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    Home/Original/inverse
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    Inverse View

    It is not the case that A proof of a sequent in first-order arithmetic gives rise to a well-founded reduction tree

    ?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    • 1.Gentzen's original 1936 proof was withdrawn precisely because its well-foundedness argument presupposed consistency in a circular way.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 for 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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    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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