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    Rosser's provability predicate cannot be used to formaliz... — Carmelics
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    Rosser's provability predicate cannot be used to formalize the consistency claim in the second incompleteness theorem.

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Using Rosser's provability predicate, one can prove the 'consistency' of a formal system F within F itself.
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    • 2.A provability predicate that allows a system to prove its own consistency is not a suitable formalization of the genuine consistency claim.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Rosser's provability predicate satisfies Hilbert-Bernays derivability conditions, which are the standard criteria for legitimate provability predicates.
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    • 2.If a predicate meets the canonical technical criteria for provability, its associated consistency statement is a genuine formalization of consistency.
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    • 3.Therefore, the unprovability of Rosser consistency constitutes a substantive, not merely nominal, consistency result within the second incompleteness framework.
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    Reason against 2 of 2
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    • 1.Feferman (1960) demonstrated that the choice of provability predicate is conventional, since differently defined but extensionally equivalent predicates yield divergent unprovability results.
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    • 2.If the second incompleteness theorem's scope is predicate-relative, then Rosser's predicate reveals the theorem's dependence on representational choices, not a defect in the predicate itself.
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    • 3.The claim that Rosser's predicate is unsuitable conflates a system's formal derivability facts with the philosophical content of consistency, which Rosser's construction legitimately captures under one coherent interpretation.
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    Related

    A provability predicate that allows a system to prove its own consistency is not...Feferman (1960) demonstrated that the choice of provability predicate is convent...If a predicate meets the canonical technical criteria for provability, its assoc...If the second incompleteness theorem's scope is predicate-relative, then Rosser'...
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    Rosser's provability predicate satisfies Hilbert-Bernays derivability conditions...The claim that Rosser's predicate is unsuitable conflates a system's formal deri...Therefore, the unprovability of Rosser consistency constitutes a substantive, no...Using Rosser's provability predicate, one can prove the 'consistency' of a forma...

    Similar

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    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
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    Giving a rigorous proof of the second theorem in a more general form that covers all such sentences, however, has turned out to be very complicated. The basic reason for this is that, unlike in the first theorem, not just any, merely extensionally adequate provability predicate works for the formalization of the consistency claim. The manner of presentation makes all the difference. For example, Rosser’s provability predicate mentioned above would not do; one can prove the “consistency” of \(F\)
    Extraction notes

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

    Details

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