught showed that inaccessible cardinals can be indexed only relative ...P1 of the supporting argument assumes κ-many inaccessibles below κ, but this req...The claim conflates ordinal indexing with cardinal hierarchy position, since 'fi...

    Similar

    A weakly compact inaccessible cardinal is exceedingly large88%If κ is inaccessible and weakly compact, then there exists a set of κ ...87%A measurable cardinal κ cannot be the least strongly inaccessible card...86%If κ is inaccessible and weakly compact, then there is a set of κ many...86%

    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