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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 consistency statement Cons(F) cannot be provable in 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.The claim implicitly assumes ω-consistency or 1-consistency of F, but Gödel's original proof required stronger consistency assumptions than mere consistency.
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    • 2.Rosser's 1936 strengthening shows simple consistency suffices, yet this move introduces its own metatheoretic commitments that are not internal to F.
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    • 3.Therefore the unprovability of Cons(F) within F depends on metatheoretic assumptions that cannot themselves be fully formalized within the system under evaluation.
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    Reason for 2 of 2
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    • 1.Wittgenstein argued in 'Remarks on the Foundations of Mathematics' that the diagonal construction conflates object-language provability with metatheoretic truth in a way that begs the question.
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    • 2.If Cons(F) is expressed via a specific arithmetical encoding, different encodings can yield extensionally equivalent but proof-theoretically distinct statements, as Feferman demonstrated in 1960.
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    • 3.Hence the unprovability result is sensitive to the choice of provability predicate, making the claim contingent rather than an absolute limitative result about F.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If Cons(F) were provable in F, then the Gödel sentence G_F would also be provable in F by simple logic
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    • 2.If G_F were provable in F, that would contradict Gödel's first incompleteness theorem
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    • 3.Gödel's first incompleteness theorem holds for F
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    Strongest counterpoint
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