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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 second incompleteness theorem is an intensional result, not merely an extensional one.

    ?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.Extensional adequacy, properly construed, already entails that a provability predicate captures the intended proof relation, making intensionality redundant.
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    • 2.The Hilbert-Bernays-Löb derivability conditions are themselves extensionally specifiable constraints on syntactic proof predicates, not genuinely semantic or intensional requirements.
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    • 3.A predicate satisfying the HBL conditions provably mirrors all provability facts, so the distinction between 'really meaning' consistency and merely expressing it collapses under reflection.
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    Reason for 2 of 2
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    • 1.Quine's thesis that intensional contexts are eliminable in favor of extensional paraphrase applies here: Con(F) is a purely syntactic, recursively defined sentence whose meaning just is its extension over arithmetic.
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    • 2.The failure of G2 for Rosser-style predicates shows sensitivity to syntactic form, not to intensional meaning, since Rosser's predicate lacks the HBL conditions — a structural, not semantic, deficiency.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The consistency statement Con(F) must genuinely express that F is consistent — it must really mean that F is consistent, not merely be extensionally equivalent to it.
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    • 2.The first incompleteness theorem and its relatives are extensional results that hold for any extensionally adequate provability predicate.
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    • 3.The second incompleteness theorem fails for some extensionally adequate provability predicates (e.g., Rosser's).
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