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
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    There is an intrinsic connection between Gentzen's ordina... — Carmelics
    Home/Proof of definition segments
    HistoryEditSee Inverse

    There is an intrinsic connection between Gentzen's ordinal assignment to deductions in PA and the standard ordinal assignment to infinite deductions in PA_ω

    All sources support itProof of definition segments
    ?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.Gentzen's method assigned ordinals to purported proofs of the empty sequent
      ?

      Think about whether this reason is strong or weak

    • 2.The infinitary approach assigns ordinals to infinite deductions in PA_ω
      ?

      Think about whether this reason is strong or weak

    • 3.Later work by Buchholz (1997) and others revealed that these two assignment methods are intrinsically connected
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The connection Buchholz establishes is a technical correspondence, not an intrinsic metaphysical relationship between distinct proof-theoretic frameworks.
      ?

      Think about whether this reason is strong or weak

    • 2.Intrinsicness requires that the connection hold in virtue of the nature of the objects themselves, but ordinal assignments are conventional choices within formal systems.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Gentzen's ordinals are assigned to finite syntactic objects while PA_ω ordinals are assigned to infinite trees, making them objects of fundamentally different ontological categories.
      ?

      Think about whether this reason is strong or weak

    • 2.A correspondence between entities of different ontological categories can at most be extrinsic and representational, as Kreisel's work on informal rigor suggests proof-theoretic notions must be grounded in their specific domains.
      ?

      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

    Proof of definition segmentsAll sources support it

    Connections

    1 topic

    Truth & Knowledge2 linked

    Related

    A correspondence between entities of different ontological categories can at mos...Gentzen's method assigned ordinals to purported proofs of the empty sequentGentzen's ordinals are assigned to finite syntactic objects while PA_ω ordinals ...Intrinsicness requires that the connection hold in virtue of the nature of the o...
    +3 moreShow less
    Later work by Buchholz (1997) and others revealed that these two assignment meth...The connection Buchholz establishes is a technical correspondence, not an intrin...The infinitary approach assigns ordinals to infinite deductions in PA_ω

    Similar

    The infinitary approach assigns ordinals to infinite deductions in PA_...84%Epsilon_0 is also the proof-theoretic ordinal of Peano Arithmetic77%The ordinals less than epsilon_0 are well-founded (there is no infinit...74%If PA were inconsistent, there would exist a proof P of the empty sequ...74%

    Source

    AI-extracted1/3 agreementValid
    SEP: proof-theory
    View source passageHide passage
    Gentzen did not deal explicitly with infinite proof trees in his second published proof of the consistency of PA (Gentzen 1938b). However, in the unpublished first consistency proof of Gentzen 1974 he aims at showing that a proof of a sequent in first-order arithmetic gives rise to a a well-founded reduction tree; that tree can be identified with a cut-free proof in the sequent calculus with the \(\omega\)-rule. The infinitary version of PA with the \(\omega\)-rule was investigated by Schütte (1
    Extraction notes

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

    Details

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