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    Among the unsolvable decision problems of recursively enu... — Carmelics
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    Home/Modality & Possibility
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    Among the unsolvable decision problems of recursively enumerable sets, there is a highest degree of unsolvability.

    Modality & PossibilityTruth & Knowledge
    ?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.The theory of recursively enumerable sets admits a primary problem of determining degrees of unsolvability of their unsolvable decision problems.
      ?

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    • 2.The structure of reducibility among these problems yields at least one problem to which all others are reducible.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Post's problem (1944) demonstrated that intermediate degrees exist between decidable and complete r.e. sets, undermining claims of a single highest degree.
      ?

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    • 2.Friedberg and Muchnik (1956) independently proved incomparable r.e. degrees exist, showing the r.e. degrees form a non-linearly-ordered structure.
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    • 3.A non-linear partial order with incomparable elements cannot possess a unique maximal element in any straightforward sense without further qualification.
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    Reason against 2 of 2
    ?
    • 1.The halting problem is Turing-complete for r.e. sets, but 'highest degree' conflates m-reducibility and Turing reducibility, which diverge extensionally.
      ?

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    • 2.Lachlan and Yates showed the r.e. degrees lack a definable well-ordering, making 'highest' a structurally ambiguous predicate absent a fixed reducibility notion.
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    Related

    A non-linear partial order with incomparable elements cannot possess a unique ma...Friedberg and Muchnik (1956) independently proved incomparable r.e. degrees exis...Lachlan and Yates showed the r.e. degrees lack a definable well-ordering, making...Post's problem (1944) demonstrated that intermediate degrees exist between decid...
    +3 moreShow less
    The halting problem is Turing-complete for r.e. sets, but 'highest degree' confl...The structure of reducibility among these problems yields at least one problem t...The theory of recursively enumerable sets admits a primary problem of determinin...

    Similar

    The theory of recursively enumerable sets admits a primary problem of ...93%The decision problem for many-sorted logic is undecidable.76%The set of validities of XL is recursively enumerable73%Without completeness, there exist semantic consequences that cannot be...72%

    Source

    AI-extracted1/3 agreementValid
    SEP: recursive-functions
    View source passageHide passage
    Related to the question of solvability or unsolvability of problems is that of the reducibility or non-reducibility of one problem to another. Thus, if problem \(P_1\) has been reduced to problem \(P_2\), a solution of \(P_2\) immediately yields a solution of \(P_1\), while if \(P_1\) is proved to be unsolvable, \(P_2\) must also be unsolvable. For unsolvable problems the concept of reducibility leads to the concept of degree of unsolvability, two unsolvable problems being of the same degree of
    Extraction notes

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

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit