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    The expressive hierarchy Σ^1_n ∪ Π^1_n of second-order lo... — Carmelics
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    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

    Philosophy of LanguageTruth & Knowledge
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    Reasons For

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    • 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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    Reasons Against

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    Reason against 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 against 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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    Related

    Higher levels Σ^1_n and Π^1_n do not increase these model-theoretic numbers beyo...If higher levels Σ^1_n yield genuinely new interpolation and preservation failur...Kreisel and Barwise demonstrated that compactness and interpolation properties d...Löwenheim and Hanf numbers measure only the cardinality thresholds for satisfiab...
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    The Löwenheim number and Hanf number of full second-order logic equal those of t...The Σ^1_n hierarchy exhibits strictly increasing definability strength for arith...Therefore, collapsing expressive hierarchy to Π^1_1 for model-theoretic cardinal...

    Similar

    The Lévy hierarchy Σ_n ∪ Π_n in set theory does not collapse at the fi...79%Standard second-order logic has serious expressive limitations as a fo...78%The LST-number of second-order logic is the Löwenheim–Skolem–Tarski nu...78%In second-order logic there is no single axiom like the Power Set Axio...77%

    Source

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    SEP: logic-higher-order
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    This means that the Löwenheim number[4] and the Hanf number[5] of the entire second-order logic are the same as those of the fragment \(\Pi^1_1\). Summing up, upon first inspection the levels \(\Sigma^1_n\) and \(\Pi^1_n\) of the hierarchy of second-order formulas grow strictly in expressive power as n increases, but a more careful analysis reveals that already the first level \(\Sigma^1_1\cup \Pi^1_1\) has the power of all the levels \(\Sigma^1_n, \Pi^1_n\) even if the power is somewhat im
    Extraction notes

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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit