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    A proof of a sequent in first-order arithmetic gives rise... — Carmelics
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    A proof of a sequent in first-order arithmetic gives rise to a well-founded reduction tree

    Proof of definition segmentsTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

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    Reason for
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    • 1.Gentzen's first consistency proof aims to show that any proof of a sequent in first-order arithmetic produces a reduction tree
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    • 2.That reduction tree can be identified with a cut-free proof in the sequent calculus with the ω-rule
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    Reasons Against

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    Reason against 1 of 2
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    • 1.Gentzen's original 1936 proof was withdrawn precisely because its well-foundedness argument presupposed consistency in a circular way.
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    • 2.The transfinite induction up to ε₀ required to establish well-foundedness cannot itself be proven within first-order arithmetic, per Gentzen's own 1943 result.
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    • 3.A reduction tree that requires resources exceeding the system being analyzed cannot serve as an internal proof of that system's consistency.
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    Reason against 2 of 2
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    • 1.Kreisel and Takeuti demonstrated that the assignment of ordinal notations to proof-steps depends on a prior interpretation of the ordinals that is not proof-theoretically neutral.
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    • 2.If the ordinal assignment presupposes a semantic model of well-ordering, the 'well-founded' character of the reduction tree is inherited from that model, not derived from the proof structure itself.
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    Topics

    Truth & KnowledgeProof of definition segments

    Related

    A reduction tree that requires resources exceeding the system being analyzed can...Gentzen's first consistency proof aims to show that any proof of a sequent in fi...Gentzen's original 1936 proof was withdrawn precisely because its well-foundedne...If the ordinal assignment presupposes a semantic model of well-ordering, the 'we...
    +3 moreShow less
    Kreisel and Takeuti demonstrated that the assignment of ordinal notations to pro...That reduction tree can be identified with a cut-free proof in the sequent calcu...The transfinite induction up to ε₀ required to establish well-foundedness cannot...

    Similar

    Gentzen's first consistency proof aims to show that any proof of a seq...89%That reduction tree can be identified with a cut-free proof in the seq...83%There exists a reduction procedure R on proofs P of the empty sequent ...72%The Church–Rosser theorem guarantees that the final result of a series...70%

    Source

    AI-extracted1/3 agreementValid
    SEP: proof-theory
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    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