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    Kreisel's squeezing argument shows that informal notions ... — Carmelics
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    Challenges→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

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

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    Reasons For

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    Reason for
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    • 1.Church-Turing thesis captures computable functions, yet Kreisel showed informal notions extend to intuitive calculability beyond formal systems.
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    • 2.Different formal systems (Turing machines, lambda calculus, recursion theory) prove equivalent, yet mathematicians recognize computable tasks outside all.
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    • 3.Gödel's incompleteness implies no single formal criterion can capture all truths about what is computable in our informal mathematical practice.
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    Reasons Against

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    • 1.Church-Turing thesis remains unrefuted: every intuitive notion of computability proposed has reduced to Turing-computable functions.
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    • 2.Kreisel's squeezing argument conflates 'provability about computability' with 'computability itself'—capturing the former doesn't require capturing the latter.
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    • 3.Claims about informal notions resisting capture lack empirical support; no concrete counterexample of intuitively computable yet formally uncomputable functions exists.
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    Related

    A function f(x) is in FP if and only if it is definable by a Σ^B₁-formula relati...Church-Turing thesis captures computable functions, yet Kreisel showed informal ...Church-Turing thesis remains unrefuted: every intuitive notion of computability ...Claims about informal notions resisting capture lack empirical support; no concr...
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    Different formal systems (Turing machines, lambda calculus, recursion theory) pr...Gödel's incompleteness implies no single formal criterion can capture all truths...Kreisel's squeezing argument conflates 'provability about computability' with 'c...

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