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    Made withinDC&Austin
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    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that A function f(x) is in FP if and only if it is definable by a Σ^B₁-formula relative to which it is provably total in V¹

    ?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.Provability in V¹ presupposes a fixed background meta-theory, yet the choice of meta-theory is not itself provably neutral with respect to what counts as 'total'.
      ?

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    • 2.Kreisel's squeezing argument shows that informal notions of computability resist full capture by any single formal provability criterion.
      ?

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    • 3.If the biconditional holds only relative to an assumed consistency of V¹, it cannot serve as a foundational characterization without circularity.
      ?

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    Reason for 2 of 2
    ?
    • 1.Σ^B₁-definability is a syntactic criterion, but polynomial-time computability is an extensional, machine-relative notion—Benacerraf's identification problem applies here.
      ?

      Think about whether this reason is strong or weak

    • 2.Two functions can be co-extensional over all inputs while differing in their Σ^B₁-definability status under alternative but equally valid formalization choices, undermining the biconditional's necessity claim.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The second-order theories V^i characterize the levels of the Polynomial Hierarchy
      ?

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    • 2.Σ^B₁-definability in V¹ captures exactly the polynomial-time computable functions
      ?

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