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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The semantics of second-order logic depend on the metatheory with respect to the Axiom of Choice.

    ?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.Standard semantics for second-order logic fixes the power set interpretation absolutely, making the range of predicate variables a matter of logic, not set theory.
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    • 2.Shapiro's full semantics treats second-order quantifiers as ranging over the actual power set of the domain, so metatheoretic AC-independence reflects set-theoretic indeterminacy, not semantic ambiguity in the logic itself.
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    • 3.The dependence on AC exposed by Cohen's models is a feature of Henkin semantics, where comprehension is restricted, not of second-order logic under its intended, unrestricted interpretation.
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    Reason for 2 of 2
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    • 1.Kreisel's squeezing argument supports the view that our pre-theoretic notion of logical validity is determinate even when formal metatheories disagree, so Cohen-style independence results do not destabilize the semantics of second-order logic.
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    • 2.The metatheoretic variation Cohen's forcing reveals is a pathology of first-order set-theoretic surrogates for second-order semantics, not evidence that second-order semantic content is itself indeterminate.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.By Cohen's results, there exist models N and N' both satisfying the axioms of ZF without the Axiom of Choice such that the sentence θ expressing the Axiom of Choice for continuum-size families of subsets of the reals holds in N but not in N'.
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    • 2.Whether 'N ⊨ θ' holds therefore depends on which model of ZF is taken as the metatheory, not on the sentence θ alone.
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