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    Made withinDC&Austin
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    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that 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

    ?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.The connection Buchholz establishes is a technical correspondence, not an intrinsic metaphysical relationship between distinct proof-theoretic frameworks.
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    • 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.
      ?

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    Reason for 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.
      ?

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    • 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.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Gentzen's method assigned ordinals to purported proofs of the empty sequent
      ?

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    • 2.The infinitary approach assigns ordinals to infinite deductions in PA_ω
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    • 3.Later work by Buchholz (1997) and others revealed that these two assignment methods are intrinsically connected
      ?

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