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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that For all natural numbers n, F proves the negation of Prf_F(n, ⌜G_F⌝)

    ?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.Strong representability of Prf_F presupposes ω-consistency, yet F's ω-consistency cannot be established within F by Gödel's own second theorem.
      ?

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    • 2.Without verified ω-consistency, the inference from each numeral-instance unprovability to universal provability of negations conflates object-level and meta-level quantification.
      ?

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    • 3.Rosser's 1936 result shows weaker consistency suffices for incompleteness, but the specific claim about provable negations for all n retains sensitivity to the consistency assumption's justificatory source.
      ?

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    Reason for 2 of 2
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    • 1.The claim quantifies over all natural numbers using the standard meta-theoretic ℕ, but ultrafinitists like Yessenin-Volpin deny that this totality is well-defined.
      ?

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    • 2.If 'all natural numbers' is not a determinate domain, the universal claim lacks a truth-condition independently of a background theory that itself requires justification.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.No natural number n is the Gödel number of a proof of G_F in F
      ?

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    • 2.The proof relation Prf_F is strongly representable in F
      ?

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    • 3.Strong representability entails that unprovability of each instance is provable in F
      ?

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