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    Carmelics

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    Made withinDC&Austin
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    Inverse View

    It is not the case that The notion of a computable set generalizes effective decidability: a relation R is computable just in case there is an algorithm for deciding whether R holds of any tuple of natural numbers that always returns an answer after a finite (though potentially unbounded) number of steps

    ?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.Hypercomputation models (Malament-Hogarth spacetimes, Turing's o-machines) permit decision procedures that transcend finite-step termination constraints.
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    • 2.If physically realizable processes can decide undecidable sets, then 'computable' cannot be conceptually identified with finite-step algorithmic decidability sans further qualification.
      ?

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    • 3.Church's Thesis is an empirical conjecture about physical computation, not a logical necessity, as Copeland and Proudfoot argue, leaving the definition stipulative rather than revelatory.
      ?

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    Reason for 2 of 2
    ?
    • 1.Wittgenstein's rule-following considerations entail that no finite symbolic procedure fixes a unique interpretation of 'algorithm,' undermining the determinacy of 'always returns an answer.'
      ?

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    • 2.If the extension of a computable relation depends on how we interpret the algorithm's steps, then effective decidability is norm-relative rather than a mind-independent mathematical fact.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Church's Thesis equates computability with effective algorithmic decidability
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    • 2.Primitive recursive relations are computable
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    • 3.A computable set is one decidable by an algorithm that always terminates in finitely many steps
      ?

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    Strongest counterpoint
    Explore the most compelling reason on the other side.