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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that The universal partial computable function υ(i,x) is not total (i.e., not computable everywhere)

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The diagonalization argument presupposes a completed enumeration of all partial computable functions, which requires an actual infinity that constructivists like Brouwer reject.
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    • 2.If only potentially infinite sequences are admissible, no fixed index j can be assigned to d(x), dissolving the contradiction before it arises.
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    • 3.The claim thus inherits classical set-theoretic commitments that are not themselves computationally justified, making it question-begging against a constructivist opponent.
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    Reason for 2 of 2
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    • 1.Wittgenstein's rule-following considerations (Philosophical Investigations §201) imply that no finite syntactic index determines a unique total function extension.
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    • 2.If indices underdetermine totality, then the inference from 'υ(i,x) is partial computable' to 'it cannot be total' conflates a representational limitation with a metaphysical one.
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    • 3.The non-totality result may therefore reflect limits of the formal index-coding scheme rather than an intrinsic property of the underlying computational process.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.υ(i,x) is a universal partial computable function
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    • 2.Define d(x) = υ(x,x) + 1, which is partial computable since υ(i,x) is partial computable
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    • 3.Since d(x) is partial computable, d(x) ≃ φ_j(x) for some index j
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