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    The Gödel sentence G_F is true (when F is consistent and ... — Carmelics
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    Home/Modality & Possibility
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    The Gödel sentence G_F is true (when F is consistent and the provability predicate is a Σ⁰₁-formula)

    Modality & PossibilityTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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
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    • 2.Universal formulas of this form can be proved false whenever they are in fact false
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    • 3.If G_F were false, there would be a number n such that F ⊢ Prf_F(n̲, ⌈G_F⌉)
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Truth in mathematics requires interpretation within a model, and G_F is true only relative to the standard model ℕ, not absolutely.
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    • 2.The standard model ℕ is not definable within F itself, so appealing to it smuggles in metatheoretic commitments F cannot sanction.
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    • 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.
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    Reason against 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.
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    • 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.
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    Modality & PossibilityTruth & Knowledge

    Related

    A formalist or deflationist (following Wittgenstein's Remarks on the Foundations...F is assumed to be consistentF ⊢ Prf_F(n̲, ⌈G_F⌉) would contradict Gödel's incompleteness theoremG_F is provably equivalent to the universal formula ∀x¬Prf_F(x, ⌈G_F⌉) when the ...
    +6 moreShow less
    Hilbert's program and its successors (e.g., Detlefsen's 'Hilbertian instrumental...If G_F were false, there would be a number n such that F ⊢ Prf_F(n̲, ⌈G_F⌉)The consistency assumption Con(F) is itself unprovable within F by Gödel's secon...The standard model ℕ is not definable within F itself, so appealing to it smuggl...Truth in mathematics requires interpretation within a model, and G_F is true onl...Universal formulas of this form can be proved false whenever they are in fact fa...

    Similar

    Using Rosser's provability predicate, one can prove the 'consistency' ...85%All provable sentences are true84%By the Completeness Theorem, provable sentences are valid, but not all...83%A provability predicate that allows a system to prove its own consiste...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
    View source passageHide passage
    In fact, in favourable circumstances, it can be shown that \(G_F\) is true, provided that \(F\) is indeed consistent. This is the case if, for example, the provability predicate \(\Prov_F (x)\) has been chosen as a \(\Sigma^{0}_1\)-formula: The Gödel sentence is then provably equivalent to the universal formula \(\forall x\neg\Prf_F (x, \ulcorner G_F\urcorner)\). Such formulas can be proved false whenever they in fact are false: if false, there would be a number \(\boldsymbol{n}\) such that \(F
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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