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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that A weakly compact inaccessible cardinal cannot be the first, second, or any finitely indexed inaccessible cardinal

    ?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.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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • 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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