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    The polynomial-time functions being 'total' in M means on... — Carmelics
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    Challenges→There exists a nonstandard model M in which all polynomial-time computable functions are total but the exponential function is not total.

    The polynomial-time functions being 'total' in M means only that M satisfies their defining axioms, not that they are total over the genuine natural numbers, making the claim modal-epistemically ambiguous.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Model M's axioms may be consistent yet incomplete regarding standard arithmetic, so 'totality in M' underdetermines reality.
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    • 2.Distinguishing formal provability from metaphysical truth requires acknowledging that satisfying axioms doesn't entail genuine totality.
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    • 3.Polynomial-time functions' behavior depends on which model interprets them, creating irreducible epistemic ambiguity about their actual scope.
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    Reasons Against

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    Reason against
    ?
    • 1.If totality-in-M has no truth conditions independent of what M satisfies, the distinction between 'in M' and 'genuine' becomes meaningless.
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    • 2.Polynomial-time function definitions are recursively constructive and model-independent; their totality doesn't depend on which axioms we posit.
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    • 3.Calling this 'modal-epistemically ambiguous' conflates semantic indeterminacy with our ignorance—the functions are either total or not.
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    Key Terms

    Ambiguous (modal-epistemically)(in philosophical logic)
    Unclear or having multiple possible meanings when it comes to both what is necessary/possible and what we can actually know about it.
    Epistemically(as how you should evaluate your own beliefs)
    In a way that relates to knowledge—meaning you understand something based on actual information or justified reasons, not just guessing.
    Genuine natural numbers(in mathematical philosophy)
    The 'real' or standard set of counting numbers (0, 1, 2, 3, ...) that mathematicians refer to in the real world, as opposed to numbers that only exist within a particular mathematical system.
    M (in formal logic)(in mathematical logic)
    A mathematical model or system—basically a set of rules and objects that logicians study to test whether certain claims are true within that system.
    Polynomial-time functions(in mathematics and computer science)
    Mathematical functions that can be computed relatively quickly by a computer, where the time needed grows at a reasonable rate as the input size increases.
    Total (in mathematics)(in mathematical logic)
    A function is 'total' if it can produce an answer for every possible input. If it's not total, there are some inputs where it doesn't work or gives no answer.
    axioms(Stumpf, 1891)
    Propositions that we assume to be true and necessary, originating in the content of judgments.
    modal(in logic and metaphysics)
    Dealing with possibility and necessity—questions about what could be true, what must be true, and what's merely contingent (could go either way).

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Calling this 'modal-epistemically ambiguous' conflates semantic indeterminacy wi...Distinguishing formal provability from metaphysical truth requires acknowledging...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    If totality-in-M has no truth conditions independent of what M satisfies, the di...
    Model M's axioms may be consistent yet incomplete regarding standard arithmetic,...
    +3 moreShow less
    Polynomial-time function definitions are recursively constructive and model-inde...Polynomial-time functions' behavior depends on which model interprets them, crea...There exists a nonstandard model M in which all polynomial-time computable funct...