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    P1 of the supporting argument assumes κ-many inaccessible... — Carmelics
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    Challenges→A weakly compact inaccessible cardinal cannot be the first, second, or any finitely indexed inaccessible cardinal

    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.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Weak compactness is defined via combinatorial properties without explicit large cardinal assumptions, so invoking κ-many inaccessibles goes beyond its definitional content.
      ?

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    • 2.The argument conflates what weak compactness entails directly with what consistency strength is needed to prove certain theorems about weakly compact cardinals.
      ?

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    • 3.Standard results show κ-inaccessibility below κ requires additional assumptions beyond the weak compactness axiom itself.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Weak compactness is known to imply that κ is inaccessible, so inaccessibles below κ follow naturally from the definition without extra assumptions.
      ?

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    • 2.The claim conflates consistency strength of a theory with logical consequences; weak compactness directly yields inaccessibility results in standard set theory.
      ?

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    • 3.No gap exists between weak compactness' definition and its standard consequences in ZFC—the argument overstates what 'beyond definition' actually requires.
      ?

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    Connections

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    Related

    A weakly compact inaccessible cardinal cannot be the first, second, or any finit...No gap exists between weak compactness' definition and its standard consequences...Standard results show κ-inaccessibility below κ requires additional assumptions ...The argument conflates what weak compactness entails directly with what consiste...
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    The claim conflates consistency strength of a theory with logical consequences; ...Weak compactness is defined via combinatorial properties without explicit large ...Weak compactness is known to imply that κ is inaccessible, so inaccessibles belo...

    Details

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