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    Therefore, collapsing expressive hierarchy to Π^1_1 for m... — Carmelics
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    Challenges→The expressive hierarchy Σ^1_n ∪ Π^1_n of second-order logic already has its full power concentrated at the first level Σ^1_1 ∪ Π^1_1 with respect to Löwenheim and Hanf numbers

    Therefore, collapsing expressive hierarchy to Π^1_1 for model-theoretic cardinals misrepresents the orthogonal dimension of definitional complexity.

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    1 reason for
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    Reasons For

    1 perspective
    Reason for
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    • 1.Π^1_1 formulas capture only second-order existential quantification, leaving higher-order definitional stratification unrepresented.
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    • 2.Model-theoretic cardinals exhibit definability properties orthogonal to arithmetical hierarchy, requiring independent complexity metrics.
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    • 3.Collapsing distinct expressive dimensions into single framework obscures structural insights about definitional independence.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Π^1_1 is sufficient to express all relevant model-theoretic cardinal properties in standard set-theoretic foundations.
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    • 2.The claimed 'orthogonal dimension' lacks clear formal definition independent of the hierarchy being rejected.
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    • 3.Empirically, working mathematicians successfully use standard complexity classifications without the proposed refinement.
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    Connections

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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Collapsing distinct expressive dimensions into single framework obscures structu...Empirically, working mathematicians successfully use standard complexity classif...Model-theoretic cardinals exhibit definability properties orthogonal to arithmet...The claimed 'orthogonal dimension' lacks clear formal definition independent of ...
    +3 moreShow less
    The expressive hierarchy Σ^1_n ∪ Π^1_n of second-order logic already has its ful...Π^1_1 formulas capture only second-order existential quantification, leaving hig...Π^1_1 is sufficient to express all relevant model-theoretic cardinal properties ...

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    claim
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    2 (1 for, 1 against)
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