Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    The second incompleteness theorem is an intensional resul... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Philosophy of Language
    HistoryEditSee Inverse

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

    Philosophy of LanguageTruth & 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.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.
      ?

      Think about whether this reason is strong or weak

    • 2.The first incompleteness theorem and its relatives are extensional results that hold for any extensionally adequate provability predicate.
      ?

      Think about whether this reason is strong or weak

    • 3.The second incompleteness theorem fails for some extensionally adequate provability predicates (e.g., Rosser's).
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Extensional adequacy, properly construed, already entails that a provability predicate captures the intended proof relation, making intensionality redundant.
      ?

      Think about whether this reason is strong or weak

    • 2.The Hilbert-Bernays-Löb derivability conditions are themselves extensionally specifiable constraints on syntactic proof predicates, not genuinely semantic or intensional requirements.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Topics

    Philosophy of LanguageTruth & Knowledge

    Related

    A predicate satisfying the HBL conditions provably mirrors all provability facts...Extensional adequacy, properly construed, already entails that a provability pre...Quine's thesis that intensional contexts are eliminable in favor of extensional ...The Hilbert-Bernays-Löb derivability conditions are themselves extensionally spe...
    +4 moreShow less
    The consistency statement Con(F) must genuinely express that F is consistent — i...The failure of G2 for Rosser-style predicates shows sensitivity to syntactic for...The first incompleteness theorem and its relatives are extensional results that ...The second incompleteness theorem fails for some extensionally adequate provabil...

    Similar

    The second incompleteness theorem requires additional conditions on th...89%The second incompleteness theorem fails for some extensionally adequat...86%Gödel's second incompleteness theorem shows that Con(PA) cannot be pro...85%By Gödel's second incompleteness theorem, if a theory T implies Con(ZF...85%

    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
    View source passageHide passage
    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