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    Inverse View

    It is not the case that 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

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Löwenheim and Hanf numbers measure only the cardinality thresholds for satisfiability, not the full expressive power of a logical fragment.
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    • 2.The Σ^1_n hierarchy exhibits strictly increasing definability strength for arithmetic and analytic sets, as established by descriptive set theory.
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    • 3.Therefore, collapsing expressive hierarchy to Π^1_1 for model-theoretic cardinals misrepresents the orthogonal dimension of definitional complexity.
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    Reason for 2 of 2
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    • 1.Kreisel and Barwise demonstrated that compactness and interpolation properties diverge across the analytical hierarchy, tracking Σ^1_n distinctions.
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    • 2.If higher levels Σ^1_n yield genuinely new interpolation and preservation failures, the claim that expressive power is 'concentrated' at level one is operationally misleading.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The Löwenheim number and Hanf number of full second-order logic equal those of the fragment Π^1_1
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    • 2.Higher levels Σ^1_n and Π^1_n do not increase these model-theoretic numbers beyond what Π^1_1 already achieves
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