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 consistency statement Cons(F) cannot be provable in F — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Truth & Knowledge
    HistoryEditSee Inverse

    The consistency statement Cons(F) cannot be provable in F

    Truth & 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.If Cons(F) were provable in F, then the Gödel sentence G_F would also be provable in F by simple logic
      ?

      Think about whether this reason is strong or weak

    • 2.If G_F were provable in F, that would contradict Gödel's first incompleteness theorem
      ?

      Think about whether this reason is strong or weak

    • 3.Gödel's first incompleteness theorem holds for F
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The claim implicitly assumes ω-consistency or 1-consistency of F, but Gödel's original proof required stronger consistency assumptions than mere consistency.
      ?

      Think about whether this reason is strong or weak

    • 2.Rosser's 1936 strengthening shows simple consistency suffices, yet this move introduces its own metatheoretic commitments that are not internal to F.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore the unprovability of Cons(F) within F depends on metatheoretic assumptions that cannot themselves be fully formalized within the system under evaluation.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      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

    Truth & Knowledge

    Connections

    1 topic

    Modality & Possibility1 linked

    Related

    Gödel's first incompleteness theorem holds for FHence the unprovability result is sensitive to the choice of provability predica...If Cons(F) is expressed via a specific arithmetical encoding, different encoding...If Cons(F) were provable in F, then the Gödel sentence G_F would also be provabl...
    +5 moreShow less
    If G_F were provable in F, that would contradict Gödel's first incompleteness th...Rosser's 1936 strengthening shows simple consistency suffices, yet this move int...The claim implicitly assumes ω-consistency or 1-consistency of F, but Gödel's or...Therefore the unprovability of Cons(F) within F depends on metatheoretic assumpt...Wittgenstein argued in 'Remarks on the Foundations of Mathematics' that the diag...

    Similar

    The consistency statement Con(F) must genuinely express that F is cons...86%A provability predicate that allows a system to prove its own consiste...85%The Geach–Boolos consistency proof for Frege Arithmetic works only whe...84%Rosser's provability predicate cannot be used to formalize the consist...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
    View source passageHide passage
    where \(G_F\) is the Gödel sentence for \(F\) provided by the first theorem. If \(\Cons(F)\) were provable in \(F\), so would be \(G_F\), by simple logic. This would contradict Gödel’s first theorem. Consequently, \(\Cons(F)\) cannot be provable in \(F\) either.
    Extraction notes

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

    Details

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