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    P ≠ NP is unlikely to be independent of strong formal the... — Carmelics
    Home/Modality & Possibility
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    P ≠ NP is unlikely to be independent of strong formal theories such as PA or ZFC

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.P ≠ NP can be formulated as an arithmetical statement of the form ∀x∃y ψ(x,y) with only bounded numerical quantifiers
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    • 2.Statements of this logical form are generally believed not to be independent of theories like PA that approximate the mathematical axioms used in practice
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    • 3.We currently possess no reason to suspect that P ≠ NP is more likely to be independent of PA or ZFC than other currently open number-theoretic statements
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Gödelian incompleteness demonstrates that syntactic complexity of a statement's form is a poor guide to its provability within a given formal system.
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    • 2.Harvey Friedman's work on independence results shows that arithmetically simple statements (e.g., finite Ramsey variants) can be independent of PA despite their Π₂ or Σ₁ form.
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    • 3.The absence of known independence proofs for P≠NP reflects our limited proof-theoretic tools, not the intrinsic provability of the statement within PA or ZFC.
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    Reason against 2 of 2
    ?
    • 1.Scott Aaronson and others have shown that known proof techniques (algebrization, relativization, natural proofs) are formally blocked from resolving P≠NP, suggesting the statement resists standard mathematical machinery.
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    • 2.A statement that systematically evades all current proof strategies within a formal system is precisely the kind of statement independence results historically attach to, as Gödel's original incompleteness construction illustrates.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    2 topics

    Proof of definition segments2 linkedSkepticism1 linked

    Related

    A statement that systematically evades all current proof strategies within a for...Gödelian incompleteness demonstrates that syntactic complexity of a statement's ...Harvey Friedman's work on independence results shows that arithmetically simple ...P ≠ NP can be formulated as an arithmetical statement of the form ∀x∃y ψ(x,y) wi...
    +4 moreShow less
    Scott Aaronson and others have shown that known proof techniques (algebrization,...Statements of this logical form are generally believed not to be independent of ...

    Similar

    It is believed that P ≠ NP is unlikely to be independent of strong the...92%Such statements are generally believed unlikely to be independent of s...88%P ≠ NP is not currently suspected to be formally independent of accept...84%Although its logical form cannot exclude independence from PA or ZFC, ...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
    View source passageHide passage
    For note that although this statement originates in theoretical computer science, it may be easily formulated as statements about natural numbers. In particular, \(\textbf{P} \neq \textbf{NP}\) is equivalent to the statement that for all indices \(e\) and exponents \(k\), there exists a propositional formula \(\phi\) such that the deterministic Turing machine \(T_e\) does not correctly decide \(\phi\)’s membership in \(\sc{SAT}\) in \(\lvert \phi\rvert^k\) steps. e. a statement \(\Theta\) of the
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    The absence of known independence proofs for P≠NP reflects our limited proof-the...
    We currently possess no reason to suspect that P ≠ NP is more likely to be indep...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit