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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 The Gödel sentence G_F is true (when F is consistent and the provability predicate is a Σ⁰₁-formula)

    ?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.Truth in mathematics requires interpretation within a model, and G_F is true only relative to the standard model ℕ, not absolutely.
      ?

      Think about whether this reason is strong or weak

    • 2.The standard model ℕ is not definable within F itself, so appealing to it smuggles in metatheoretic commitments F cannot sanction.
      ?

      Think about whether this reason is strong or weak

    • 3.A formalist or deflationist (following Wittgenstein's Remarks on the Foundations of Mathematics) can reject that 'true but unprovable' is coherent without a privileged intended model.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.The consistency assumption Con(F) is itself unprovable within F by Gödel's second incompleteness theorem, making the argument circular from F's internal perspective.
      ?

      Think about whether this reason is strong or weak

    • 2.Hilbert's program and its successors (e.g., Detlefsen's 'Hilbertian instrumentalism') deny that unprovability-relative-to-F licenses truth-attribution absent an independent consistency proof.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.G_F is provably equivalent to the universal formula ∀x¬Prf_F(x, ⌈G_F⌉) when the provability predicate Prov_F(x) is chosen as a Σ⁰₁-formula
      ?

      Think about whether this reason is strong or weak

    • 2.Universal formulas of this form can be proved false whenever they are in fact false
      ?

      Think about whether this reason is strong or weak

    • 3.If G_F were false, there would be a number n such that F ⊢ Prf_F(n̲, ⌈G_F⌉)
      ?

      Think about whether this reason is strong or weak

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    Strongest counterpoint
    Explore the most compelling reason on the other side.