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    Because this bridge is proven rather than merely conjectu... — 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.

    Because this bridge is proven rather than merely conjectured, a function defined by primitive recursion over a computable base inherits computability in any model proven equivalent to the recursive functions, removing the alleged 'a priori' gap.

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    Reasons For

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    Reason for
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    • 1.Model equivalence proofs (Church-Turing thesis) establish formal isomorphisms, not mere conjecture, between recursive functions and other computational frameworks.
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    • 2.Primitive recursion's computability is intrinsic to its definition; model equivalence transfers this property deductively, not inductively or by assumption.
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    • 3.The 'a priori gap' assumes computability requires external validation; proven equivalence dissolves this by showing the property holds across all equivalent formalisms.
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    Reasons Against

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    • 1.Church-Turing equivalence proofs themselves rely on informal intuitions about 'computation,' making them meta-theoretically circular rather than foundationally transparent.
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    • 2.Transferring computability via model equivalence assumes the property is *structural* and *model-invariant*; this itself requires justification independent of the equivalence proof.
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    • 3.An 'a priori gap' may persist epistemically: even if formally proven equivalent, why believe the models capture *all* computability? Proof doesn't address scope questions.
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    Key Terms

    Model (in logic)(in formal logic and semantics)
    An imaginary scenario or description of a possible world where certain statements are true or false—used to test whether logical arguments work.
    a priori(Frege treats 'analytic' as entailing 'a priori' for arithmetic.)
    Knowable independently of empirical experience; here treated as a consequence of analyticity.
    computable(As employed in the technical literature discussed in this passage)
    Computable by an effective method
    equivalent (logical equivalence)(as used in logic)
    Two statements are equivalent when they mean exactly the same thing and always have the same truth value—if one is true, the other must be true too.
    primitive recursion(computability theory / recursive function theory)
    A restricted kind of recursion in which a function h with first argument n+1 is defined in terms of h with first argument n, with all other arguments unchanged.
    recursive functions(Distinguished from the concrete operations used to compute them.)
    Abstract relations defined on natural numbers that can in principle be defined without any reference to space and time.

    Connections

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    Truth & Knowledge1 linkedCausation1 linked

    Related

    An 'a priori gap' may persist epistemically: even if formally proven equivalent,...Church-Turing equivalence proofs themselves rely on informal intuitions about 'c...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    If a model of computation does not natively support recursion, then defining a f...
    Model equivalence proofs (Church-Turing thesis) establish formal isomorphisms, n...
    +3 moreShow less
    Primitive recursion's computability is intrinsic to its definition; model equiva...The 'a priori gap' assumes computability requires external validation; proven eq...Transferring computability via model equivalence assumes the property is *struct...