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    Carmelics

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

    It is not the case that A semantic argument in set theory is convertible to a syntactic formal proof only if the argument is valid in every model of ZFC, not merely in some preferred model

    ?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.Gödel's completeness theorem applies to first-order logic, but ZFC's intended semantics presupposes a background set theory that is itself model-relative.
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    • 2.The claim smuggles in a privileged notion of 'all models of ZFC' that cannot be cashed out without a metatheory subject to the same underdetermination.
      ?

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    • 3.Hilbert and Bernays demonstrated that syntactic derivability and semantic validity come apart precisely when the metatheory is not fixed independently of the object theory.
      ?

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    Reason for 2 of 2
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    • 1.Forcing constructions, as developed by Cohen, produce extensions of ground models where new syntactic proofs become available that were unavailable in the original model.
      ?

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    • 2.A semantic argument valid only in a forcing extension is nonetheless convertible to a syntactic proof relative to the axioms of that extension, which contradicts the claim's universal quantification over all ZFC models.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The Completeness Theorem is applicable only when a semantic argument works in whatever model of set theory is being used
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    • 2.Model-specific properties such as CH, ◇, or 2^ℵ₀ = ℵ₂ must be stated separately as assumptions when used
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    • 3.Non-first-order properties of models cannot be used at all if the Completeness Theorem is to apply
      ?

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