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    Primitive recursion is a paradigm case of finite, effecti... — Carmelics
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    Challenges→If a model of computation does not natively support recursion, then defining a function h(y) by primitive recursion over a base function g(y) computable in that model provides no a priori assurance that h(y) is itself computable in that model.

    Primitive recursion is a paradigm case of finite, effective procedure, so any function defined by it over a computable base is already Turing-computable by Church's Thesis.

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    Reasons For

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    • 1.Primitive recursion uses only bounded loops and composition, operations executable by any Turing machine without modification.
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    • 2.Church's Thesis claims all intuitively computable functions are Turing-computable; primitive recursion exemplifies intuitive computability.
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    • 3.No primitive recursive function requires oracles, infinite lookahead, or non-mechanical steps—all mechanically simulable.
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    Reasons Against

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    • 1.Church's Thesis is an unprovable philosophical claim about intuitive computability, not a theorem that validates the argument's conclusion.
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    • 2.The claim conflates 'paradigm case of finite procedure' with 'provably equivalent to Turing computability'—distinct logical steps.
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    • 3.Some primitive recursive functions are not efficiently Turing-computable in practice, challenging the functional equivalence claim.
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    Related

    Church's Thesis claims all intuitively computable functions are Turing-computabl...Church's Thesis is an unprovable philosophical claim about intuitive computabili...If a model of computation does not natively support recursion, then defining a f...No primitive recursive function requires oracles, infinite lookahead, or non-mec...
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    Primitive recursion uses only bounded loops and composition, operations executab...Some primitive recursive functions are not efficiently Turing-computable in prac...The claim conflates 'paradigm case of finite procedure' with 'provably equivalen...

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