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    The universal partial computable function υ(i,x) is not t... — Carmelics
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    The universal partial computable function υ(i,x) is not total (i.e., not computable everywhere)

    Skepticism
    ?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.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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Topics

    SkepticismTruth & Knowledge

    Connections

    3 topics

    Modality & Possibility2 linkedProof of definition segments1 linkedCausation1 linked

    Related

    Define d(x) = υ(x,x) + 1, which is partial computable since υ(i,x) is partial co...If indices underdetermine totality, then the inference from 'υ(i,x) is partial c...If only potentially infinite sequences are admissible, no fixed index j can be a...If υ(i,x) were total, then d(j) would be defined, leading to a contradiction via...
    +6 moreShow less
    Since d(x) is partial computable, d(x) ≃ φ_j(x) for some index jThe claim thus inherits classical set-theoretic commitments that are not themsel...

    Similar

    υ(i,x) is a universal partial computable function90%Define d(x) = υ(x,x) + 1, which is partial computable since υ(i,x) is ...83%The Recursion Theorem (Theorem 3.5) holds for all computable functions...80%Since g_i(x) is always defined (total), u_k(i,x) is not merely partial...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: recursive-functions
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    Having just seen that there is a universal partial computable function \(\upsilon(i,x)\), a natural question is whether this function is also computable (i.e., total). A negative answer is provided immediately by observing that by using \(\upsilon(i,x)\) we may define another modified diagonal function \(d(x) = \upsilon(x,x) + 1\) which is partial computable (since \(\upsilon(i,x)\) is). This in turn implies that \(d(x) \simeq \phi_j(x)\) for some \(j\). But now note that if \(\upsilon(i,x)\) we
    Extraction notes

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

    Details

    The diagonalization argument presupposes a completed enumeration of all partial ...
    The non-totality result may therefore reflect limits of the formal index-coding ...
    Wittgenstein's rule-following considerations (Philosophical Investigations §201)...
    υ(i,x) is a universal partial computable function
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit